text:
Written Answers 8 to 12 February 2010
Home » Written Answers 8 to 12 February 2010
9 Feb 2010 | Parliament, Parliamentary Questions
Mr. Goodwill: To ask the Secretary of State for Northern Ireland what mechanisms have been established to enable data sharing in respect of fixed penalty motoring offences between the Police Service of Northern Ireland and the Irish Garda in accordance with the principle of mutual recognition of financial penalties. [315998]
Paul Goggins: Roads Policing Policy in Northern Ireland is a devolved matter for the Department of Environment. The sharing of data between the Police Service of Northern Ireland and An Garda Siochana in relation to motoring offences is an operational matter for the Chief Constable. I have asked the Chief Constable to reply directly to the hon. Member, and a copy of his letter will be placed in the Library of the House.
Mr. Sanders: To ask the Secretary of State for Justice how many convictions for (a) an offence of drink driving and (b) each other motoring offence in (i) Torbay constituency, (ii) Devon and (iii) England resulted in a custodial sentence in each of the last 10 years. [314213]
Claire Ward: The number of defendants found guilty at all courts in the Devon and Cornwall police force area and England for driving etc. after consuming alcohol or taking drugs and other motoring offences (by offence type) is given in Tables 1 and 2 from 1999 to 2008 (latest available). Sentences of immediate custody imposed at all courts are given in Tables 3 and 4.
Court proceedings data are not available at parliamentary constituency level. Data for Devon cannot be separately identified from within the Devon and Cornwall police force area.
Data for 2009 are expected to be published in the autumn 2010.
http://www.publications.parliament.uk/pa/cm200910/cmhansrd/cm100208/text/100208w0025.htm#10020854000009
http://www.publications.parliament.uk/pa/cm200910/cmhansrd/cm100208/text/100208w0026.htm
Driving Offences: Disqualification
Mr. Sanders: To ask the Secretary of State for Justice how many people in (a) Torbay constituency, (b) the South West and (c) England were disqualified from driving as a result of an offence of (i) drink-driving, (ii) speeding and (iii) dangerous or reckless driving in each of the last 10 years. [313334]
Claire Ward: The number persons disqualified from driving as a result of convictions for (i) drink driving, (ii) speeding and (iii) dangerous or careless driving in the South West and England in each of the last 10 years shown in tables 1 and 2 as follows.
Court proceedings data are not available at parliamentary constituency level. Data for 2009 are expected to be published in the autumn 2010.
Tuesday 9 February 2010
Mr. David Anderson: To ask the Minister of State, Department for Transport if he will set out, with statistical information related as directly as possible to Blaydon constituency, the effects on Blaydon of his Department's policies and actions since 2000. [315299]
Mr. Khan: The Transport Act 2000, as amended by the Local Transport Act 2008, has provided a new policy framework benefiting all local transport authorities. The framework gives greater certainty of funding, while encouraging more strategic transport planning with local consultation, and increasing local flexibility and discretion over resources. It was accompanied by a significant increase in capital funding: support from the Department for transport investment in the Metropolitan Borough of Gateshead, within which transport authority Blaydon falls, has more than doubled over the last decade.
Investment in Tyne and Wear's Joint Local Transport Plan, to which Gateshead Council is a partner authority, has delivered a number of improvements to the quality, safety and accessibility of the local transport network. Between 2005 and 2008, bus patronage per head of population increased by 8 per cent, and the number of people killed or seriously injured on the local highway network decreased by 22 per cent. in the period 2001-07.
In 2005, the Tyne and Wear Integrated Transport Authority, Nexus, submitted a successful bid to the Department for 'Kickstart' revenue funding of the 639 Gateshead Orbital bus service, operated by Go North-East. The £336,256 award facilitated the doubling of the daytime and early-evening frequency of the service from hourly to half-hourly. The improved service, featuring fully accessible vehicles, has significantly improved links between the Queen Elizabeth Hospital and areas of western Gateshead, including Blaydon.
The £17 million Centrelink scheme was completed in 2005, providing enhanced public transport links between Gateshead town centre and the MetroCentre. The project included the construction of an £8.4 million busway and the reorganisation of the public transport interchange in Gateshead town centre to facilitate interchange between bus and Metro services. The project has had the complementary effect of removing large numbers of buses from shopping streets, thereby reducing congestion and improving the environment for pedestrians and shoppers.
In July 2007, Gateshead council completed a £1 million upgrade to the A184 Felling bypass. The project included the widening of the road to a three-lane carriageway to provide space for a dedicated bus priority lane. The scheme has provided bus passengers with greater journey time reliability and forms part of Tyne and Wear's 'Superoute' network of Quality Bus Corridors, which generated patronage growth of 14 per cent. between 2002 and 2007.
On 3 February 2010 the Department announced that it would provide around £580 million to the Tyne and Wear Metro for ongoing operating support and major infrastructure renewals to 2021.
Bridges: Accidents
Mr. Hollobone: To ask the Minister of State, Department or Transport what information his Department holds on (a) the number of bridge strikes in other EU countries where metric height and width restriction signs are required and (b) the number of such strikes involving UK lorry drivers in the last 12 months. [316208]
Mr. Khan [holding answer 8 February 2010]: The Department for Transport does not hold this information.
Driver and Vehicle Licensing Agency
Anne Milton: To ask the Minister of State, Department for Transport what he expects the timetable to be for implementation of the Driver and Vehicle Licensing Agency's draft enforcement policy. [316048]
Paul Clark: The Driver and Vehicle Licensing Agency's revised enforcement strategy document is scheduled to be completed by the end of March. This strategy will seek to maximise collection of vehicle excise duty at minimum cost through encouraging compliance and using proportionate enforcement action where necessary. Most of the initiatives contained in the strategy are already in place. When finalised the strategy document will also set out proposed timescales for the implementation of new initiatives.
Driver Information Systems: Bridges
Mr. Hollobone: To ask the Minister of State, Department for Transport what steps his Department has taken to ensure that accurate information on the (a) location and (b) height in metric and imperial measurements of low bridges is accessible to producers of satellite navigation systems. [316207]
Mr. Khan [holding answer 8 February 2010]: None. Producers of satellite navigation systems are responsible for the accuracy of information provided in their products. Moreover, the onus remains on drivers to adhere to warnings and restrictions indicated by traffic signs.
Jon Trickett: To ask the Minister of State, Department for Transport if he will set out, with statistical information related as directly as possible to Hemsworth constituency, the effects on Hemsworth of his Department's policies and actions since 2000. [316304]
Mr. Khan: The Transport Act 2000, as amended by the Local Transport Act 2008, has provided a new policy framework benefiting all local transport authorities. The framework gives greater certainty of funding, while encouraging more strategic transport planning with local consultation, and increasing local flexibility and discretion over resources. It was accompanied by a significant increase in capital funding: support from the Department for transport investment in the Metropolitan borough of Wakefield, within which transport authority Hemsworth falls, has doubled over the last decade.
Investment in West Yorkshire's Joint Local Transport plan, to which Wakefield council is a partner authority, has delivered a number of improvements to the quality, safety and accessibility of the local transport network. Between 2006 and 2008, the percentage of local authority roads requiring structural maintenance decreased by 58 per cent. and the number of people killed or seriously injured on the local highway network decreased by 30 per cent. in the period 2001-07.
In December 2003, the West Yorkshire Integrated Transport Authority, Metro, submitted a successful £18.7 million bid to the Department for funding of the 'Mybus' project, which provides dedicated school transport to 133 schools throughout West Yorkshire, including Hemsworth Arts and Community college. Mybus has delivered a number of benefits, including reduced congestion, air pollution and accidents around schools, improved pupil behaviour on journeys to and from school, and enhanced educational performance and attendance. More than 10,000 pupils currently use the service, 64 per cent. of which previously travelled to school by car.
The Wakefield 'Free City Bus' service was introduced by Metro and Wakefield council in April 2007, providing residents and visitors with a frequent and fully accessible service between the city's rail and bus interchanges and main shopping areas. Patronage of the service has far exceeded expectations, particularly for journeys to and from the Ings Road and Westgate Retail Parks, which were previously inaccessible by public transport. The introduction of the Free City Bus has resulted in a 5 per cent. increase in the number of people living within 30 minutes travel time of Wakefield city centre by public transport.
The £24.2 million Hemsworth to A1 Link Road was completed in November 2009. The project was commissioned to promote the economic regeneration of the former coal and steel communities in the area by significantly improving their access to the strategic road network. The new section of road, between Hemsworth and the A638 at Upton, includes a link to the proposed South Kirkby Business Park.
Invalid Vehicles
Mr. Harper: To ask the Minister of State, Department for Transport pursuant to the answer of 9 September 2009, Official Report, column 1902W, on invalid vehicles: insurance, when the survey his Department is procuring to help assess the number of mobility scooter users and the extent to which their use may have injured people will be (a) completed and (b) published. [316099]
Mr. Khan [holding answer 5 February 2010]: Fieldwork for this survey took place during January 2010 and the Department expects to receive the survey data set during March 2010. It is likely that results will be published during the first half of the 2010-11 financial year, although a firm date has not yet been fixed.
The survey focused primarily on public attitudes to regulatory issues around mobility scooters. Although a question was asked on injuries caused by mobility scooters, it is not yet clear whether the sample size will be sufficient to allow the publication of robust estimates on this issue.
Motorways: Accidents
Mrs. Villiers: To ask the Minister of State, Department for Transport what recent research his Department and its agencies have undertaken on the average number of patrol vehicles required to attend incidents on motorways. [309453]
Chris Mole: The Highways Agency is currently undertaking operational research and development to provide a new software model to identify the optimum number of traffic officer patrols required to attend incidents on motorways. This operational research and development is due for delivery towards the end of 2010.
Mrs. Villiers: To ask the Minister of State, Department for Transport what information his Department and its agencies collect on the clearing of motorways following road traffic incidents. [309455]
Chris Mole: The Highways Agency routinely gathers the following information regarding the clearing of motorways following road traffic incidents where the Traffic Officer Service is involved in the management of the incident:
the time the incident log is created;
the time a traffic officer arrives at the scene of an incident;
the type of incident;
the time(s) that the incident scene is transferred to direct police control;
the time(s) that the incident scene is transferred from direct police control;
the time(s) that the carriageway is adversely affected by an incident, if the arrival of a traffic officer resource at the scene occurs before the time(s) that the carriageway became adversely affected;
the time that the carriageway is reopened by the Traffic Officer Service.
Mrs. Villiers: To ask the Minister of State, Department for Transport what data his Department and its agencies collect on the average time taken to reopen a motorway following a closure resulting from a collision. [309456]
Chris Mole: The Highways Agency does not collect data on the average time taken to reopen a motorway following a closure resulting from a collision.
The Agency sets Key Performance Indicators for its Traffic Officer Service. The Key Performance Indicator for incident clearance is the most accurate proxy measure of restoring traffic flow after an incident, i.e. when the road is reopened to traffic.
The target for clear-up on heavily trafficked roads is 80 per cent. of incidents cleared within 40 minutes and 95 per cent. cleared within 90 minutes.
Mrs. Villiers: To ask the Minister of State, Department for Transport what recent estimate his Department has made of the average time taken between notification of an incident on the motorway and the reinstatement of normal traffic flow following that incident in the last 12 months for which figures are available. [309457]
Chris Mole: The Highways Agency does not have the data to estimate the average time taken between notification of an incident on the motorway and the reinstatement of normal traffic flow following that incident.
The Agency has found that after most incidents have been cleared, some congestion is highly likely, which diminishes over time until normal flow is resumed. This varies according to factors including the time of day and the location.
Mrs. Villiers: To ask the Minister of State, Department for Transport what recent steps his Department has taken to improve arrangements for communication between traffic officers and the police in respect of traffic incidents on the motorway. [309462]
Chris Mole: On-road traffic officers and the police utilise the Airwave radio service to communicate at incident scenes. The police also use Airwave to communicate directly with Highways Agency regional control centres. To add to this capability, the agency is progressing a programme of work to co-locate police service staff at agency regional control centres.
The Highways Agency and the Association of Chief Police Officers recently launched a Traffic Incident Management Guidance Framework which outlines the roles and responsibilities of both parties when managing motorway incidents.
Regular multi-agency exercises test operational capability, capture lessons learned and improve working relationships between the Highways Agency and other incident responders on a regional basis.
Mrs. Villiers: To ask the Minister of State, Department for Transport what key performance indicators have been set by his Department and its agencies for traffic officers on setting signs and signals to inform drivers of a motorway incident; and what recent assessment he has made of the effectiveness of the system of setting signals to inform drivers of a motorway incident. [309463]
Chris Mole: Signs and signals setting by traffic officers is measured by two key performance indicators within the Highways Agency:
Signal and sign setting performance indicators
Set 90 per cent. of appropriate signs and signals within three minutes of a traffic officer/police request being received and changed/removed within two minutes.
All operators to achieve 90 per cent. pass on quality compliance checks for incident handling.
The agency is unable to fully measure the effectiveness of the system of setting signs and signals to inform drivers of a motorway incident.
But the agency has monitoring processes in place at the National Traffic Control Centre, through which it reviews the accuracy of variable message signs and puts corrective action in place when errors are identified.
The agency undertakes National Road Users Satisfaction Surveys on a regular basis which includes questions regarding the accuracy of messages shown on variable message signs. These results are used to inform improvements to accuracy of messages.
The Highways Agency information line also gathers real time feedback from road users and is able to check the accuracy of the set signs and signals.
Mrs. Villiers: To ask the Minister of State, Department for Transport what recent assessment his Department and its agencies have made of the effects on motorway incident clear-up times of the division of roles and responsibilities between the Highways Agency and the police; and if he will make a statement. [309464]
Chris Mole: There has been no assessment of how motorway incident clear-up times have been affected following the realignment of roles and responsibilities of the Highways Agency and the Police in 2004.
