text:
Inhaltsverzeichnis / Verlag
Die weltweit erste und allgemeinverständliche Gesamtdarstellung der Biophotonenforschung.
Physiker entdecken
das neue Bild vom Leben:
Wir sind Lichtwesen
Das Licht in unseren Zellen
Marco Bischof's widely acclaimed book has already sold some 38'000 German-language copies (12th printing) since its publication in March 1995, and the success is continuing. It is the first comprehensive book on the world market for the general and scientific public on one of the hottest fields of frontier science which is about to lead to major conceptual breakthroughs and many useful applications in biophysics, biomedical science, biology, biotechnology, environmental science and food technology. Thousands of medical doctors, scientists, and interested laypersons in Germany, Switzerland and Austria who from the many newspaper and magazine articles and from several TV features in the last couple of years were aware of this development of potential breakthroughs in a number of scientific disciplines and wanted to obtain more precise and broadly accessible information, have been waiting for this book that will remain the definitive publication on the topic for many years to come. Russian and Chinese translations are in preparation. The book has been awarded the 1995 Book Price by the Scientific and Medical Network (U.K.) and the Swiss Award 1997 by the Swiss Parapsychological Foundation.
What are biophotons ?
Biophotons, or ultraweak photon emissions of biological systems, are weak electromagnetic waves in the optical range of the spectrum - in other words: light. All living cells of plants, animals and human beings emit biophotons which cannot be seen by the naked eye but can be measured by special equipment developed by German researchers.
This light emission is an expression of the functional state of the living organism and its measurement therefore can be used to assess this state. Cancer cells and healthy cells of the same type, for instance, can be discriminated by typical differences in biophoton emission. After an initial decade and a half of basic research on this discovery, biophysicists of various European and Asian countries are now exploring the many interesting applications which range across such diverse fields as cancer research, non-invasive early medical diagnosis, food and water quality testing, chemical and electromagnetic contamination testing, cell communication, and various applications in biotechnology.
According to the biophoton theory developed on the base of these discoveries the biophoton light is stored in the cells of the organism - more precisely, in the DNA molecules of their nuclei - and a dynamic web of light constantly released and absorbed by the DNA may connect cell organelles, cells, tissues, and organs within the body and serve as the organism's main communication network and as the principal regulating instance for all life processes. The processes of morphogenesis, growth, differentiation and regeneration are also explained by the structuring and regulating activity of the coherent biophoton field. The holographic biophoton field of the brain and the nervous system, and maybe even that of the whole organism, may also be basis of memory and other phenomena of consciousness, as postulated by neurophysiologist Karl Pribram an others. The consciousness-like coherence properties of the biophoton field are closely related to its base in the properties of the physical vacuum and indicate its possible role as an interface to the non-physical realms of mind, psyche and consciousness.
The discovery of biophoton emission also lends scientific support to alternative and unconventional methods of healing based on concepts of homeostasis (self-regulation of the organism), such as various somatic therapies, homeopathy and acupuncture. The "ch'i" energy flowing in our bodies' energy channels (meridians) which according to Traditional Chinese Medicine regulates our body functions, may be related to node lines of the organism's biophoton field. The "prana" of Indian Yoga physiology may be a similar regulating energy force that has a basis in weak, coherent electromagnetic biofields.
First discovered in 1923 by Russian medical scientist Professor Alexander G.Gurvich (who named them "mitogenetic rays") and in the 1930s widely researched in Europe and the USA, biophotons have been rediscovered and backed since the 1970s by ample experimental and theoretical evidence by European scientists. In 1974, German biophysicist Fritz-Albert Popp has proved their existence, their origin from the DNA and later their coherence (laser-like nature), and has developed biophoton theory to explain their possible biological role and the ways in which they may control biochemical processes, growth, differentiation etc. Popp's biophoton theory leads to many startling insights into the life processes and may well provide one of the major elements of a future theory of life and holistic medical practice based on such an approach. The importance of the discovery has been confirmed by eminent scientists such as Herbert Froehlich and Nobel laureate Ilya Prigogine. Since 1992, the International Institute of Biophysics, a network of 22 research laboratories in 14 countries, based in Germany, is coordinating research in this field which promises rapid development in the coming decade (see http://www.lifescientists.de).
Aims of the book
To date the few books about biophotons have been highly technical and being addressed only to a scientific public. Not even among these, there was any single book integrating all that is known today about this fascinating field of science which is likely to become soon a much discussed topic also in the English-speaking world. The author, who in 1994-95 has served as Managing Director of the International Institute of Biophysics at Neuss (Germany) and still is a member of the Board of Directors of this institute, has been involved in biophoton research since 1977 and so was predestined to write the first comprehensive account of the subject ever made. His aim was to reach a wider public among scientists, medical doctors and the scientifically aware. The book which embeds the more technical parts in a popular treatment of the historical antecedents of the concept of "energy bodies", "life energies" and biolectricity, and to the ages-old scientific controversy between vitalistic and mechanistic trends in biology and medicine, also appeals to a general readership interested in new developments in the biological and physical sciences and in medicine and in their connections with consciousness research and New Age ideas.