An assessment of how the realignment of the roles and responsibilities has impacted motorway clear-up times would require performance data indicative of how effectively motorway clear-up was performed prior to the realignment of roles and responsibilities. This data is not available to the Agency and therefore it is not possible to make a comparison.
Mrs. Villiers: To ask the Minister of State, Department for Transport (1) what his most recent estimate is of the average time taken to remove a vehicle from a motorway following a collision; [309465]
(2) what his most recent estimate is of the average time taken to remove an abandoned vehicle from a motorway. [309466]
Chris Mole: Where a traffic officer arranges for a vehicle to be removed from the carriageway as part of the National Vehicle Recovery Programme, performance targets are in place.
In November 2009 81 per cent. of light vehicles classified as 'immediate priority' were removed within 45 minutes. 99.9 per cent. of all vehicles classified as 'immediate priority' were removed within 120 minutes. 82 per cent. of light vehicles classified as 'routine' were removed within 45 minutes. 100 per cent. of all vehicles classified as 'routine' were removed within 120 minutes.
The response time information only relates to the vehicle removal component of the incident, for those incidents involving a vehicle where a traffic officer has arranged for removal on the Strategic Road Network. The Department for Transport does not hold response time information for those vehicles removed by the police or where the owner of the vehicle makes their own suitable recovery arrangements.
Mrs. Villiers: To ask the Minister of State, Department for Transport what key performance indicators his Department has set in respect of the duration of motorway closures following traffic incidents; and what recent steps his Department has taken to reduce the time taken to re-open a motorway after such an incident. [309467]
Chris Mole: The Highways Agency has set key performance indicators for the traffic officer service in respect of incident response times and incident clearance times.
The Agency has introduced a number of initiatives aimed at reducing the incident time-line. These include loan of collision investigation equipment to the police, introduction of the Traffic Incident Management Guidance Framework, Enhanced Incident Support Units, availability of specialist equipment and resources and the National Vehicle Recovery Contract.
Mrs. Villiers: To ask the Minister of State, Department for Transport what key performance indicators have been set by his Department and its agencies for traffic officers in respect of incident clear-up times on motorways following traffic incidents. [309471]
Chris Mole: The Highways Agency has set the following key performance indicators in respect of incident clear-up times on motorways following traffic incidents:
Incident clearance times performance indicators( 1)
Between 6 am and 10 pm on heavily trafficked roads, clear 80 per cent. of immediate graded(2) incidents from all live lanes within 40 minutes.
Between 6 am and 10 pm on lightly trafficked roads, clear 80 per cent. of immediate graded incidents from all live lanes within 45 minutes.
Between 6 am and 10 hrs on heavily trafficked roads, clear 95 per cent. of immediate graded incidents from all live lanes within 90 minutes.
Between 6 am and 10 am on lightly trafficked roads, clear 95 per cent. of immediate graded incidents from all live lanes within 95 minutes.
(1) "Incident clearance" is defined and recorded where Traffic Officer Service is involved in the management of the incidents.
(2) An "immediate graded incident" is one where there is an obstruction on or related to the live carriageway that is likely to cause congestion or threaten safety.
Mrs. Villiers: To ask the Minister of State, Department for Transport what the average time taken was between the notification of a regional control centre of a motorway traffic incident and the arrival of the traffic officer service at the scene of that incident in each of the last 10 years; and what performance indicators are in place in respect of that response time. [309472]
Chris Mole: The traffic officer service has only been fully operational nationally since 2007. While some regional data exists on response time prior to October 2006, the incomplete data cannot be used to calculate the average time.
The following table shows the average time taken between the notification of a regional control centre of a motorway traffic incident and the arrival of the traffic officer service:
Average immediate response times over a 24 hour period (minutes)
1 October 2006 to 31 March 2007
The figures illustrated in the table above show the average response time for the Traffic Officer Service to attend incidents classified as 'immediate'. An 'immediate graded incident' is one where there is an obstruction on or related to the live carriageway that is likely to cause congestion or threaten safety.
The Highways Agency has not set a key performance indicator for the average time taken between the notification to a regional control centre of a motorway traffic incident and the arrival of the traffic officer service at the scene. This aspect of performance is addressed as part of the indicators for the clearance of incidents from all live lanes.
Mrs. Villiers: To ask the Minister of State, Department for Transport how many staff of his Department are employed (a) full-time and (b) part-time on research for the purposes of reducing incident clear-up times on motorways. [309474]
Chris Mole: The Highways Agency manages research into incident clear up times on motorways. The Highways Agency currently has six full-time staff who spend on average approximately 10 per cent. of their time working on research into reducing incident clear up times on motorways.
Mrs. Villiers: To ask the Minister of State, Department for Transport what trials using laser-surveying equipment his Department has undertaken for the purposes of reducing motorway incident clear-up times. [309475]
Chris Mole: The Highways Agency is currently working closely with Humberside, Sussex and the Metropolitan police forces to trial laser-surveying equipment. This trial will establish what benefits can be provided with respect to reducing the time it takes the police to carry out the investigation of fatal or serious injury incidents on the motorway. The final evaluation of the laser-surveying trial will be completed in 2011.
Mrs. Villiers: To ask the Minister of State, Department for Transport how much his Department and its agencies have spent on (a) research and (b) trials of equipment for the purposes of reducing motorway incident clear-up times on motorways in the last five years. [309476]
Chris Mole: The Highways Agency manages research into incident clear up times on motorways. In the last five years (from financial year 2005-06 up to and including the current financial year) the Highways Agency has invested approximately £0.507 million on research and £0.154 million on trials to improve incident clear up times on motorways.
Mr. Andrew Smith: To ask the Minister of State, Department for Transport what recent assessment his Department has made of the effects on accident rates of spray on (a) motorways and (b) other principal roads. [316103]
Chris Mole: There were 421 personal injury accidents on our trunk road network in the three year period 2006 to 2008 where spray was noted by the police as a contributory factor. Of the 421 personal injury accidents, there were 262 on motorways and 159 on 'A Class' trunk roads associated with 'spray from other vehicles'.
There is no ongoing or recently completed research into the effects of spray on accident rates on motorways or other principal roads.
In 2008, joint research by the Highways Agency and the road surfacing industry attempted to measure spray levels on different road surfaces on motorways and trunk roads with varying rainfall conditions but was inconclusive as the available technology could not provide a feasible way of measuring spray levels.
Roads: Accidents
Norman Baker: To ask the Minister of State, Department for Transport how many road accidents resulted in the death or serious injury of (a) an adult and (b) a child in (i) Lewes constituency and (ii) East Sussex in each year since 1994. [315770]
Paul Clark: The information requested can be found in the following tables.
Speed Limits: Cameras
Mr. Greg Knight: To ask the Minister of State, Department for Transport if he will require each local authority seeking to install speed cameras on motorways passing through that local authority's area to use the money raised in consequence of the operation of such cameras to install lighting on motorways throughout that area. [316010]
Chris Mole: The Highways Agency is responsible for maintaining and operating the motorway network including installing speed cameras and lighting.
Lighting on motorways would only be considered where safety benefits gained from the lighting will exceed the whole-life costs of installation, operating and maintaining it.
Fine revenue from speeding offences goes directly to HM Treasury.
Traffic Officers
Mrs. Villiers: To ask the Minister of State, Department for Transport how many patrol vehicles were in operation on the motorway network on the latest date for which figures are available. [309459]
Chris Mole: Figures for week commencing Monday 21 December 2009 show that 113 patrol vehicles were deployed for each of the two day shifts and 64 patrol vehicles were deployed for the night shift.
The patrol routes include the motorway network and a small number of specific sections of all purpose trunk roads in England.
Mrs. Villiers: To ask the Minister of State, Department for Transport what assessment has been made of the performance of the traffic officer service since its implementation; what recent steps his Department and its agencies have taken to increase the effectiveness of that service; and when he next plans to review the roles of traffic officers. [309468]
Chris Mole: The traffic officer service was established between 2004 and 2007. A review of the service was completed in April 2009 and recognised that performance of the service was in line with expectations in terms of the contribution towards reduced congestion, improved safety and the freeing up of police time.
The effectiveness of the service has been enhanced since becoming fully operational with the introduction of a number of initiatives. These include:
the introduction of key performance indicators;
the national vehicle recovery manager contract which has resulted in improvements to the removal of abandoned and collision damaged vehicles; the co-location of the police control room staff in the regional control centres enabling improved incident management; and
an award-winning training scheme improving the traffic officer service capability.
There are no current plans to further review the roles of the traffic officers.
Traffic Officers: Expenditure
Mrs. Villiers: To ask the Minister of State, Department for Transport how much his Department and its agencies spent on (a) fuel and (b) maintenance for patrol vehicles in each of the last three years. [309452]
Chris Mole: The Vehicle Operator Services Agency and the Highways Agency use patrol vehicles; however, spend information on fuel and maintenance for patrol vehicles is not recorded separately and can be provided only at disproportionate cost. Where information is available it has been set out as follows:
The Vehicle Operator Services Agency spend on fuel for patrol vehicles from October 2008 to September 2009 was £161,467 and spend on maintenance of patrol vehicles from October 2008 to September 2009 was £164,343.
The Highways Agency spend on fuel for patrol vehicles was £2,340,000 in 2006-07, £2,016,000 in 2007-08 and £2,772,000 in 2008-09. Vehicles are provided on a lease basis which includes the lease and maintenance costs which cannot be separated. The lease cost for patrol vehicles was £2,305,000 in 2006-07, £2,337,000 in 2007-08 and £3,118,000 in 2008-09. For 2008-09 approximately one-third of the lease cost was for maintenance. The increase in costs is largely as a result of the growth in the number of traffic officer patrols as the service was rolled out across the motorway network in England.
The Maritime and Coastguard Agency does not use patrol vehicles but has vehicles that are used in response to incidents.
Mrs. Villiers: To ask the Minister of State, Department for Transport what the cost of patrol vehicles maintenance has been in each year since the introduction of the traffic officer service. [309460]
Chris Mole: Traffic officer vehicles are provided on a lease basis and the Highways Agency is charged a monthly fee that comprises costs for both lease and maintenance. The annual spend since 2004 is shown in the following table.
Financial year Lease/maintenance (£000)
For 2008-09 approximately one third of the lease cost was for maintenance.
The increase in costs is largely as a result of the growth in the number of traffic officer patrols as the service was rolled out across the motorway network in England.
Mrs. Villiers: To ask the Minister of State, Department for Transport how much his Department and its agencies had spent on the traffic officer service on the latest date for which figures are available. [309470]
Chris Mole: The expenditure on the traffic officer service is not separately identified in the annual report and accounts of the Highways Agency. An extract of the management accounts has been used to provide the information requested.
The Highways Agency has spent the following on the provision of the traffic officer service, from inception to the latest audited accounts.
Financial year Expenditure (£ million)
The early years include the set-up costs for the service. The service was rolled out across the regions, reaching full coverage in October 2006 with the first full year of operation being 2007-08.
Traffic Officers: Training
Mrs. Villiers: To ask the Minister of State, Department for Transport what his most recent estimate is of the average cost to his Department and its agencies of training a traffic officer to the standard required to discharge duties effectively in the latest period for which figures are available. [309469]
Chris Mole: Traffic officers operate primarily either on the road or in a regional control centre and the training requirement for each role is different. The average cost of training for each role is:
This cost covers initial off the job training, followed by on the job coaching to ensure that new recruits can operate safely and competently in a live environment.
Mr. Greg Knight: To ask the Minister of State, Department for Transport how many fatalities there were on the M1 in (a) sections with side run-off protection and (b) sections without side run-off protection in each of the last three years for which figures are available. [316447]
Chris Mole: The following table shows validated figures for all fatalities for the M1 in each of the last three years for which figures are available.
Mr. Greg Knight: To ask the Minister of State, Department for Transport what proportion of the length of the M1 has side run-off protection; and what estimates he has made of the cost of installing such protection for the entire length of the M1. [316448]
Chris Mole: Approximately 45 per cent. of the M1 has side run-off protection.
To date there has been no estimate made of the cost of installing side run-off protection on the entire length of the M1. Such systems are put in place on sections of road that satisfy specific criteria as set out in Highways Agency standards. The need for such vehicle restraints is under constant review as part of operational requirements.
Driving Offences: Convictions
Chris Grayling: To ask the Secretary of State for Justice how many hit and run traffic incidents led to an individual being convicted in (a) England and Wales and (b) each police force area in each year since 1998. [315061]
Maria Eagle: The number of hit and run incidents cannot be directly linked to prosecutions and convictions at court. Therefore, it is not possible to provide the information requested.
The information given in the following table shows the number of offenders convicted at all courts for the offence of "failing to stop after an accident" in England and Wales, by police force area, from 1998 to 2008 (latest available).
http://www.publications.parliament.uk/pa/cm200910/cmhansrd/cm100210/text/100210w0024.htm#1002111000022
Driving Offences: Fines
Mr. Grieve: To ask the Secretary of State for Justice what the average fine imposed for the offence of using a motor vehicle uninsured against third-party risks was in 2008. [316323]
Claire Ward: I refer the hon. Gentleman to my answer on 2 February 2010, Official Report, column 262W.
Driving Offences: Suffolk
Mr. Ruffley: To ask the Secretary of State for Justice (1) how many drivers have been prosecuted for offences under sections (a) 3 and (b) 34 of the Road Traffic Act 1988 in Suffolk in each year since 1997; [314875]
10 Feb 2010 : Column 1075W
(2) how many drivers have been prosecuted for failing to stop after an accident under the Road Traffic Act 1988 in Suffolk in each year since 1997. [314882]
Claire Ward: The number of proceedings at the magistrates courts for the offence of driving without
due care and attention, driving on footway and failing to stop after an accident for the years 1997 to 2008 are shown in the following tables.


text:
EORÐESLAJYR Ambient Metal / Co. Dublin
EORÐESLAJYR are an experimental/heavy music group from Dublin Ireland. For fans of Noise/Drone/Dark/Ambient/Avante-Garde… it's all thrown into the mix.
I literally just stumbled across this outfit while browsing through my newsfeeds. Their debut album "HÄXAN" (link) was released on the 31st October 2016 and is available to buy from the band as a limited edition cassette or digital download.
Doesn't really qualify as 'METAL', per se, but this kind of stuff normally finds it's way into many dark metal fans playlists. Most of it I wouldn't normally be too bothered with, but it's still an interesting find and worth checking out if your into the more sinister and off-kilter 'Noise' side of the musical spectrum.
EORÐESLAJYR – HÄXAN
EORÐESLAJYR – VIGIL
FRACTURED: "Altered States – A Fractured Mix"…
Ambient Metal, Atmospheric Metal, Avante-Garde, Drone/Noise
Sam Burton - Voice
Simon Bird - Synth/Electronics
Tom Morris - Guitars

redpajama_set_name: RedPajamaCommonCrawl


Haven't found the right listing? There are 69 other Salons in Hendersonville. Click here!
Guys and Dolls Styling Salon can be found at 501 Brookside Camp Rd . The following is offered: Salons . In Hendersonville there are 69 other Salons. An overview can be found here.


"This recipe has been used for years in our church for Communion bread. Easy to make and stores well in fridge. Watch carefully when baking, ovens vary in time needed."
Mix flour, oil, and salt together in a bowl; add water and mix using a pastry cutter until dough is soft. Form dough into 6 balls and press into disks onto the prepared baking sheet using your hands.
Bake in the preheated oven until bread is cooked, 8 to 10 minutes.
Make authentic Cuban bread for awesome sandwiches!
I use a scant 1/3 cup oil and 1/3 cup water.
I made this for church according to recipe and everyone loved it!
i had a bible study at my house, so i decided to bake these up for my gals. this recipe works great! i really felt gods presence within these tokens of bread.god bless ??!
I tried this, but with whole wheat flour, olive oil, coconut oil, garlic powder and thyme! Great recipe, thanks!


Lessons from a botched haircut.
You know I find a lesson in everything, right? So why would my haircut be any different? When it comes to my 'do, it's rare that I keep up with it. Sure, I style it up, but for the past few years it gets a d.i.y. coloring job and highlights compliments of the O'ahu sun. So when it was time for my yearly haircut, I was sadly excited and impatient, and hopped on the first appointment I could get. The salon was fancy, offering a mix of my favorite Aveda products (I had been there before, but sadly had forgotten my experience a year prior) and green tea with way too much splenda. I showed about 8 different photos of the hair I had been coveting, and was met with what felt like confidence behind the scissors. Hours later the chair whirled around as the stylist flipped my locks in front of my shoulders and announced I was "all set!". I glanced at the mirror in front of me in total shock. My hair was lighter, but not over $200 worth, and my 'feathery layers that could be bangs' seemed like hunks of hair bluntly hanging by my cheeks. Fighting back the mix of emotions, I paid my balance, left tips, and hustled to my car, pulling my hair into a bun. I tried every kind of hairstyle I could in hopes my hair wasn't that bad, but the added "layers" that seemed more like a 5 year old was appointed to give me my new look were hard to conceal. Blonde, brown or wavy, even my sweet husband who always declares "don't change your hair, I LIKE it!" agreed: "yeah, maybe you should get a refund...". In the meantime I frantically texted another stylist Bride (hey Lauren!) had been referred to via instagram (alongside with the salon I left in tears). It was two weeks worth of my hair in topknot later until Kitt was kind enough to get me in and work magic on my botched 'do. As I sat in his chair I was absolutely CONFIDENT that he would be able to work wonders--and I had never even met the guy before, just stalked out his social media. I knew that with so many girls left leaving his salon that he had to be my savior. As he got to work brightening up my hair and evening out my layers we shared conversation and I got a glimpse into his passion. This guy wasn't just someone who went to school for a craft and does it to make a living, he is passionate about it. Working 12 hour days 6 days a week, he never grows tired of giving precise cuts, honing his eye and being one of the best at what he does. As talked about business he commented on how using different mediums and tools help you see and create things different, using the comparison of an artist picking up a different brush and being free to make a stroke they never even knew they could create. Astounded by the sole marketing power of his instagram account, I asked this business school graduate turned independent hairstylist what he 'secret' to killin it was.
"Do good work and people will talk about you".
Wanna set up an appointment with this mastermind? He is an independent stylist based at KISSA salon by Ala Moana. Kitt, @khietluu on Instagram.


This is an application that allows the user to conduct different tasks on different Facebook pages. The user can post updates and photographs and also respond to the different pictures in the form of comments. The tool will also generate push notifications when there is any activity on Facebook and also the latest insights on the several pages that have been saved. With this application, the user can add a photo to a specific album, attach a picture to a status update and much more.
There are also features which will enable sharing of links and pictures from other applications and websites. What impresses the most is that the user can also save drafts of a post using this application which proves to be a boon for the people who upload statuses regularly. Advanced pictures like emojis and stickers are also available. The interface of the application is easy and the tasks are conducted smoothly. Facebook Pages Manager for PC can be downloaded and installed from the BlueStacks App Player.


Amazon.com : Custom Trophy Plate / Name Plate With 3 Lines of Custom Text : Sports Award Trophies : Sports & Outdoors.
Home > Small Flexi-Brass Name Plates with Engraving. Home > Award Plaques. Includes 3 lines of engraving (25 characters/spaces per line). Custom sizes are. The plaque I ordered was perfect and arrived amazingly quick. I highly .
Crown Awards engraving plates are the perfect way to personalize and complete the look of any trophy, plaque, or medal to be awarded. Choose from our gold, .
Engraved Plate 1"x3" Custom TWO TONE Aluminum Trophy Plate Name Tag Plaque Art. YOU ARE PURCHASING ONE (1) CUSTOM ENGRAVED 1" X 3" TWO TONE ALUMINUM PLATE. Engraved Plate Color Choices: Gold, Black, Red, Blue, Copper, Silver, Violet, Purple, Green, Red-Marbled or Black-Marbled.
Customize these engravable plates for your occasion today.. The non-coated silver and gold enameled surface engravable name plates are. Items 1-3 of 3.
Home · Awards and Trophies; 3" x 6" Laser Engraved Name Plate. Custom Sized Laser Engraved Name Plate "up to" 3×5 $7.50. 3" x 1.5" Laser Engraved .
3" x 1.5" Laser Engraved Name Plate $6.50. 2.5 x 1 Laser. Photo Frames, Awards, Plaques, Boy Scouts, Taxidermy, Wood Workers, Businesses and more.
Any Size Laser Engraved Name Plates up to 3" x 5" or 15 Square inches. . 3×1" Personalized Engraved Silver Name Plate, Name Plaque, Name Tag, Trophy .
Skyframe provides custom plaques and custom plaque plates for its gallery, commercial and residential. 7/8 x 2 3/4" Black Perpetual Plaque Name Plate .