Marco Bischof was born in Switzerland in 1947 and is a leading scholar, science writer and consultant in frontier areas of science and holistic medicine in the German-speaking countries. He has published a number of scientific papers and has been a regular contributor to many popular magazines in Germany (e.g. Esotera, Sphinx, Bioenergetik, German OMNI, German Penthouse) and abroad. He has studied cultural and medical anthropology and comparative religions at Zurich University, has been a member of the research council of the Laboratory of Biophysical Balneology at Zurzach, Switzerland, from 1990 to 1992, and of the board of directors of the International Institute of Biophysics at Kaiserlautern and Neuss, Germany, since 1992. 1992-1994 he has served as Director of the Institute for Synthesis, Coordination and Documentation (ISCD) at Biberist, Switzerland. In 1992 he was visiting scholar at the Center for Frontier Sciences, Temple University, Philadelphia. From December 1994 to December 1995 he has served as Managing Director of the International Institute of Biophysics at Neuss, Germany. From December 1995 to 1998 he has been a scientific consultant to the "Patienteninformation für Naturheilkunde", a company offering documentation, research and consulting in complementary medicine and the frontier sciences. He is now living in Berlin as an independent scholar, science writer and consultant. At present he is preparing a review on unconventional technologies for the German government and has just published a book on the controversial topic of "subtle energies". He is a full member of the Scientific and Medical Network (U.K.) and the Society for Scientific Exploration (USA).
"Unsere Seele kann fliegen" (Our soul can fly), Verlag im Waldgut, Switzerland (1985);
"Wasser" (Documentation on Water Research), in "Dokumentation der besonderen Therapierichtungen und natürlichen Heilweisen in Europa (Documentation on the Special Therapies and Natural Ways of Healing in Europe), Vol.II, Verlag fuer Ganzheitsmedizin, Essen, Germany (1990);
"The History of Bioelectromagnetics", in Bioelectrodynamics and Biocommunication, M.W.Ho, F.A.Popp, and U.Warnke, eds., World Scientific, Singapore 1994;
"Vitalistic and Mechanistic Concepts in the History of Bioelectromagnetics", in Biophotonics - Non-Equilibrium and Coherent Systems in Biology, Biophysics and Biotechnology, eds. L.Beloussov and F.A.Popp, Bioinform Services, Moscow 1996;
"Some Remarks on the History of Biophysics and its Future", in Current Development of Biophysics, eds. C.L.Zhang, F.A.Popp and M.Bischof. (Proceedings of the 1st Hombroich Symposium of Biophysics, Oct.3-6, 1995, Neuss, Germany). Hangzhou University Press, Hangzhou, China, 1996;
"Holism and Field Theories in Biology", in Biophotons, ed. J.J.Chang, J.Fisch and F.A.Popp (Proceedings of the Second Hombroich Symposium on Biophysics, Neuss, Germany, June 1997), Kluwer Academic Publishers, Dordrecht, The Netherlands 1998;
"Energy from Space - Crazy Dream or Real Possibility ?". Diwan Magazine - The Online Magazine of the DATADIWAN, Issue No.1, February, 1998. Http://www.datadiwan.de;
"Skalarwellen und Quantenfelder als mögliche Grundlage biologischer Information" (Scalar waves and quantum fields as a possible base for biological information). Erfahrungsheilkunde, Vol.47, No.5 (May, 1998), pp.295-300;
"The history of biological holism and field theories in the 20th century". MISAHA Newsletter, No.22-23 (July-December, 1998), 7-11. Monterey Institute for the Study of Alternative Healing Arts, Carmel, CA, USA.
"Field concepts and the emergence of a holistic biophysics", in Beloussov, L.V., Popp, F.A., Voeikov, V.L., and Van Wijk, R., (eds.): Biophotonics and Coherent Structures. Moscow University Press, Moscow 2000, S.1-25.
"Biofield concepts and the emergence of a holistic biophysics". MISAHA Newsletter, No.28-29 (January-June 2000), S.3-4 [excerpt from the above].
"Introduction to Integrative Biophysics". In: Popp, Fritz-Albert / Beloussov, Lev V. (eds.): Intregrative Biophysics - Biophotonics. Kluwer Academic Publishers, Dordrecht 2003, pp.1-115.·
"Biophotons and Beyond - The Question of Ultraweak Light Emission from Organisms: 'Superfluous' Light or Electromagnetic Bio-Information ?" (with B. Rubik), sponsored by the Institute of Noetic Sciences, Sausalito, USA, in preparation, Gordon & Breach Publishers, New York-London.
"Selected and Annotated Bibliography on the Pleomorphism of Microbes", in Proceedings of the Symposium on Pleomorphism, Nov.4-7, 1993, Philadelphia, ed. B.Rubik, in preparation, Nova Publishers, New York.
What the critics have said about his work
"Sober, rich in material and well written, the book contains any amount of explosives for engrained notions. To me it is one of the most important books of the last several years" (Hans-Curt Flemming in Connection Magazine (Germany), on "Biophotons").
"The book is a must for any leading edge scientist or researcher or alternative specialist who wants a scientific overview, and seeks to understand the variety of theories or possible explanations for a part of reality which has just begun to be measured - the interface between the material world and virtual reality. (...) We consider the book a work of reference with regard to the New Paradigm of science and philosophy in general and specifically with regard to the exciting area of biophoton research. (...) The biophoton theory may be one of the links between spiritual and esoteric empiricism and the New Paradigm science. (...) We find the book enlightening and recommendable. It provides scientific foundation for the developing New Paradigm within the sciences. The book is a treasure of information and details, rich in illustrations, and the glossary contains definitions of concepts normally difficult to define. This book was awarded the SMN Book Price in 1995 - an obvious choice". (Ole and Suzette van Hauen-Drucker, in The Scientific and Medical Network Review (U.K.), No.62, December 1996, on "Biophotons").
"A fascinating and unique book which does not fit into any current scheme" (Christian Bachmann, in Tages-Anzeiger, Zürich, Switzerland, on "Biophotons")
"A spirited and highly interesting discourse" (Dennis Stillings in Artifex (Minneapolis) on "Unsere Seele kann fliegen").
"Marco Bischof disposes of a language that through its simplicity makes complex relationships accessible also to the layman" (Bruno Kesseli in Limmat-Zeitung (Switzerland) on "Unsere Seele kann fliegen").
"Marco Bischof belongs to a tradition mainly found in Anglo-Saxon writing whose most known representative is maybe Arthur Koestler. Like Aldous Huxley, and today Colin Wilson or Robert Anton Wilson, he soberly investigates border areas of rationality without abusing cheap catchwords or neboulous expectations. What characterizes these authors is the ability to present difficult scientific concepts in a simple way without misrepresenting them. By proceeding and thinking in an interdisciplinary way, they arrive at surprising and new insights. Marco Bischof's book is a good exemple of this" (Guido Stefani in Luzerner Neuste Nachrichten and Badener Tagblatt (Switzerland) on "Unsere Seele kann fliegen").
Biophotons