I don't have a kid, but if I did, this is the record I think I would be least annoyed by if my kid got totally obsessed with it. The vocals (courtesy of indie rock legend Sarah Shannon of Velocity Girl) fall somewhere between riot grrrl and the B-52s, and the guitars just rip. The lyrics are also kind of punk: "Done with the Science Fair" is all about a child who just can't take it anymore, and decides to just throw some tape on cardboard and call it good. However, this surrender doesn't come from a place of apathy. "Don't get me wrong, I love science, I really do," she assures the listener; she is just against the academic bureaucracy that bogs it down. As someone who has seen friends lose interest in their fields due to the tedium of writing countless grants for access to ever-diminishing funds and pushing against the patriarchal barriers held in place by academia, this is a moral I can get behind.
Engagement is a thread that actually runs solidly through the Not-Its!' work. "Traffic Jam" is a track about turning the most mundane and boring moments of life into catalysts for creativity, which I think is a really wonderful message.
In a genre that is intentionally gimmicky, the Not-Its! have found a really great niche. They are the party rockers, the Kiss stage show-havers, they are the first real rock n' roll show a kid is going to see. They aren't just going to teach lessons about the alphabet or basic congressional procedure; they're here to instill a sense of creativity and a love of loud guitars in a child, and that's really cool.
I will say I totally disagree with "Don't Fear the Dentist," though. Dentists are terrifying, and that's just a fact.


Internal communications within any organization are completely vital to its overall success. Communications that flows downward, upward and across company lines is encouraged for many reasons. However, if the internal communications are going to be effective all over the organization, there must be several essential elements in good practices. Hence, for those business owners and representatives who are responsible for the primary communications between individuals and groups, here are some key elements that must be in place to make up an effective internal communications plan.
One of the most critical elements in any internal communications program is consistency. Trying to improve communications in any area involves providing consistent messages to all across the board. Therefore, whenever a plan is being designed, the method used must not be tinkered with by following a new trending methods. Instead, a consistent message of communications will build a certain amount of expectations that groups of people internally will be looking for.
Line management is considered to be one of the top choices when communications are being designed and communicated. Typically, this is the case for those employees who will have questions and those that will need answers to critical situations. The line manager is tasked with providing these messages directly instead of indirectly.
Some people are known for having natural gifts in communication and they may also have a certain amount of Charisma when giving a message to both small audiences. While all of the added benefits of being a gifted speaker, these gifts cannot outweigh the need for careful planning and attention to detail. Simply put, internal communication must be accurate in order for the receiver of the message to gain trust in what is being said.
When internal communications are being planned, the provider of the message must consider a wide variety of crucial factors including the timing of its presentation. So, the channel that is used must also be considered a big part of distributing the message to those who need to know. Bad timing can destroy all communications and can lead to issues within the organization. Hence, the message cannot be too late or given prematurely. Also, the channels of communications planned must be available for use at the time that they are needed.
Internal communication is not a one-way method of getting information to a group of employees. In fact, once the information has been provided to the group, there needs to be a way to obtain feedback from those who have received what has been written or orally presented. By getting feedback, the management will have an opportunity to act on what else can be done. For instance, if a new procedure is being communicated to an operation, the management will need feedback to see how well it is working and if changes need to be made.
What Does The Time period AV Stand For?


1. You will receive an e-receipt from PayPal when you process your payment.
2. If the email that you enter to process your payment is different from the email you used to complete the online workshop registration, please click here to inform us. We will send you a confirmation email and bonus download.
3. This offer is only available if you have registered and attend Saturday's workshop.
*Details regarding the schedule and registration will be provided during the 10/20 workshop.
Get your copy now at this special $25 price. Save $10 off the retail price, plus save on shipping & handling. We will have your workbook available at the sign-in table when you arrive at the workshop.


text:
\section*{Introduction}

\indent
The quaternionic geometry is, informally, that part of the geometry where the points are spheres. This means that the objects
of quaternionic geometry are parameter spaces for families of (Riemann) spheres embedded into CR manifolds, the latter,
being called the \emph{twistor spaces} (of the corresponding objects). When the twistor spaces are complex manifolds
and the normal bundles of the twistor spheres are nonnegative we obtain what we call $\rho$-quaternionic manifolds \cite{Pan-qgfs}\,.\\
\indent
This paper grew out from our aim to understand the $\rho$-quaternionic manifolds $M$ whose twistor spaces are homogeneous.
As $M$ is not necessarily homogeneous (see Example \ref{exm:nil_orbits}\,, below), one of the main tasks is to describe
the intrinsic geometry of the corresponding (local) orbits. This is, further, motivated by the idea that, because the corresponding twistor space
is homogeneous, each such orbit should, somehow, determine the geometry of $M$.\\
\indent
We are, thus, led to consider the \emph{quaternionic-like manifolds} (note that, this expression was previously used with a different meaning)
which, at the linear level, are given by embeddings of the sphere into the complex Grassmannian of a vector space
(these embeddings are required to be compatible with the conjugations; see Section \ref{section:q-like_linear}\,, below, for details).
Then, similarly to the classical linear quaternionic structures, the dual of a linear quaternionic-like structure
is quaternionic-like. Furthermore, the class of quaternionic-like manifolds contains two important subclasses: the CR quaternionic manifolds of \cite{fq}\,,
and the $\rho$-quaternionic manifolds of \cite{Pan-qgfs} (in particular, the co-CR quaternionic manifolds of \cite{fq_2}\,).
Moreover, the notion of \emph{heaven space} (see \cite{fq}\,) finds its adequate level of generality in this setting (Theorem \ref{thm:heaven_space_for_q-like}\,).
For example, if $M$ is a $\rho$-quaternionic manifold whose twistor space is homogeneous then each orbit, of the induced local action on $M$,
is endowed with a quaternionic-like structure whose heaven space is $M$.\\
\indent
We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres (with nonnegative normal bundles)
which is then characterized and illustrated with several classes of examples (Section \ref{section:constr_hqo}\,).
This emphasizes the abundance of the quaternionic-like manifolds.

\section{Quaternionic-like vector spaces} \label{section:q-like_linear}

\indent
This section is dedicated to the study of the following notions.

\begin{defn} \label{defn:q-like}
A \emph{linear quaternionic-like structure} on a (real) vector space $U$ is a holomorphic embedding of the sphere
into the complex Grassmannian ${\rm Gr}_k\bigl(U^{\mathbb{C}\,\!}\bigr)$ which intertwines the antipodal map and the conjugation determined by
the conjugation of $U^{\mathbb{C}\,\!}$, where $k\in\mathbb{N}$\,, $k\leq\dim U$.\\
\indent
A \emph{quaternionic-like vector space} is a vector space endowed with a linear quaternionic-like structure.
\end{defn}

\indent
On identifying, as usual, an embedding with its image, if $(U,t)$ is a quaternionic-like vector space, we denote by $U^z$ the subspace of $U^{\mathbb{C}\,\!}$
which is the image of $z\in t$ through the corresponding embedding. Then the \emph{dual} linear quaternionic-like structure on $U^*$ is
given by associating to each $z\in t$ the annihilator of $U^z$. Also, a \emph{quaternionic-like linear map},
between the quaternionic-like vector spaces $(U,t_U)$ and $(V,t_V)$\,, is a pair $(\psi,T)$\,, where $\psi:U\to V$ is linear
and $T:t_U\to t_V$ is an orientation preserving isometry
such that $\psi\bigl(U^z\bigr)\subseteq V^{T(z)}$\,, for any $z\in t_U$\,.

\begin{prop} \label{prop:basic_q-like_corresp}
There exists a functorial (bijective) correspondence between linear qua\-ter\-ni\-o\-nic-like structures on $U$
and nontrivial (that is, nonsplitting) exact sequences
$$0\longrightarrow\mathcal{U}_-\longrightarrow t\times U^{\mathbb{C}\,\!}\longrightarrow\mathcal{U}_+\longrightarrow0$$
of holomorphic vector bundles over the sphere $t$\,, endowed with conjugations covering the antipodal map.\\
\indent
Furthermore, under this correspondence, dual linear quaternionic-like structures correspond to dual exact sequences.
\end{prop}
\begin{proof}
If $(U,t)$ is a quaternionic-like vector space then $\mathcal{U}_-$ is the restriction to $t$
of the tautological vector bundle over the corresponding Grassmannian.
\end{proof}

\indent
Note that, if $E$ is a (classical) quaternionic vector space (see, for example, \cite{fq}\,) and $t$ is the corresponding space
of compatible linear complex structures then $t\to{\rm Gr}_{2k}\bigl(E^{\mathbb{C}\,\!}\bigr)$\,, $J\mapsto{\rm ker}(J+{\rm i})$\,, $(J\in t)$\,,
is a linear quaternionic-like structure, where $\dim E=4k$\,.\\
\indent
More generally, we have the following classes of examples.

\begin{exm}
Let $E$ be a quaternionic vector space and let $\rho:E\to U$ be a linear map such that $({\rm ker}\rho)\cap J({\rm ker}\rho)=\{0\}$\,,
for any compatible linear complex structure $J$ on $E$. Then, on denoting by $t$ the space of compatible linear complex structures on $E$,
we obtain the linear quaternionic-like structure on $U$ given by $t\to{\rm Gr}_{2k}\bigl(E^{\mathbb{C}\,\!}\bigr)$\,, $J\mapsto\rho\bigl({\rm ker}(J+{\rm i})\bigr)$\,,
$(J\in t)$\,, where $\dim E=4k$\,.\\
\indent
We call $(E,\rho)$ a \emph{linear $\rho$-quaternionic structure} \cite{Pan-qgfs} on $U$. Dualizing, we obtain what we shall call a
\emph{linear $\rho^{*\!}$-quaternionic structure}.
\end{exm}

\indent
Note that, if we assume $\rho$ and $\iota$ surjective and injective, respectively, then we obtain the notions of linear co-CR quaternionic and
linear CR quaternionic structures, introduced in \cite{fq}\,.\\
\indent
To understand how the linear $\rho$-quaternionic structures are placed among the linear qua\-ter\-ni\-o\-nic-like structures,
we use Proposition \ref{prop:basic_q-like_corresp}\,. Then a quaternionic-like vector space $(U,t)$ is $\rho$-quaternionic if and only if
in the Birkhoff--Grothendieck decomposition of $\mathcal{U}_-$ appear only terms of Chern number $-1$\,; consequently, $(U,t)$ is, in this case,
determined by $\mathcal{U}_+$  which we call its holomorphic vector bundle.\\
\indent
Moreover, we have the following basic result (in which the relevant morphisms are defined in the obvious way, and $\mathcal{U}_{\pm}$
are as in Proposition \ref{prop:basic_q-like_corresp}\,).

\begin{thm} \label{thm:basic_q-like_corresp}
There exist functorial correspondences between the following, where $U$ is a vector space:\\
\indent
\quad{\rm (i)} Linear quaternionic-like structures on $U$.\\
\indent
\quad{\rm (ii)} Pairs $(U_+,\psi_+)$\,, where $U_+$ is endowed with a linear $\rho$-quaternionic structure $(E_+,\rho_+)$\,,
and $\psi_+:U\to U_+$ is a linear map such that $\bigl(U_+\bigr)^z+\bigl({\rm im}\,\psi_+\bigr)^{\mathbb{C}\,\!}=\bigl(U_+\bigr)^{\mathbb{C}\,\!}$,
for any $z$ in the embedded sphere of $E_+$\,.\\
\indent
\quad{\rm (iii)} Pairs $(U_-,\psi_-)$\,, where $U_-$ is endowed with a linear $\rho^{*\!}$-quaternionic structure $(E_-,\rho_-^*)$\,,
and $\psi_-:U_-\to U$ is a linear map such that $\bigl(U_-\bigr)^z\cap\bigl({\rm ker}\,\psi_-\bigr)^{\mathbb{C}\,}=\{0\}$\,, for any $z$
in the embedded sphere of $E_-$\,.\\
\indent
Moreover, if $(U_+,\psi_+)$ and $(U_-,\psi_-)$ correspond to the same linear quaternionic-like structure then $E_+=E_-$
and $\psi_+\circ\psi_-=\rho_+\circ\rho_-^*$\,. Also, the holomorphic vector bundles of $(U_+,E_+,\rho_+)$ and $(U_-,E_-,\rho_-^*)$
are $\mathcal{U}_+$ and $\mathcal{U}_-$\,, respectively.
\end{thm}

\indent
To prove Theorem \ref{thm:basic_q-like_corresp}\,, we shall need the following result (cf.\ \cite[Proposition 3.1]{Qui-QJM98}\,).

\begin{prop} \label{prop:from_Quillen_3.1}
Any nonnegative holomorphic vector bundle $\mathcal{U}$ over the sphere corresponds uniquely (up to canonical isomorphisms) and functorialy,
to each of the following exact sequences (where, for example, $\mathcal{O}$ is the canonical function sheaf of the sphere (identified with
the sheaf of holomorphic section of the trivial line bundle $t\times\mathbb{C}\,$)\,, $\mathcal{O}(-1)$ is the tautological line bundle over the sphere (identified with the
complex projective line), and $\mathcal{U}(-1)=\mathcal{U}\otimes\mathcal{O}(-1)$\,):
\begin{equation} \label{e:first_canonical}
0\longrightarrow \mathcal{O}(-1)\otimes H^0\bigl(\mathcal{U}(-1)\bigr)\longrightarrow\mathcal{O}\otimes H^0(\mathcal{U})\longrightarrow\mathcal{U}\longrightarrow0\;,
\end{equation}
\begin{equation} \label{e:second_canonical}
0\longrightarrow\mathcal{O}\otimes H^0\bigl(\mathcal{U}(-2)\bigr)\longrightarrow\mathcal{O}(1)\otimes H^0\bigl(\mathcal{U}(-1)\bigr)\longrightarrow\mathcal{U}
\longrightarrow\mathcal{O}\otimes H^1\bigl(\mathcal{U}(-2)\bigr)\longrightarrow0\;.
\end{equation}
\end{prop}
\begin{proof}
\indent
The sequence \eqref{e:first_canonical} is the exact sequence of \cite[Proposition 3.1]{Qui-QJM98}\,. It is useful, however, to recall how it is obtained.
Firstly, as $\mathcal{U}$ is nonnegative, the canonical morphism $\mathcal{O}\otimes H^0(\mathcal{U})\longrightarrow\mathcal{U}$ is surjective, and the induced exact sequence
of cohomology implies that its kernel is of the form $\mathcal{O}(-1)\otimes V$ for some (complex) vector space $V$. Then, on tensorising
$0\longrightarrow \mathcal{O}(-1)\otimes V\longrightarrow\mathcal{O}\otimes H^0(\mathcal{U})\longrightarrow\mathcal{U}\longrightarrow0$ with $\mathcal{O}(-1)$
and passing to the cohomology exact sequence we obtain that $V=H^0\bigl(\mathcal{U}(-1)\bigr)$\,.\\
\indent
To construct \eqref{e:second_canonical}\,, note that, there exists a canonical exact sequence
\begin{equation} \label{e:for_second_canonical_1}
0\longrightarrow\mathcal{U}_1\longrightarrow\mathcal{U}\longrightarrow\mathcal{U}_0\longrightarrow0\;,
\end{equation}
where $\mathcal{U}_1$ is positive and $\mathcal{U}_0$ is trivial; moreover, $\mathcal{U}_0=\mathcal{O}\otimes H^1\bigl(\mathcal{U}(-2)\bigr)$\,.
Then $\mathcal{U}_1(-1)$ is nonnegative and therefore \eqref{e:first_canonical} holds for it. As
$H^0\bigl(\mathcal{U}(-j)\bigr)=H^0\bigl(\mathcal{U}_1(-j)\bigr)$\,, for any $j\in\mathbb{N}\setminus\{0\}$\,, on tensorising with $\mathcal{O}(1)$
the exact sequence \eqref{e:first_canonical}\,, applied to $\mathcal{U}_1(-1)$\,, we obtain
\begin{equation*}
0\longrightarrow \mathcal{O}\otimes H^0\bigl(\mathcal{U}(-2)\bigr)\longrightarrow\mathcal{O}(1)\otimes H^0\bigl(\mathcal{U}(-1)\bigr)\longrightarrow\mathcal{U}_1\longrightarrow0\;,
\end{equation*}
which, together with \eqref{e:for_second_canonical_1}\,, gives \eqref{e:second_canonical} (with its uniqueness and functoriality).
\end{proof}

\begin{rem} \label{rem:ro_from_canonical}
Let $\mathcal{U}$ be a nonnegative holomorphic vector bundle over the sphere, endowed with a conjugation covering the antipodal map.
Then the complexification of the corresponding linear $\rho$-quaternionic structure is obtained by applying the functor $H^0$ to the morphism
$\mathcal{O}(1)\otimes H^0\bigl(\mathcal{U}(-1)\bigr)\longrightarrow\mathcal{U}$ from \eqref{e:second_canonical}\,.
\end{rem}

\begin{proof}[Proof of Theorem \ref{thm:basic_q-like_corresp}\,]
If we dualize (ii) we obtain (iii) applied to $U^*$. Furthermore, if (ii)/(iii) holds then there exists a unique linear quaternionic-like structure on $U$,
with respect to which $\mathcal{U}_{\pm}$ is the holomorphic vector bundle of $U_{\pm}$\,, and $\psi_{\pm}$ is quaternionic-like linear and
induces the identity morphisms on $\mathcal{U}_{\pm}$.\\
\indent
To prove the converse, for simplicity, we work in the category of complex vector spaces. Thus, we shall ignore the conjugations, and, in particular,
we shall denote in the same way $U$ and its complexification. Then endow $U$ with a linear quaternionic-like structure.
By Proposition \ref{prop:basic_q-like_corresp}\,, this means that we have an exact sequence of holomorphic vector bundles over the sphere, as follows
\begin{equation} \label{e:exact_seq_for_q-like}
0\longrightarrow\mathcal{U}_-\longrightarrow\mathcal{O}\otimes U\longrightarrow\mathcal{U}_+\longrightarrow0\;.
\end{equation}
\indent
Let $U_+=H^0(\mathcal{U}_+)$ and let $\psi_+:U\to U_+$ the induced linear map. By duality, we, also, obtain $\psi_-:U_-\to U$.
Further, on tensorising \eqref{e:exact_seq_for_q-like} with $\mathcal{O}(-1)$ and by passing to the cohomology sequence we obtain
\begin{equation} \label{e:E+=E-}
H^0\bigl(\mathcal{U}_+(-1)\bigr)=H^1\bigl(\mathcal{U}_-(-1)\bigr)=H^0\bigl(\mathcal{U}_-^*(-1)\bigr)^*\;,
\end{equation}
where the second equality is given by the Serre duality. Together with Remark \ref{rem:ro_from_canonical}\,, this implies that $E_+=E_-$
(as (complex-)quaternionic vector spaces)\,, where $(E_+,\rho_+)$ and $(E_-,\rho_-^*)$ are the linear $\rho$-quaternionic and the linear $\rho^*\!$-quaternionic
structures of $U_+$ and $U_-$\,, respectively.\\
\indent
Now, we claim to have the following commutative diagram, in which all of the vertical and horizontal sequences are exact.

\begin{displaymath}
\xymatrix{
                 &          0 \ar[d]                                                                &   0  \ar[d]                                                              &                  &  \\
   0 \ar[r]   &   \mathcal{O}\otimes H^0(\mathcal{U}_-) \ar[d] \ar[r]^{=}     &   \mathcal{O}\otimes H^0(\mathcal{U}_-) \ar[d] \ar[r]      &   0 \ar[d]    &  \\
   0 \ar[r]   &   \mathcal{U}_- \ar[d] \ar[r]                                        &   \mathcal{O}\otimes U \ar[d] \ar[r]          & \mathcal{U}_+ \ar[d]^{=} \ar[r]   &   0  \\
   0 \ar[r]   &   \mathcal{O}(-1)\otimes H^0\bigl(\mathcal{U}_+(-1)\bigr) \ar[d] \ar[r]   &   \mathcal{O}\otimes U_+ \ar[d] \ar[r]          & \mathcal{U}_+ \ar[d] \ar[r]   &   0   \\
   0 \ar[r]   &   \mathcal{O}\otimes H^1(\mathcal{U}_-) \ar[d] \ar[r]^{=}       &   \mathcal{O}\otimes H^1(\mathcal{U}_-) \ar[d] \ar[r]      &   0           &   \\
                 &                0                                                                      &                      0                                                      &                 &   }
\end{displaymath}

\indent
Indeed, the second horizontal sequence is just \eqref{e:exact_seq_for_q-like}\,, the third horizontal sequence is \eqref{e:first_canonical}
applied to $\mathcal{U}_+$\,, whilst the morphisms, corresponding to vertical arrows, between these two sequences are induced by $\psi_+$\,.\\
\indent
Further, the second vertical sequence is the cohomology sequence of \eqref{e:exact_seq_for_q-like}\,. By using the fact that $\psi_+$ is injective
if and only if $\mathcal{U}_-$ is negative we, also, obtain the first horizontal sequence, and by passing to cokernels we may complete the diagram
with its fourth horizontal sequence.\\
\indent
Now, by using the Serre duality, and \eqref{e:E+=E-} we deduce that the first vertical sequence of the diagram is just the dual of \eqref{e:second_canonical}
applied to $\mathcal{U}_-^*$\,. Thus, if we dualize the diagram, pass to the cohomology sequence of the horizontal sequences, and then dualize, again
(recall, also, Remark \ref{rem:ro_from_canonical}\,), we deduce the following facts:\\
\indent
\quad(a) $\psi_+\circ\psi_-=\rho_+\circ\rho_-^*$\,,\\
\indent
\quad(b) $\rho_-^*$ maps ${\rm ker}\,\psi_-$ isomorphically onto ${\rm ker}\rho_+$\,,\\
\indent
\quad(c) $\psi_-$ maps ${\rm ker}\rho_-^*$ isomorphically onto ${\rm ker}\,\psi_+$\,,\\
\indent
\quad(d) $\rho_+$ maps ${\rm coker}\rho_-^*$ isomorphically onto ${\rm coker}\,\psi_+$\,,\\
\indent
\quad(c) $\psi_+$ maps ${\rm coker}\,\psi_-$ isomorphically onto ${\rm coker}\rho_+$\,.\\
\indent
Finally, properties (b) and (c)\,, above, quickly complete the proof of statement (iii)\,, and the proof of the theorem is complete.
\end{proof}