text:
The Ostrov Theatre () is a private theatre on the corner of Kamennoostrovsky Prospect in Saint Petersburg. It was founded in 1990.

References

External links

Theatres in Saint Petersburg
Kamennoostrovsky Prospekt

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text:
Бережінфо — газета. Виходить у м. Бережани від 2006 українською мовою раз на тиждень.

Засновник і видавець – ПП В. Ціпліцький. Головний редактор В. Ціпліцький. Наклад — 3000 прим.

Джерела
 Мельничук Б. Бережінфо //

Посилання
 Про нас  // Береж інфо. — 2015. — 21 квітня.

Засновані в Україні 2006
Бережани


Not sure who reads my blog, nor why they do so...what I do know, is that I tract my visitors, the links that bring them here. One of these visits caught my eye today for several reasons. First, not sure how wise it is for a Navy Commander to have a whole lot of information on where he was stationed at on his Linked Profile...further, wondered how it is that one of the Google Searches that brought someone to my Washington Scandal Blog took me to Commander Dave Waterman's LINKED page. Have a look folks, and tell me how wise you think it is for a Naval Officer to be putting some much identifying information up onto the public web. Maybe this has something to do with those Lincoln Group articles I wrote so long ago, or perhaps more to do with Eric Prince stepping down over at Blackwater (Xe)...he was a former Seal.
Naval Flight Officerat VA-95 "Green Lizards"
The Multi-National Force - Iraq Media Operations Center (MOC) provides 24/7 operational situational awareness of the Western, Iraqi and Pan Arab media battlespace about Coalition and Iraqi Security Forces and its impact to Iraq and across the globe. The MOC consists of media monitors, visual information experts, press desk functions, outreach and press conference preparation for spokespersons. By fully understanding current news events and strategic messaging, the MOC team educates and provides public affairs advice and counsel to senior MNF-I leadership and subordinate commands.
Key spokesperson for all U.S. Navy in Japan, working with national and international media, and interacting with local communities and host nation government at all levels. Managed crisis communications varying from serious crimes, to environmental issues, to sensitive military issues. Specific accomplishments include writing the Strategic Communication plan for the arrival of USS George Washington, the first nuclear-powered aircraft carrier to be forward deployed to Japan. Utilizing effects-based communications, successfully bridged the cultural divide in informing the general Japanese public about the safety of the ship by using manga, the famous Japanese comic style, to produce "CVN73", which can be found at http://www.cnfj.navy.mil/Manga.html for free download.
Key player in the transformation of NATO Public Information (PI). Principle contributor to NATO PI policy redesign, including MC458, NATO's Education, Training, Exercise and Evaluation Policy. Established PI as a key training objective in NATO exercises. Introduced new injects which exercise new Rules of Engagement (ROE) for Commanders engaging media on the battlefield and embedded media with forward forces. Drafted the first-ever Bilateral Strategic Commanders (BiSC) Strategic Communcation Plan for NATO. Author of the Public Information Joint Functional Area Training Guide (PI JFATG), the primary PI training document in NATO. Guest speaker at the NATO School in Oberammergau, GE. Designed the PI training plan for the NATO Response Force (NRF) as well as many projects at NATO conferences around the world.
Primary mission was to develop and maintain relations with media networks in New York, coordinating NYC Fleet Weeks and representing the Navy at many diverse events, ranging from movie premiers to national awards ceremonies. Was the DoD and Navy primary spokesperson on Sept 11, 2001 and for days afterwards. Designed the Navy's City to the Sea program, reaching 65,000 inner city youth. Orchestrated the Navy Book Program, providing a focal point for authors to gain approval for Navy support of future books. Was liasion to many local organizations, including NYSE and NYY. Organized Navy guests for NYSE openings and closing as well as first pitches or coin tosses at area games. Organized the first-ever jet aircraft flyover of Yankee Stadium, successfully coordinating the closure of the three area airports with the FAA, City of New York and involved Navy squadrons.
As Battle Group Public Affairs Officer, authored the first CVBG Collateral Duty Public Affairs Officer handbook, and coordinated numerous PA opportunities for events such as RIMPAC, CAPSTONE, JCOC, WESTPAC, and Sea Fair. Was key USN PA in Operation Desert Fox. Hosted numerous media on at-sea embarks with great success. Hosted 60,000+ guests, ranging from the Secertary of the Navy, to international royalty to school children.
District Public Affairs office for NDW, dealing with topics such as Environmental Law, BRAC moves, expansion, and regionalization within the Nation's Capital. Was key organizer in Navy's role in Joint Service Open House for two years in a row. Always an innovator, generated great PA opportunities for Special Warfare community, bringing MK V boats up the Potomac for both Washington DC media and government leadership to ride and exerience. Completely designed the award-winning NDW website; pioneered the initiative to make websites a one-stop shopping for incoming personnel, as well as having command websites written into personnel orders as an information source. Smithsonian Institute awarded the NDW website for innovation in government computing and a CD copy was put in the Smithsonian Institute's timecapsule for the year.
U.S. Government Relations, Lobbying / Lobbyists; Global Public Relations, Communications; Industry / Association Relations; Government Affairs / Grassroots Advocacy in 50 States; Public Affairs; Crisis Communications; Polling / Public Opinion Research; Advocacy Advertising and lobbyist networking.
By registering to be a member of the A U.S. Government Relations & Public Affairs Group group, you consent to have your name and your email address be accessible by the official representative of the A U.S. Government Relations & Public Affairs Group group, and to be identified as a A U.S. Government Relations & Public Affairs Group group member in your profile and the LinkedIn search results. If you do not wish to give such consent, please do not register.
Join the Executive Suite to connect with over 100,000 US-based executive and recruiter members. Your email address will be used to send you our free, biweekly career trend newsletter, Executive Insider, and other relevant information. You can quickly and easily unsubscribe at any time.
By registering to be a member of the Executive Suite group, you consent to have your name and your email address be accessible by the official representative of the Executive Suite group, and to be identified as a Executive Suite group member in your profile and the LinkedIn search results. If you do not wish to give such consent, please do not register.
Innovate with innovative marketing, public relations, promotions sales & selling professionals skilled at using creative marketing methods. Join thousands of members to learn & share best practices & advice. LinkedIn's largest Marketing & PR Innovation Group. Organized by Gerald "Solutionman" Haman.
By registering to be a member of the Innovative Marketing, PR, Sales, Word-of-Mouth & Buzz Innovators group, you consent to have your name and your email address be accessible by the official representative of the Innovative Marketing, PR, Sales, Word-of-Mouth & Buzz Innovators group, and to be identified as a Innovative Marketing, PR, Sales, Word-of-Mouth & Buzz Innovators group member in your profile and the LinkedIn search results. If you do not wish to give such consent, please do not register.
This is a network for US military professionals engaged in New Media. This is the forum to ask questions, exchange lessons learned and share best business practices on leveraging New Media and Web 2.0 trends within Dept. of Defense commands and agencies.
By registering to be a member of the Defense 2.0 group, you consent to have your name and your email address be accessible by the official representative of the Defense 2.0 group, and to be identified as a Defense 2.0 group member in your profile and the LinkedIn search results. If you do not wish to give such consent, please do not register.
This is a group of professionals that are strategic communicators or have a genuine interest in strategic communication. The purpose of this group is to provide a forum for sharing best practices and collaboration.
By registering to be a member of the Strategic Communication group, you consent to have your name and your email address be accessible by the official representative of the Strategic Communication group, and to be identified as a Strategic Communication group member in your profile and the LinkedIn search results. If you do not wish to give such consent, please do not register.
Stars and Stripes , the daily newspaper of the United States Armed Forces, reports that the public affairs office for the Commander, Naval Forces Japan (CNFJ) will be producing a manga to explain its mission to the Japanese public. According to its spokesperson, Commander David Waterman, the goal is reach Japanese people in their twenties and thirties. "We used comics as a means to introduce a summary of [a defense white paper] that is easily understandable for young people." The office has already produced trading cards for Japanese children.
The CNFJ is still developing the storyline, and it will contract out the artwork and production. As planned, the 40-page, all-Japanese booklet will feature a male and a female sailor on the U.S.S. George Washington aircraft carrier and their experiences both abroad in various countries and at Yokosuka Naval Base, the carrier's port in Japan.
Japan's own Ministry of Defense has created three manga booklets for its mascot Prince Pickles to explain its mission to its citizens, and the Ministry of Foreign Affairs has Detective Conan explaining its mission. The Maritime Self-Defense Force, Japan's nominally non-military fleet, has recruited new sailors with idol group and sentai imagery.
The magazine-sized books feature everything from classics to romances to crime novels. The Navy produced 25,000 copies of the GW manga at a cost of $3 apiece to be handed out free and they were snapped up in hours by Japanese who waited in long lines in Yokosuka to get copies.
The U.S. Navy said Wednesday it has imposed limits on travel and alcohol consumption for personnel at its base near Tokyo following the killing of a Japanese taxi driver.
The limits would take effect immediately, said Cmdr. David Waterman, a Navy spokesman.
No alcohol will be sold due to "a period of heightened sensitivity" over the stabbing of a 61-year-old Japanese taxi driver last month in Yokosuka, just south of Tokyo and home to the U.S. naval base, Waterman said. Off-base travel and activities will be limited, he said.
The announcement came hours after Japanese authorities questioned a U.S. sailor over his suspected role in the killing of taxi driver Masaaki Takahashi.
The sailor, a 22-year-old crew member of the Yokosuka-based ship USS Cowpens, has been in U.S. custody since he was apprehended on a desertion charge earlier this month in Tokyo.
[PDF] SUPREME COURT OF THE UNITED STATES .
... S UPREME C OURT D ECISIONS Page 2. United States v. David M. Hicks ... Id., at 110-111.
Seriously? Someone followed a link to your site, and you're all "there's a conspiracy here"? And since when is it a problem to list your former duty stations? It's not. Apparently you have no knowledge of the military. Dude, you need to get a life.