\section{Quaternionic-like manifolds and their heaven spaces}

\indent
We are interested in homogeneous complex manifolds $Z$ endowed with a (holomorphically) embedded (Riemann) sphere $t\subseteq Z$
with nonnegative normal bundle. For the corresponding differential geometry to be \emph{real} we have to assume that $Z$ is endowed with an involutive
antiholomorphic involution $\tau$ without fixed points, such that $\tau(t)=t$\,, and the complex Lie group $G$ acting holomorphically
and tranzitively on $Z$ is the complexification of the (real) Lie subgroup of $G$ formed of those elements whose induced
holomorphic diffeomorphisms on $Z$ commute with $\tau$.\\
\indent
Along the way, we shall see that such pairs $(Z,t)$ are abundant. In this section, we answer to the following question:
\emph{what geometric structure is induced on the (local) orbits of the $\rho$-quaternionic manifold
whose twistor space is $Z$ and for which $t$ is a twistor sphere}? The importance of this problem
is strengthened by the proof of the following simple proposition, and its consequences.

\begin{prop} \label{prop:homog_nonnegative}
If $Z$ is a homogeneous complex manifold then any sphere embedded in $Z$ has nonnegative normal bundle.
\end{prop}
\begin{proof}
Let $t\subseteq Z$ be an embedded sphere, and let $G$ be a complex Lie group acting holomorphically on $Z$.
Denote by $G_t$ the closed subgroup of $G$ formed of the elements preserving $t$\,. Thus,
each point $aG_t\in G/G_t$ corresponds to the sphere $at\subseteq Z$.\\
\indent
 The map $G\times t\to Z\times G/G_t$\,, $(a,z)\mapsto(az,at)$\,, induces an embedding of $G\times_{G_t}t$ into $Z\times G/G_t$
 whose image is formed of those pairs $(w,s)\in Z\times G/G_t$ such that $w\in s$\,.\\
\indent
Suppose, now, that $G$ acts tranzitively on $Z$. Then $G\times_{G_t}t\to Z$\,, $[a,z]\mapsto az$\,, is a surjective holomorphic submersion.
Consequently, the normal bundle of $t$ in $Z$ is a quotient of the trivial bundle over $t$ with typical fibre the holomorphic tangent space
to $G/G_t$ at its `origin'. Hence, the normal bundle of $t$ in $Z$ is nonnegative, and the proof is complete.
\end{proof}

\begin{rem}
1) It is easy to see that any (generalized) flag manifold admits an embedded sphere. Thus, Proposition \ref{prop:homog_nonnegative} and \cite{Pan-qgfs}\,,
implies that any flag manifold is the twistor space of a (nonunique) $\rho$-quaternionic manifold.\\
\indent
2) With the same notations as in the proof of Proposition \ref{prop:homog_nonnegative}\,, and assuming that $G$ acts tranzitively on $Z$,
we have that $G$ acts tranzitively on $G\times_{G_t}t\subseteq Z\times G/G_t$ if and only if $G_t$ acts tranzitively on $t$\,.
If this holds, then $G\times_{G_t}t=G/(K\cap G_t)$\,, where $K\subseteq G$ is the isotropy group
of the action of $G$ on $Z$ at some point of $t$\,.
\end{rem}

\indent
Now, to answer to the above mentioned question we have to introduce the following notion.

\begin{defn} \label{defn:almost_q-like_str}
Let $M$ be a manifold and let $k\in\mathbb{N}$\,, $k\leq\dim M$. An \emph{almost quaternionic-like structure} on a $M$ is a pair $(Y,t)$\,,
where $Y$ is an oriented sphere bundle over $M$, and $t:Y\to{\rm Gr}_k\bigl(T^{\mathbb{C}\,\!}M\bigr)$ is a bundle morphism such that
$t_x$ is a linear quaternionic-like structure on $T_xM$, for any $x\in M$.\\
\indent
An \emph{almost quaternionic-like manifold} is a manifold endowed with an almost qua\-ter\-ni\-o\-nic-like structure.
\end{defn}

\indent
Let $(M,Y,t)$ be an almost quaternionic-like manifold. We denote by $\mathcal{T}^{\pm}M$ the complex vector bundles obtained as follows
(cf.\ Proposition \ref{prop:basic_q-like_corresp}\,). Firstly, $\mathcal{T}^-M$ is the pull-back by $t$ of the tautological bundle over
${\rm Gr}_k\bigl(T^{\mathbb{C}\,\!}M\bigr)$ and then $\mathcal{T}^+M$
is the cokernel of the inclusion $\iota:\mathcal{T}^-M\to\pi^*\bigl(T^{\mathbb{C}\,\!}M\bigr)$\,, where $\pi:Y\to M$ is the projection.\\
\indent
From \cite[Theorem 2.3]{KodSpe-I_II}\,, \cite[Theorem 9]{KodSpe-III}\,, and Theorem \ref{thm:basic_q-like_corresp}\,, we deduce
the existence of vector bundles $T^{\pm}M$ and $E\,(=E_{\pm})$\,, and morphisms $\psi_{\pm}$ and $\rho_{\pm}$
which determine the given almost quaternionic-like structure.\\
\indent
Let $c:\mathcal{T}^-M\to T^{\mathbb{C}\,\!}Y$ be a morphism of complex vector bundles over $Y$ which intertwines the conjugations,
and such that $\widetilde{\dif\!\pi}\circ c=\iota$\,,
where  $\widetilde{\dif\!\pi}:T^{\mathbb{C}\,\!}Y\to\pi^*\bigl(T^{\mathbb{C}\,\!}M\bigr)$ is the morphism of complex vector bundles induced by $\dif\!\pi$\,.
We shall denote $\mathcal{C}=({\rm ker}\dif\!\pi)^{0,1}\oplus({\rm im}\,c)$\,.

\begin{defn} \label{defn:q-like_str}
The almost quaternionic-like structure $(Y,t)$ is \emph{integrable, with respect to $c$\,,} if the space of smooth sections of $\mathcal{C}$
is closed under the usual bracket. Then $(Y,t,c)$ is a \emph{quaternionic-like structure} and $(M,Y,t,c)$ is a \emph{quaternionic-like manifold}.\\
\indent
Suppose, further, that there exists a surjective submersion $\varphi:Y\to Z$, where $Z$ is endowed with a CR structure $\mathcal{C}_Z$ such that:\\
\indent
\quad(1) $\dif\!\varphi(\mathcal{C})=\mathcal{C}_Z$\,,\\
\indent
\quad(2) $({\rm ker}\dif\!\varphi)^{\mathbb{C}\,\!}=\mathcal{C}\cap\overline{\mathcal{C}}$\,,\\
\indent
\quad(3) $\varphi$ restricted to each fibre of $\pi$ is injective,\\
\indent
\quad(4) the antipodal map on $Y$ descends to an (anti-CR) involution $\tau$ on $Z$.\\
\indent
Then the CR manifold $(Z,\mathcal{C}_Z)$ is the \emph{twistor space} of $(M,Y,t,c)$\,.
\end{defn}

\indent
Note that, in Definition \ref{defn:q-like_str}\,, as $\mathcal{C}$ is integrable, a sufficient condition for $Z$\,, with the required properties, to exist, is
that $\mathcal{C}\cap\overline{\mathcal{C}}$ be the complexification of the tangent bundle of a foliation given by a submersion $\varphi$ with connected fibres.\\
\indent
Also, if, in Definition \ref{defn:q-like_str}\,, $(Y,t)$ is (given by) an almost CR quaternionic structure then, by applying \cite[Theorem 2.1]{Pan-twistor_(co-)cr_q}\,,
we retrieve the notion of CR quaternionic manifold of \cite{fq}\,. Moreover, we shall, next, see that, assuming real analyticity, the notion of `heaven space'
appears naturally in the current, more general, setting. In particular, if $(Y,t)$ is an almost $\rho$-quaternionic structure
then, up to a complexification, we retrieve the $\rho$-quaternionic manifolds of \cite{Pan-qgfs}\,.\\
\indent
In the next result, by a `Hausdorff manifold' we mean a manifold whose underlying topological space is Hausdorff (not necessarily paracompact).

\begin{thm} \label{thm:heaven_space_for_q-like}
Let $(M,Y,t,c)$ be a real analytic quaternionic-like manifold with twistor space $(Z,\mathcal{C}_Z)$\,, given by the surjective submersion $\varphi$\,.
Then there exists a (germ) unique Hausdorff quaternionic-like manifold $\bigl(M^+,Y^+,t^+,c^+\bigr)$\,, with twistor space
$\bigl(Z^+,\mathcal{C}^+\bigr)$\,, given by $\varphi^+$, and a unique quaternionic-like map $(\psi,\Psi)$ from $M$ to $M^+$
such that the following assertions are satisfied:\\
\indent
{\rm (i)} $\bigl(Y^+,t^+\bigr)$ is an almost $\rho$-quaternionic structure (equivalently, $TM^+=T^+M^+$),\\
\indent
{\rm (ii)} $\psi^*\bigl(TM^+\bigr)=T^+M$ and the morphism from $TM$ to $T^+M$ induced by $\dif\!\psi$ is $\psi_+$\,,\\
\indent
{\rm (iii)} $\Psi$ induces a CR embedding $\psi_Z:Z\to Z^+$ such that $\psi_Z\circ\varphi=\varphi^+\circ\Psi$.\\
\noindent
(The `germ unicity', here, means that, up to diffeomorphisms, the germ of $M^+$ (as a quaternionic-like manifold) along the image of $\psi$ is uniquely determined.)
\end{thm}
\begin{proof}
By \cite{AndFre}\,, we may assume that $Z$ is a CR submanifold of a complex manifold $\widetilde{Z}$ such that the inclusion
$T^{\mathbb{C}\,\!}Z\subseteq T^{\mathbb{C}\,\!}\widetilde{Z}$ induces an isomorphism between the following two complex vector bundles:\\
\indent
\quad(1) the restriction to $Z$ of the holomorphic tangent bundle of $\widetilde{Z}$, and\\
\indent
\quad(2) the quotient of $T^{\mathbb{C}\,\!}Z$ through $\mathcal{C}_Z$\,.\\
(In fact, we may, essentially, assume that $Z$ admits a complexification such that $\mathcal{C}_Z$ is the restriction to $Z$ of a holomorphic foliation whose leaf space is
$\widetilde{Z}$.)\\
\indent
Moreover, we may assume that $\tau$ is the restriction to $Z$ of an antiholomorphic involution $\widetilde{\tau}$ on $\widetilde{Z}$, without fixed points.\\
\indent
Let $\widetilde{Y}$ be the complexification of $Y$ as a sphere bundle over $M$. Then $\widetilde{Y}$ is a sphere bundle, whose projection we denote by
$\widetilde{\pi}$\,, over a complexification $\widetilde{M}$ of $M$\,; in particular, $\widetilde{\pi}|_Y=\pi$\,.
(Note that, $\widetilde{Y}$ is not a complexification of $Y$, as a real analytic manifold, but the `complexification' of $Y$ endowed with the CR structure
given by the antiholomorphic tangent bundles of its fibres.) We may assume that the antipodal map extends to an involution
without fixed points on $\widetilde{Y}$ which covers, through $\widetilde{\pi}$\,, the conjugation of $\widetilde{M}$ (whose fixed point set is $M$).\\
\indent
By passing, if necessary, to open neighbourhoods of $\widetilde{Z}$ and $\widetilde{M}$ we may assume that $\varphi$ extends
to a holomorphic surjective submersion $\widetilde{\varphi}:\widetilde{Y}\to\widetilde{Z}$ which intertwines the conjugations.
We claim that we may, further, assume that the restriction of $\widetilde{\varphi}$ to each fibre of $\widetilde{\pi}$ is an embedding.
For this, we need the deformation theory of compact complex submanifolds (see \cite{Nam-83}\,, \cite{Ro-LeB_nonrealiz}\,; cf.\ \cite{Kod}\,).
Let $S$ be the moduli space of compact complex one-dimensional submanifolds of $\widetilde{Z}$. This is a Hausdorff reduced complex analytic space
endowed with a conjugation induced by $\widetilde{\tau}$\,. Moreover, as $M$ is a parameter space for a real-analytic family of embedded spheres
in $\widetilde{Z}$ there exists a unique map $\psi:M\to S$ corresponding to $\varphi$ and the embedding $Z\subseteq\widetilde{Z}$\,.
Now, the isomorphism between (1) and (2)\,, above, implies that, for any $x\in M$, the normal bundle of the sphere $\varphi\bigl(\pi^{-1}(x)\bigr)$ in $\widetilde{Z}$
is isomorphic to $\mathcal{T}^+_xM$ which, because is nonnegative, has the property that $H^1\bigl(\mathcal{T}^+_xM\bigr)=0$\,.
Consequently, the image of $\psi$ is formed of smooth points of $S$\,. Moreover, we may assume that $\psi$ admits a holomorphic extension
$\widetilde{\psi}$ over $\widetilde{M}$ corresponding to a holomorphic extension of the CR map $\varphi:Y\to Z$, and whose image is formed of smooth points.
By unicity this should correspond to $\widetilde{\varphi}$\,, after passing, if necessary, to an open neighbourhood of $M$ in $\widetilde{M}$.
As the image of $\widetilde{\psi}$ is formed of smooth points (corresponding to spheres embedded into $\widetilde{Z}$\,) this proves our claim.\\
\indent
Finally, by applying \cite{Pan-qgfs}\,, an open neighbourhood of the image of $\psi$ is the complexification of a Hausdorff $\rho$-quaternionic manifold
$M^+$, as required, thus, completing the proof.
\end{proof}

\indent
We call the $\rho$-quaternionic manifold $M^+$ of Theorem \ref{thm:heaven_space_for_q-like} the \emph{heaven space} of $M$.

\begin{exm}
Any quaternionic-like vector space is a real analytic quaternionic-like manifold, in an obvious way, whose heaven space is
given by the $\rho$-quaternionic vector space $U^+$.
\end{exm}

\begin{exm}
As already mentioned, any CR quaternionic manifold $M$ is quaternionic-like. Furthermore, assuming real analyticity, the heaven space of $M$,
as a CR quaternionic manifold, is equal to the heaven of $M$, as a quaternionic-like manifold.
\end{exm}

\indent
It is straightforward to extend Theorem \ref{thm:heaven_space_for_q-like} to obtain a result similar to \cite[Theorem 5.3]{fq}\,.

\begin{exm}
With the same notations as in Proposition \ref{prop:homog_nonnegative}\,, the homogeneous complex manifold $G/G_t$
is a complex-quaternionic-like manifold. Consequently, if $M$ is a $\rho$-quaternionic manifold whose twistor space is homogeneous
then each orbit, of the induced local action on $M$, is endowed with a real-analytic quaternionic-like structure whose heaven space is $M$.
\end{exm}

\section{A construction of homogeneous twistor spaces} \label{section:constr_hqo}

\indent
We shall give and characterise a construction of homogeneous complex manifolds endowed with embedded Riemann spheres
(with nonnegative normal bundles), and we shall describe several classes of $\rho$-quaternionic manifolds whose twistor spaces
are obtained by this construction.\\
\indent
The ingredients of the construction are:\\
\indent
\quad(1) a complex Lie algebra $\mathfrak{g}$\,,\\
\indent
\quad(2) a (complex) representation $\sigma:\mathfrak{g}\to\mathfrak{gl}(E)$\,,\\
\indent
\quad(3) an injective morphism of complex Lie algebras $\tau:\mathfrak{sl}(2,\mathbb{C}\,)\to\mathfrak{g}$\,,\\
\indent
\quad(4) a complex vector subspace $U\subseteq E$ such that $A\mapsto (\sigma\circ\tau)(A)|_U$
is an irreducible nontrivial representation of $\mathfrak{sl}(2,\mathbb{C}\,)$ on $U$.

\begin{rem}
1) Condition (3)\,, above, implies that $\mathfrak{g}$ is not solvable (see \cite[\S\,6, n$^o$\,8, cor.\,1]{Bou-Lie_I}\,). Conversely, for any
Lie algebra $\mathfrak{g}$ which is not solvable we can find $\sigma$, $\tau$, $U$ as in (1)-(4)\,, above. Indeed, if $\mathfrak{g}$\,, $\sigma$\,, $\tau$
are as in (1)-(3)\,, then there exists $U$ such that (4) is satisfied if and only if $\sigma\circ\tau$ is nontrivial.\\
\indent
2) To obtain \emph{real} $\rho$-quaternionic manifolds, in (1)-(3)\,, we use complexifications (with $\mathfrak{su}(2)$\,, in (3)\,), and,
in (4)\,, we assume $U$ invariant under the conjugation of $E$.
\end{rem}

\indent
To simplify the exposition, we shall call a quadruple $(\mathfrak{g},\sigma,\tau,U)$ as in (1)-(4)\,, above, \emph{good}.
With these quadruples we can form a category whose morphisms are pairs
$(\varphi,\psi):(\mathfrak{g},\sigma_{\mathfrak{g}},\tau_{\mathfrak{g}},U_{\mathfrak{g}})\to(\mathfrak{h},\sigma_{\mathfrak{h}},\tau_{\mathfrak{h}},U_{\mathfrak{h}})$\,,
where $\varphi:\mathfrak{g}\to\mathfrak{h}$ is a morphism of Lie algebras such that $\varphi\circ\tau_{\mathfrak{g}}=\tau_{\mathfrak{h}}$,
and, if $E_{\mathfrak{g}}$ and $E_{\mathfrak{h}}$ are the representation spaces of $\sigma_{\mathfrak{g}}$ and $\sigma_{\mathfrak{h}}$, respectively, then
$\psi:E_{\mathfrak{g}}\to E_{\mathfrak{h}}$ is a $\varphi$-equivariant complex linear map
such that $\psi(U_{\mathfrak{g}})=U_{\mathfrak{h}}$ (in particular, $U_{\mathfrak{g}}$ and $U_{\mathfrak{h}}$
are isomorphic as representation spaces of $\mathfrak{sl}(2,\mathbb{C}\,)$\,).

\begin{prop}
Let $(\mathfrak{g},\sigma,\tau,U)$ be a good quadruple, and let $t\subseteq PU$ be the Veronese curve  formed of all\/ ${\rm ker}\bigl((\sigma\circ\tau)(A)|_U\bigr)$\,,
with $A\in\mathfrak{sl}(2,\mathbb{C}\,)$ nilpotent.\\
\indent
Let $Z$ be the orbit containing\/ $t$\,, of the induced action of $G$ on $PE$, where $G$ is the simply-connected complex Lie group
whose Lie algebra is $\mathfrak{g}$\,.\\
\indent
Then the normal bundle of\/ $t$ in $Z$ is nonnegative, and $(\mathfrak{g},\sigma,\tau,U)\mapsto(Z,t)$ is functorial.
\end{prop}
\begin{proof}
This follows quickly from Proposition \ref{prop:homog_nonnegative}\,.
\end{proof}

\indent
To state the next result, we recall two standard notions of algebraic geometry (see \cite{GriHar}\,).
Let $N$ be a complex manifold (not necessarily compact) and let $\mathcal{L}$ be a holomorphic line bundle over $N$. Also, let $E$ be a finite dimensional
complex vector subspace of the space of holomorphic sections of $\mathcal{L}$\,. We say that $\mathcal{L}$ is \emph{very ample, with respect to $E$},
if on associating to any $x\in N$ the subspace of $E$ formed of those sections that vanish at $x$ we obtain a well-defined injective immersion into $PE^*$.\\
\indent
On the other hand, let $N$ be a compact complex submanifold of a complex projective space $PE$, and let $\mathcal{L}$ be the dual of the tautological
over $PE$. Then $N$ is \emph{normal} if the restriction map from $E^*$ to the space of holomorphic sections of $\mathcal{L}|_N$ is surjective.

\begin{thm}
There exists a natural correspondence between {\rm (i)} good quadruples, and {\rm (ii)} triples $(Z,t,\mathcal{L})$\,, with $Z=G/K$ a homogeneous complex manifold,
$t\subseteq Z$ an embedded Riemann sphere, and $\mathcal{L}$ a holomorphic line bundle over $Z$, such that:\\
\indent
{\rm (a)} the morphism $G_t\to{\rm PGL}(2,\mathbb{C}\,)$\,, $a\mapsto a|_t$ is surjective, where $G_t\subseteq G$ is the subgroup preserving $t$\,,\\
\indent
{\rm (b)} the action of $G$ on $Z$ lifts to a holomorphic action of $G$ on $\mathcal{L}$ by automorphisms,\\
\indent
{\rm (c)} $\mathcal{L}$ is very ample, with respect to a $G$-invariant subspace $F$ of the space of holomorphic sections of $\mathcal{L}$\,,\\
\indent
{\rm (d)} the induced embedding of $t$ into $PF^*$ is normal.
\end{thm}
\begin{proof}
It is sufficient to show how to pass from (ii) to (i)\,. If $(Z,t,\mathcal{L})$ is a triple as in (ii)\,, with $Z=G/K$, let $\mathfrak{g}$ be the Lie algebra
of $G$, let $E=F^*$, and let $U$ be the dual of the space of holomorphic sections of $\mathcal{L}|_t$\,.
From (d)\,, we obtain that $U\subseteq E$, and, from (b)\,, (c)\,, we obtain a representation $\sigma$ of $\mathfrak{g}$ on $E$\,.\\
\indent
If $\mathfrak{g}_t$ be the Lie algebra of $G_t$ then, by (a)\,, we have a surjective morphism of complex Lie algebras
$\mu:\mathfrak{g}_t\to\mathfrak{sl}(2,\mathbb{C}\,)$\,. As $\mathfrak{sl}(2,\mathbb{C}\,)$
is simple, there exists a Lie algebras section $\tau$ of $\mu$ (see \cite[\S\,6, n$^o$\,8, cor.\,3]{Bou-Lie_I}\,).\\
\indent
Finally, (d) implies that the representation of $\mathfrak{sl}(2,\mathbb{C}\,)$\,, induced on $U$, is irreducible and of dimension $d+1$\,,
where $d$ is the degree of $t\subseteq PE$\,. Thus, $(\mathfrak{g},\sigma,\tau,U)$ is good, and the proof is complete.
\end{proof}