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scary things Nov. 16, 2014
Man Killed After Being Pushed Onto Subway Tracks [Updated]
Every subway rider's nightmare has once again come true: 61-year-old Wai Kuen Kwok was killed by an oncoming D train after being pushed onto the tracks on Sunday morning. While surveillance footage of the incident hasn't yet been released, the NYPD told reporters that Kwok "had not fallen on the tracks by accident or jumped into the path of the train" at the Bronx's East 167th Street station.
The train's operator saw him "fly" off the platform but was unable to stop in time to save him. "The guy was still in the air when he hit him," said a New York Daily News source, horrifyingly. The victim's wife, who was standing nearby when her husband died, was taken to a hospital unharmed. The couple had been on their way to Manhattan's Chinatown to have breakfast and do some grocery shopping.
It's not clear what led to the push, though one witness told the New York Times that he'd heard two men fighting just before it happened. The police are now searching for the attacker, who is believed to have fled via the Bx35 bus. The Daily News reports that he was wearing a dark-colored puffy coat, which isn't exactly a lot to go on. The New York Post reports that a camera caught the suspect getting off the bus and entering a bodega a few blocks away. The bodega's employees and the bus driver are being interviewed, so there should be a better description of the person who did this soon.
Man Pushed to His Death in Front of a Subway Train Yet Again [Updated]
The Worst Thing That Can Happen in the Subway Has Happened Again
Elderly Man Barely Survives Push Onto Subway Tracks [Updated]
Kwok's son told the Post that his father was an employee of a kitchen-accessories factory, where he worked six days a week. "He provided for his family. He wanted us to be successful. His plan was to work and get his sons through school, then he would return to Hong Kong and enjoy himself," he said.
This is the first time someone has died after being pushed onto the subway tracks since 2012, when it happened twice in one month. More recently, a 72-year-old man avoided the same fate thanks to his fellow commuters, who managed to pull him back onto the platform before the train arrived.
This post has been updated throughout.
NYT NBC New York
scary things
stand clear of the closing doors
Man Killed After Being Pushed Onto Subway Tracks

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Home > Administrative > How to Apply for a Nevada Business License For A For-Profit Corporation
How to Apply for a Nevada Business License For A For-Profit Corporation
A for-profit corporation is an organization which aims to earn profit through its operations and is concerned with its own interests, unlike those of the public (a non-profit corporation). Source: Wikipedia.
In this article, we will show you the steps in applying for a Nevada Business License for a for-profit corporation.
For steps in applying for a state business license for Sole Proprietorship, General Partnerships, and other Non-Title 7, please click the links below:
How to Apply for a Nevada Business License By Mail
How To Sign up for a SilverFlume Account to Manage Your Nevada Business Online
Fill out the Application Form of the State Business License For Profit Corporation
Name of Entity
Jurisdiction of Incorporation
Purpose or Profession of Practice
Authorized Shares
Officer Making Statement
Incorporators
Download the form from the www.nvsos.gov website.
Fill out the name of the business you are registering. Note that you cannot use a name that appears to be of a natural person or initials of the given name unless it will have corporate endings such as Limited, Ltd., Corporation, Corp., Incorporated, Inc., or other additions that would not make it appear as a natural person. The chosen name must be unique or is not in use by other organizations.
If applying as a foreign company, give the name registered in home jurisdiction.
Those who wish to incorporate in Nevada must elect a registered agent who lives in the state. They will need to indicate their street address for service of process. They must also have a mailing address different from their street address. Registered agents must sign the certificate of acceptance. If the registered agent cannot sign the Articles of Formation, they must submit a separate Registered Agent Acceptance form with the signature.
You must indicate if the corporation will be under a board of directors. If yes, you must supply the names and addresses of each.
You must indicate the jurisdiction of incorporation by which the corporation was created. You must also declare that the corporation is in good standing since its formation.
When you select, "Yes," this means that you are declaring the corporation as a benefit corporation pursuant to NRS Chapter 78B. The purpose for public benefit must be indicated within the formation filing.
You must fill this out if you are a benefit corporation. Otherwise, leave it blank.
It is important to state the number of shares the company has authority to sell. It must include par value. If there is no par value, indicate the number of shares in the space provided for shares without par value.
You must fill out the name of the authorized person(s) responsible for filing. Each officer must sign the form.
The form must list out all the names and addresses of all incorporators. Each one of them must sign the form as well. You may use a separate white sheet necessary.
Send the form by mail to:
BARBARA K. CEGAVSKE
202 North Carson Street
Carson City, Nevada 89701-4201
Please note that unsigned applications will be returned.
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text:
\section{Introduction

In this paper we study nondegenerate affine hypersurfaces $M^n$ into
$\mathbb R^{n+1}$, equipped with its standard affine connection $D$.
It is well known that on such a hypersurface there exists a
canonical transversal vector field $\xi$, which is called the affine
normal. With respect to this transversal vector field one can
decompose
\begin{equation}
D_X Y = \nabla_X Y +h(X,Y) \xi,
\end{equation}
thus introducing the affine metric $h$ and the induced affine
connection $\nabla$. The Pick-Berwald theorem states that $\nabla$
coincides with the Levi Civita connection $\widehat{\nabla}$ of the affine
metric $h$ if and only if $M$ is immersed as a nondegenerate
quadric. The difference tensor $K$ is introduced by
\begin{equation}
K_X Y = \nabla_X Y -\widehat{\nabla}_X Y.
\end{equation}
It follows easily that $h(K(X,Y),Z)$ is symmetric in $X$, $Y$ and
$Z$. The apolarity condition states that $\operatorname{trace} K_X =0$ for every
vector field $X$. The fundamental theorem of affine differential
geometry, Dillen, see Ref.~\cite{dinovr91} implies that an affine
hypersurface is completely determined by the metric and the
difference tensor $K$.

Deriving the affine normal, we introduce the affine shape operator $S$ by
\begin{equation}
D_X \xi =-SX.
\end{equation}

Here, we will restrict ourselves to the  case that the affine shape
operator $S$ is a multiple of the identity, i.e. $S=H I$. This means
that all affine normals are parallel or pass through a fixed point.
We will also assume that the metric is positive definite in which
case one distinguishes the following classes of affine hyperspheres:
\begin{romanlist}
\item elliptic affine hyperspheres, i.e. all affine normals pass through a fixed point and $H >0$,
\item hyperbolic affine hyperspheres, i.e. all affine normals pass through a fixed point and $H<0$,
\item parabolic affine hyperspheres, i.e. all the affine normals are parallel ($H=0$).
\end{romanlist}
Due to the work of amongst others Calabi \cite{ca72}, Pogorelov
\cite{po72}, Cheng and Yau \cite{chya86}, Sasaki \cite{sa80} and Li
\cite{li92}, positive definite affine hyperspheres which are
complete with respect to the affine metric $h$ are now well
understood. In particular, the only  complete elliptic or parabolic
positive definite affine hyperspheres are respectively the ellipsoid
and the paraboloid. However, there exist many hyperbolic affine
hyperspheres.

In the local case, one is far from obtaining a classification. The reason for this is that affine hyperspheres
reduce to the study of the Monge-Amp{\`e}re equations.
Calabi introduced a construction, called the Calabi product, which shows how to associate with one (or two)
hyperbolic affine hyperspheres a new hyperbolic affine hypersphere. This construction,
as well as the corresponding properties for the difference tensor are recalled in the next section.

In this paper we are interested in the reverse construction, i.e. how to determine using properties of the difference
tensor whether or not a given hyperbolic affine hypersphere (with mean curvature $-1$) can be decomposed
as a Calabi product of a hyperbolic affine hypersphere and a point or as a Calabi product of two hyperbolic
affine hyperspheres.

In particular we show the following two theorems:
\begin{theorem} Let $\phi: M^n \rightarrow \mathbb R^{n+1}$ be a (positive definite) hyperbolic affine hypersphere with mean
curvature $\lambda$, $\lambda<0$. Assume that there exists two
distributions $\mathcal D_1$ and $\mathcal D_2$ such that
\begin{romanlist}
\item $T_pM = \mathcal D_1 \oplus \mathcal D_2$,
\item $\mathcal D_1$ and $\mathcal D_2$ are orthogonal with respect to the affine metric $h$
\item $\mathcal D_1$ is a one dimensional distribution spanned by a unit length vector field $T$
\item there exist numbers $\lambda_1$ and $\lambda_2$ satisfying $-\lambda+\lambda_1 \lambda_2 -\lambda_2^2= 0$ such that
\begin{align*}
& K(T,T)=\lambda_1 T\\
&K(T,U) = \lambda_2 U,
\end{align*}
where $U \in \mathcal D_2$.
\end{romanlist}
Then $\phi:M^n \rightarrow \mathbb R^{n+1}$ can be decomposed as the Calabi product of a hyperbolic
affine sphere $\psi:M_1^{n-1} \rightarrow \mathbb R^{n}$ and a point.
\end{theorem}
and
\begin{theorem} \label{theoremprod2}Let $\phi: M^n \rightarrow \mathbb R^{n+1}$ be a (positive definite) hyperbolic affine hypersphere with mean
curvature $\lambda$, $\lambda<0$. Assume that there exists
distributions $\mathcal D_1$ (of dimension 1, spanned by a unit
length vector field $T$), $\mathcal D_2$
 (of dimension $n_2$) and $\mathcal D_3$ (of dimension $n_3$) such that
\begin{romanlist}
\item $1+n_2+n_3 = n$,
\item $\mathcal D_1$, $\mathcal D_2$  and $\mathcal D_3$ are mutually orthogonal
 with respect to the affine metric $h$
\item there exist numbers $\lambda_1$, $\lambda_2$ and $\lambda_3$ such that
\begin{align*}
& K(T,T)=\lambda_1 T\\
&K(T,V) = \lambda_2 V,\\
&K(T,W)= \lambda_3 W,\\
&K(V,W)=0.
\end{align*}
where $V \in \mathcal D_2$, $W \in \mathcal D_3$, $\lambda_1 =
\lambda_2 +\lambda_3$ and $\lambda_2 \lambda_3 = \lambda$.
\end{romanlist}
Then $\phi:M^n \rightarrow \mathbb R^{n+1}$ can be decomposed as the Calabi product of two hyperbolic
affine sphere immersions $\psi_1:M_1^{n_2} \rightarrow \mathbb R^{n_2+1}$ and
 $\psi_2:M_2^{n_3} \rightarrow \mathbb R^{n_3+1}$.
\end{theorem}
Note that, as explained in the next section, the converse of the above two theorems is also true.