\begin{exm}[compare \cite{Pan-qgfs}\,]
The irreducible representations of $\mathfrak{sl}(2,\mathbb{C}\,)$ are given by injective morphisms of Lie algebras $\tau_k:\mathfrak{sl}(2,\mathbb{C}\,)\to\mathfrak{sl}(U_k)$\,,
where $\dim_{\mathbb{C}\,\!}U_k=k+1$\,, $(k\in\mathbb{N})$\,; in particular, $\mathfrak{sl}(2,\mathbb{C}\,)=\mathfrak{sl}(U_1)$\,.\\
\indent
If $k\geq1$ then $\bigl(\mathfrak{sl}(U_k),{\rm Id}_{\mathfrak{sl}(U_k)},\tau_k,U_k\bigr)$ is good, and the resulting complex homogeneous manifold is $Z=PU_k$\,,
the twistor space of the space $M_k$ of Veronese curves of degree $k$\,. Note that, $M_k$ is the quotient of ${\rm PGL}(U_k)$ through ${\rm PGL}(U_1)$
and therefore is homogeneous. Thus, up to the integrability, the geometric structure of $M_k$ is determined by the type of the normal bundle of
$t\subseteq PU_k$ as a holomorphic vector bundle endowed with an action of ${\rm PGL}(U_1)$\,.
It is fairly well-known that this normal bundle is $U_{k-2}\otimes\mathcal{O}(k+2)$\,,
where $U_{-1}=\{0\}$ (if $k$ is odd, ${\rm PGL}(U_1)$ acts on $U_{k-2}\otimes\mathcal{O}(k+2)$\,, although it does not act on its factors).
Consequently, the geometry of $M_k$ is underlied by the canonical representation of ${\rm PGL}(U_1)$ on $U_{k-2}\otimes U_{k+2}$\,.
\end{exm}

\begin{exm}[compare \cite{fq_2}\,]
Let $E$ be a (real) Euclidean space and let $U\subseteq E$ be a three-dimensional vector subspace. By using the Euclidean structure, we obtain
an injective morphism of Lie algebras $\tau:\mathfrak{so}(U)\to\mathfrak{so}(E)$\,.\\
\indent
Then (the complexification of) $\bigl(\mathfrak{so}(E),{\rm Id}_{{\mathfrak{so}(E)}},\tau,U\bigr)$ is good, and the resulting complex homogeneous space
is the (isotropic) hyperquadric in $PE^{\mathbb{C}\,}$. This is the twistor space of the Grassmannian of oriented three-dimensional subspaces of $E$.
\end{exm}

\begin{exm}[compare \cite{fq_2}\,]
Let $(V,\omega)$ be a complex symplectic vector space, $\dim_{\mathbb{C}\,}V\geq4$\,, and let $p_1$ and $p_2$ be two-dimensional complex vector subspaces
of $V$ such that $V_1=p_1+p_2$ is nondegenerated (and four-dimensional), with respect to $\omega$\,, and $\omega|_{p_1}=0$\,, $\omega|_{p_2}=0$\,. Alternatively,
$p_1$ and $p_2$ are characterised by the facts that they are of dimension $2$\,, $\omega|_{p_1}=0$ and $\omega$ induces an isomorphism between $p_2$ and $p_1^*$.\\
\indent
Let $U_1=({\rm ker}\,\omega)\cap\Lambda^2V_1$\,, and, note that, $\Lambda^2p_1$ and $\Lambda^2p_2$ are subspaces of $U_1$\,.
Moreover, $\Lambda^2p_1+\Lambda^2p_2$ is a two-dimensional subspace of $U_1$ which is nondegenerated,
with respect to the (complex-)conformal structure, given by the wedge product.
Therefore the orthogonal complement $U\subseteq U_1$ of $\Lambda^2p_1+\Lambda^2p_2$\,,
with respect to this conformal structure, is nondegenerated and three-dimensional. Consequently, we have
$\mathfrak{sl}(2,\mathbb{C}\,)=\mathfrak{so}(U)\subseteq\mathfrak{so}(U_1)=\mathfrak{sp}(V_1)\subseteq\mathfrak{sp}(V)$\,;
denote by $\tau:\mathfrak{sl}(2,\mathbb{C}\,)\to\mathfrak{sp}(V)$ the obtained injective morphism of Lie algebras.\\
\indent
On denoting by $\sigma$ the respresentation of $\mathfrak{sp}(V)$ on $\Lambda^2V$ we obtain the good quadruple
$\bigl(\mathfrak{sp}(V),\sigma,\tau,U\bigr)$\,. The obtained homogeneous complex manifold is the complex hypersurface
$Z\subseteq{\rm Gr}_2(V)$ formed of the two-dimensional complex subspaces which are isotropic with respect to $\omega$\,.\\
\indent
The corresponding $\rho$-quaternionic manifold is the submanifold $M\subseteq Z\times Z$ formed of those $(q_1,q_2)\in Z\times Z$
such that $q_1+q_2$ is nondegenerated, with respect to $\omega$\,. Consequently, $M$ is the quotient of ${\rm Sp}(V)$ through
$H={\rm GL}(p_1)\times{\rm Sp}(V_2)$\,, where $V_2=(p_1\oplus p_2)^{\perp_{\omega}}$\,, and to embed $H$ into ${\rm Sp}(V)$
we have used the complex linear isomorphism $V=(p_1\oplus p_1^*)\oplus V_2$. Then $M={\rm Sp}(V)/H$ is a reductive homogeneous space,
and the action of $H$ on $\mathfrak{sp}(V)/\mathfrak{h}$\,, where $\mathfrak{h}$ is the Lie algebra of $H$,
is induced through the complex linear isomorphism
$\mathfrak{sp}(V)/\mathfrak{h}=\bigl(\odot^2p_1\bigr)\oplus\bigl(\odot^2p_1^*\bigr)\oplus\bigl((p_1\oplus p_1^*)\otimes V_2\bigr)$\,,
where $\odot$ denotes the symmetric product. It follows that the normal bundle of each twistor sphere is $2\mathcal{O}(2)\oplus2(\dim_{\mathbb{C}\,\!}V-4)\mathcal{O}(1)$\,.\\
\indent
To make this construction \emph{real}, think of $V$ as a real vector space and let $I$ be the given linear complex structure on it.
Then endow $V$ with another linear complex structure $J$ such that $JI=-IJ$ and $J^*\omega=\overline{\omega}$\,.
Now, the map $M\to M$, $(q_1,q_2)\mapsto(Jq_2,Jq_1)$\,, is a conjugation (that is, an involutive antiholomorphic diffeomorphism)
whose fixed point set is the image of the real analytic map $Z\to M$\,, $p\mapsto(p,Jp)$\,; thus, $M=Z^{\mathbb{C}\,\!}$.
This way, $Z$ is an $f$-quaternionic manifold and its twistor space is $Z$ itself.
\end{exm}

\indent
For the next example, recall \cite{Kos} that if $\mathfrak{g}$ is a complex semisimple Lie algebra then there
is a natural correspondence between nilpotent adjoint orbits and conjugation classes of Lie subalgebras of $\mathfrak{g}$
isomorphic to $\mathfrak{sl}(2,\mathbb{C}\,)$\,. The involved association is the following: if $Y\in\mathfrak{sl}(2,\mathbb{C}\,)\setminus\{0\}$
is nilpotent (equivalently, $Y^2=0$, in the canonical representation) and $\mathfrak{sl}(2,\mathbb{C}\,)$ is a Lie subalgebra of $\mathfrak{g}$
then $Y\in\mathfrak{g}$ is nilpotent, and, obviously, the adjoint orbit of $Y$ depends only of the conjugation class of
the embedding $\mathfrak{sl}(2,\mathbb{C}\,)\subseteq\mathfrak{g}$\,. Conversely, any nilpotent $Y\in\mathfrak{g}\setminus\{0\}$
is contained in a Lie subalgebra of $\mathfrak{g}$ isomorphic to $\mathfrak{sl}(2,\mathbb{C}\,)$ (this fact is, usually, called the
Jacobson--Morozov Theorem), and, moreover, any two such subalgebras are conjugated.

\begin{exm}[compare \cite{Swa-99}\,] \label{exm:nil_orbits}
Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and let ${\rm ad}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$
be its adjoint representation. Then, for any Lie algebra embedding $\tau:\mathfrak{sl}(2,\mathbb{C}\,)\to\mathfrak{g}$\,, the quadruple
$(\mathfrak{g},{\rm ad},\tau,{\rm im}\,\tau)$ is good.\\
\indent
Let $(Z,t)$ be the resulting homogeneous complex manifold, endowed with an embedded Riemann sphere.
To describe the normal bundle of $t$ in $Z$, for any $j\in\mathbb{N}$\,, let $a_j\in\mathbb{N}$ be the multiplicity with which
the irreducible component $U_j$ appears in the decomposition of the represention of $\mathfrak{sl}(2,\mathbb{C}\,)$ on $\mathfrak{g}$\,, induced by ${\rm ad}$ and $\tau$\,.
Then the normal bundle of $t$ in $Z$ is $\bigl(-2+\sum_{j\in\mathbb{N}}ja_j\bigr)\mathcal{O}(1)$ (note that, the previous sum contains only finitely
many nonzero terms).\\
\indent
It is known that any compact real form $\mathfrak{g}_{\mathbb{R}}$ of $\mathfrak{g}$ - which we assume such that
$\mathfrak{g}_{\mathbb{R}}\cap{\rm im}\,\tau=\mathfrak{su}(2)$ - determines a conjugation, with respect to which,
$Z$ is the twistor space of a quaternionic-K\"ahler manifold $M$, and $t$ is a real twistor sphere.\\
\indent
Then $M$ is a co-CR quaternionic submanifold of ${\rm Gr}_3^+\bigl(\mathfrak{g}_{\mathbb{R}}\bigr)$\,. Furthermore,
the conjugation class $N$ of the embedding $\mathfrak{su}(2)\subseteq\mathfrak{g}_{\mathbb{R}}$ is a CR quaternionic submanifold of $M$\,;
moreover, $M$ is the heaven space of $N$.
On the other hand, if there exists $j\geq3$ such that $a_j\neq0$\,, then $N$ is not a CR quaternionic submanifold of ${\rm Gr}_3^+\bigl(\mathfrak{g}_{\mathbb{R}}\bigr)$\,.\\
\indent
Finally, it is known that the following assertions are equivalent:\\
\indent
\quad(a) $M$ is compact,\\
\indent
\quad(b) $M=N$,\\
\indent
\quad(c) $M$ is a Wolf space (that is, $\mathfrak{g}$ is simple and $Z$ is the projectivisation of the adjoint orbit of a highest root vector,
with respect to a Cartan subalgebra of $\mathfrak{g}$\,).
\end{exm}

redpajama_set_name: RedPajamaArXiv