To conclude this introduction, we remark that the basic integrability conditions for a hyperbolic
 affine hypersphere with mean curvature $-1$ state that:
\begin{align}
&\hat R(X,Y)Z = -(h(Y,Z) X-h(X,Z)Y)-[K_X,K_Y]Z,\label{gauss}\\
&(\hat\nabla K)(X, Y,Z) =(\hat \nabla K)(Y,X,Z).\label{codazzi}
\end{align}

\section{The Calabi product

Let $\psi_1: M_1^{n_2} \rightarrow R^{n_2+1}$ and $\psi_2: M_2^{n_3} \rightarrow R^{n_3+1}$
be hyperbolic affine hyperspheres with mean curvature $-1$. Then we define
the Calabi product of $M_1$ with a point by
\begin{equation*}
\tilde \psi(t,p)= (c_1 e^{\tfrac{t}{\sqrt{n}}} \psi_1(p), c_2
e^{-{\sqrt{n}}t}),
\end{equation*}
where $p \in M_1$ and $t \in \mathbb R$
and the Calabi product of $M_1$ with $M_2$
by
\begin{equation*}
\psi(t,p,q))= (c_1 e^{\tfrac{\sqrt{n_3+1} t}{\sqrt{n_2+1}}}\psi_1(p),
 c_2 e^{-\tfrac{\sqrt{n_2+1} t}{\sqrt{n_3+1}}} \psi_2(q)),
\end{equation*}
where $p \in M_1$, $q \in M_2$ and $t \in \mathbb R$.

We now investigate the conditions on the constants $c_1$ and $c_2$ in order that the Calabi product
has constant mean curvature $-1$. We first do so for the Calabi product of two affine spheres.
We denote by $v_1,\dots,v_{n_2}$ local coordinates for $M_1$ and by $w_1,\dots,w_{n_3}$ local coordinates
for $M_2$. Then, it follows that
\begin{align*}
&\psi_t= (c_1 \tfrac{\sqrt{n_3+1}}{\sqrt{n_2+1}}e^{\tfrac{\sqrt{n_3+1} t}{\sqrt{n_2+1}}}\psi_1(p),
-c_2 \tfrac{\sqrt{n_2+1}}{\sqrt{n_3+1}}e^{-\tfrac{\sqrt{n_2+1} t}{\sqrt{n_3+1}}} \psi_2(q)),\\
&\psi_{tt}=(c_1 \tfrac{n_3+1}{n_2+1}e^{\tfrac{\sqrt{n_3+1} t}{\sqrt{n_2+1}}}\psi_1(p),
-c_2 \tfrac{n_2+1}{n_3+1}e^{-\tfrac{\sqrt{n_2+1} t}{\sqrt{n_3+1}}} \psi_2(q)),\\
&\psi_{tv_i}=\tfrac{\sqrt{n_3+1}}{\sqrt{n_2+1}}(c_1 e^{\tfrac{\sqrt{n_3+1} t}{\sqrt{n_2+1}}}(\psi_1)_{v_i},0),\\
&\psi_{tw_j}=-\tfrac{\sqrt{n_2+1}}{\sqrt{n_3+1}}(0,c_2 e^{-\tfrac{\sqrt{n_2+1} t}{\sqrt{n_3+1}}} (\psi_2)_{w_j}),\\
&\psi_{v_iv_j}=(c_1 e^{\tfrac{\sqrt{n_3+1} t}{\sqrt{n_2+1}}}(\psi_1)_{v_iv_j},0),\\
&\psi_{w_iw_j}=(0,c_2 e^{-\tfrac{\sqrt{n_2+1} t}{\sqrt{n_3+1}}} (\psi_2)_{w_iw_j}).
\end{align*}
If we denote by $h_2$ the affine metric on $M_2$ and by $h_3$ the
centroaffine metric introduced on $M_3$ it follows from the above
formulas that
\begin{align*}
&\psi_{tt} =\tfrac{n_3-n_2}{\sqrt{(n_2+1)(n_3+1)}} \psi_t + \psi\\
&\psi_{v_iv_j} = \tfrac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2} h_2(\partial v_i,\partial v_j) \psi+...\\
&\psi_{w_iw_j} = \tfrac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2}
h_3(\partial w_i,\partial w_j) \psi+...
\end{align*}
From \cite{nosa94} we see that $M$ is an affine hypersphere with
mean curvature $-1$ if and only if
\begin{equation*}
det[\psi,\psi_t,\psi_{v_1},\dots,\psi_{v_{n_2}},\psi_{w_1},\dots,\psi_{w_{n_3}}]^2
=h(\partial_t,\partial_t) det[h(\partial v_i,\partial v_j)]
det[h(\partial w_i,\partial w_j)].
\end{equation*}
Taking into account that $\psi_1$ and $\psi_2$ are already affine
spheres with mean curvature $-1$ we must have that
\begin{equation}
(c_1)^{n_2+1} (c_2)^{n_3+1} = \left
(\tfrac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2}\right )^{n_2+n_3+2}.
\end{equation}
Hence we can take
\begin{align*}
&c_1=\tfrac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2} d_1\\
&c_2=\tfrac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2} d_2,
\end{align*}
where
\begin{equation}
(d_1)^{n_2+1} (d_2)^{n_3+1} = 1.
\end{equation}
Hence by applying an equiaffine transformation we may assume that
$d_1=d_2=1$ and therefore that the Calabi product of two hyperbolic
affine spheres with mean curvature $-1$ is an hyperbolic affine
sphere with mean curvature $-1$ if and only if
\begin{equation*}
\psi(t,p,q))= \tfrac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2}(
e^{\tfrac{\sqrt{n_3+1} t}{\sqrt{n_2+1}}}\psi_1(p),
 e^{-\tfrac{\sqrt{n_2+1} t}{\sqrt{n_3+1}}} \psi_2(q)),
\end{equation*}
up to an equiaffine transformation.

For the Calabi product of a hyperbolic affine sphere and a point, we
proceed in the same way to deduce the following. The Calabi product
of a hyperbolic affine spheres with mean curvature $-1$ and a point
is an hyperbolic affine sphere with mean curvature $-1$ if and only
if
\begin{equation*}
\tilde \psi(t,p)= \tfrac{\sqrt{n}}{n+1}( e^{\tfrac{t}{\sqrt{n}}}
\psi_1(p), e^{-\sqrt{n} t}),
\end{equation*}
up to an equiaffine transformation.

\begin{remark} A straightforward calculation shows that the Calabi
product of two hyperbolic affine spheres has parallel cubic form
(with respect to the Levi Civita connection) if and only if both
original hyperbolic affine spheres have parallel cubic forms.
Similarly one has that the Calabi product of a hyperbolic affine
sphere and a point has parallel cubic form if and only if the
original affine sphere has parallel cubic form.
\end{remark}

\section{Characterisation of the Calabi product of two hyperbolic affine spheres and the proof of Theorem 2

Throughout this section we will assume that $\phi:
M^n\longrightarrow\mathbb R^{n+1} $ is a hyperbolic affine
hypersphere. Without loss of generality we may assume that
$\lambda=-1$ by applying a homothety. We will now prove Theorem
\ref{theoremprod2}. Therefore, we shall also assume that $M$ admits
three mutually orthogonal differential distributions $\mathcal D_1$,
$\mathcal D_2$ and $\mathcal D_3$ of dimension $1$, $n_2> 0$ and
$n_3> 0$ respectively with $1+n_2+n_3=n$, and, for all vectors $V\in
\mathcal D_2$, $W\in \mathcal D_3$,
\begin{gather*}
K(T,T) = \lambda_1 T,\;\;\;\;\;\;
K(T,V) = \lambda_2 V,\\
K(T,W) = \lambda_3 W,\;\;\;\;\;\; K(V,W) = 0.
\end{gather*}
By the apolarity condition we must have that
\begin{equation}
\lambda_1 +n_2 \lambda_2 +n_3 \lambda_3 = 0,
\end{equation}
Moreover, we will assume that
\begin{align}
&\lambda_1 = \lambda_2+\lambda_3\\
&\lambda_2 \lambda_3 = -1.
\end{align}

The above conditions imply that $\lambda_1$, $\lambda_2$ and $\lambda_3$ are constants and
can be determined explicitly in terms of the dimension $n$.

As $M$ is a hyperbolic affine sphere we have that the difference
tensor is a symmetric tensor with respect to the Levi Civita
connection $\hat \nabla$ of the affine metric. In that case, as also
$h(K(X,Y),Z)$ is totally symmetric, the information of Lemma 1 and
Lemma 2 of \cite{BRV} remains valid and can be summarized in the
following lemma:
\begin{lemma} \label{lemme1}
We have
\begin{enumerate}
\item $\hat \nabla_{\mathcal D_1} \mathcal D_1 \subset \mathcal D_1$
\item $\hat \nabla_{\mathcal D_2} \mathcal D_2 \subset \mathcal D_2
\oplus \mathcal D_3$
\item $\hat \nabla_{\mathcal D_3} \mathcal D_3
\subset \mathcal D_2 \oplus \mathcal D_3$
\item $h(\hat \nabla_T
W,V) = h(\hat \nabla_W T,V)=-h(\hat \nabla_V T,W)$, for any $V
\in\mathcal D_2, W\in\mathcal D_3$
\end{enumerate}
\end{lemma}

Similarly using the
information of the previous lemma, Lemma 3 of \cite{BRV} reduces to
\begin{lemma} \label{lemme2} We have
\begin{enumerate}
\item $(\lambda_3-\lambda_2) h(\hat \nabla_{V} \tilde V, W)
= h( K(V,\tilde V),  \hat\nabla_T W)
 =  h(K(V,\tilde V),  \hat \nabla_W T)$
\item  $(\lambda_2-\lambda_3) h(\hat\nabla_{W} \tilde W, V)
=  h(K(W,\tilde W),  \hat\nabla_T V)
 =  h(K(W,\tilde W), \hat \nabla_V T)$
\end{enumerate}
\end{lemma}

We denote now by $\{V_1,\dots,V_{n_2}\}$, respectively  $\{W_1,\dots,W_{n_3}\}$ an
orthonormal basis of $\mathcal D_2$ (resp. $\mathcal D_3$) with respect to the affine
metric $h$. Then, we have
\begin{lemma} \label{lemme3} Let $V, \tilde V \in \mathcal D_2$. Then
$$h(\hat \nabla_V T, \hat \nabla_{\tilde V} T) =0.$$
\end{lemma}
\begin{proof} Using the Gauss equation, we have that
\begin{align*}
h(\hat R(V,T)T,\tilde V)&= -h(V,\tilde V) - h(K(T,T),K(V,\tilde V))
 + h(K(T,V),K(T,\tilde V))\\
&= (-1- \lambda_1 \lambda_2 +\lambda_2^2) h( V,\tilde V)\\
&=(-1- \lambda_3 \lambda_2) h(V,\tilde V)=0.
\end{align*}
On the other hand, by a direct computation using the previous lemmas, we have
\begin{align*}
h(\hat R(V,T)T,\tilde V)&=h(\hat \nabla_{V} \hat \nabla_{T}T -\hat
\nabla_{T}\hat \nabla_{V} T
 -\hat \nabla_{\hat \nabla_{V}T -\hat \nabla_{T}V} T,\tilde V)\\
&=h(-\hat\nabla_{T} \hat\nabla_V T,\tilde V ) -\sum_{k=1}^{n_3} h(\hat \nabla_{V}T - \hat \nabla_T V,W_k) h(\hat
\nabla_{W_k} T, \tilde V)\\
&=h(-\hat\nabla_{T} \hat \nabla_V T,\tilde V) \qquad \text{by Lemma \ref{lemme1} (iv)}\\
&=-\sum_{k=1}^{n_3} h(\hat\nabla_V T,W_k) h(\hat \nabla_{T} W_k,\tilde V)\\
&=\sum_{k=1}^{n_3} h(\hat \nabla_V T,W_k) h(\hat \nabla_{\tilde V} T,W_k)\\
&=h(\hat \nabla_V T, \hat \nabla_{\tilde V} T).
\end{align*}
\end{proof}

Similarly, we have
\begin{lemma} \label{lemme4} Let $W, \tilde W \in \mathcal D_3$. Then
$$h(\hat\nabla_W T, \nabla_{\tilde W} T) = 0.$$
\end{lemma}

Combining the two previous lemmmas with  Lemma \ref{lemme2} and
Lemma \ref{lemme1} we see that the distributions determined by
$\mathcal D_2$ and $\mathcal D_3$ are totally geodesic. It also
implies that $h(\hat \nabla_V T,W) = h(\hat \nabla_W T, V) = 0$.

This is sufficient to conclude that locally $(M,h)$ is isometric with
$I \times M_1 \times M_2$ where $T$ is tangent to $I$, $\mathcal D_2$ is tangent
to $M_1$ and $\mathcal D_3$ is tangent to $M_2$.

The product structure of $M$ implies the existence of local
coordinates $(t,p,q)$ for $M$ based on an open subset containing
the origin of $\mathbb R \times \mathbb R^{n_2} \times \mathbb R^{n_3}$,
 such that $\mathcal D_1$ is given by $dp=dq=0$,
$\mathcal D_2$ is given by $dt=dq=0$, and $\mathcal D_3$ is given by
$dt=dp=0$. We may also assume that $T = \tfrac{\partial}{\partial
t}$. We now put

\begin{equation}\label{eqphi2phi3}
\quad \phi_2= -f \lambda_3 \phi + f T, \quad \phi_3 =
 g \lambda_2 \phi - g T,
\end{equation}
where the functions $f$ and  $g$, which depend only on the variable $t$,  are determined by
\begin{align*}
f'=f (\lambda_3  - \lambda_1) ,\\
g'=g(\lambda_2 - \lambda_1).
\end{align*}
It is clear that solutions are given by
\begin{equation*}
f(t) = d_1 e^{(\lambda_3-\lambda_1)t} \qquad \text{and} \qquad g(t)
= d_2 e^{(\lambda_2-\lambda_1)t},
\end{equation*}
where $d_1$ and $d_2$ are constants. Of course, as $\lambda_1 =
\lambda_2+\lambda_3$ we can rewrite the above equation as
\begin{equation*}
f(t) = d_1 e^{-\lambda_2 t} \qquad \text{and} \qquad g(t) = d_2
e^{-\lambda_3t}.
\end{equation*}
Computing $\lambda_1$, $\lambda_2$ and $\lambda_3$ explicitly, where if necessary by changing the sign of $E_1$ we
may assume that $\lambda_2 \ge 0$ we find that
\begin{align*}
&\lambda_2 = \tfrac{\sqrt{n_3+1}}{\sqrt{n_2+1}},\\
&\lambda_3 = -\tfrac{\sqrt{n_2+1}}{\sqrt{n_3+1}}.
\end{align*}
Solving now the above equations for the immersion $\phi$ we find that
\begin{align*}
\phi &=\tfrac{1}{f (\lambda_2-\lambda_3)}\phi_2 - \tfrac{1}{g (\lambda_2-\lambda_3)} \phi_3\\
&=(\tfrac{1}{d_1} e^{\tfrac{\sqrt{n_3+1}}{\sqrt{n_2+1}} t} \phi_2
 + \tfrac{1}{d_2} e^{-\tfrac{\sqrt{n_2+1}}{\sqrt{n_3+1}} t} \phi_3)(\frac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2}).
\end{align*}

A straightforward computation, using \eqref{eqphi2phi3}, now shows that
\begin{align*}
D_T (\phi_2)&=D_T(-f \lambda_3 \phi + f T)\\
&=f(\lambda_3-\lambda_1) (-\lambda_3\phi+T)+ f (-\lambda_3 T +(K(T,T)+\phi))\\
&=f(\lambda_3-\lambda_1)(-\lambda_3\phi+T)+ f ((\lambda_1-\lambda_3) T +\phi))\\
&=f (\lambda_2 \lambda_3 +1)\phi =0.
\end{align*}
Similarly
\begin{align*}
&D_W (\phi_2)=f(- \lambda_3 W +K(W,T))=0,\\
&D_T (\phi_3)=0,\\
&D_V (\phi_3)=0.
\end{align*}
The above implies that $\phi_2$ reduces to a map of $M_1$ in $\mathbb R^n$ whereas
$\phi_3$ reduces to a map of $M_2$ in $\mathbb R^n$. As we have
that
\begin{align*}
&d\phi_2(V)=D_V (\phi_2)=f(- \lambda_3 V +K(V,T))=f(-\lambda_3+\lambda_2)V,\\
&d\phi_3(W)=D_W(\phi_3)=g(\lambda_2 W -K(W,T))=g
(\lambda_2-\lambda_3)W,
\end{align*}
these maps are actually immersions. Moreover, denoting by $\nabla^1$ the
$\mathcal D_2$ component of $\nabla$, we find that
\begin{align*}
D_{V}d\phi_2(\tilde V)&=f(-\lambda_3+\lambda_2)D_{V}\tilde V\\
&= f(-\lambda_3+\lambda_2)\nabla_{V}\tilde V+ f(-\lambda_3+\lambda_2)h(V,\tilde V)\phi \\
&= f(-\lambda_3+\lambda_2)\nabla^1_{V}\tilde V+f(-\lambda_3+\lambda_2)(h(K(V,\tilde V),T)T+h(V,\tilde V)\phi)\\
&=d\phi_2(\nabla^1_{V}  \tilde V)+ f(-\lambda_3+\lambda_2)h(V,\tilde V)(\lambda_2 T +\phi)\\
&=d\phi_2(\nabla^1_{V} \tilde V)+f (-\lambda_3+\lambda_2)\lambda_2 h(V,\tilde V)(T-\lambda_3 \phi)\\
&=d\phi_2(\nabla^1_{V} \tilde V)+ (-\lambda_3+\lambda_2)\lambda_2
h(V,\tilde V)\phi_2.
\end{align*}
The above formulas imply that $\phi_2$ can be interpreted as a
centroaffine immersion contained in an $n_2+1$-dimensional vector
subspace of $\mathbb R^{n+1}$ with induced connection $\nabla^1$ and
affine metric $h_1 = (-\lambda_3+\lambda_2)\lambda_2 h$. Similarly,
we get that $\phi_3$ can be interpreted as a centroaffine immersion
contained in an $n_3+1$-dimensional vector subspace of $\mathbb
R^{n+1}$ with induced connection $\nabla^2$ (the restriction of
$\nabla$ to $\mathcal D_3$) and affine metric $h_2 =g
(\lambda_3-\lambda_2)\lambda_3 h$. Of course as both spaces are
complementary, we may assume by a linear transformation that the
$n_2+1$ dimensional space is spanned by the first $n_2+1$
coordinates of $\mathbb R^{n+1}$ whereas the $n_3+1$ dimensional
space is spanned by the last $n_3+1$ coordinates of $\mathbb
R^{n+1}$.

Moreover, taking $V_1,\dots, V_{n_2}$ as before, we find that
\begin{align*}
\sum_{i=1}^{n_2} (\nabla^1 h_1) (V,V_i,V_i)&=
\lambda_2(\lambda_2-\lambda_3)\sum_{i=1}^{n_2} (\nabla^1 h)
(V,V_i,V_i)\\
&=-2\lambda_2(\lambda_2-\lambda_3)\sum_{i=1}^{n_2} h(\nabla^1_V V_i,V_i)\\
&=-2\lambda_2(\lambda_2-\lambda_3)\sum_{i=1}^{n_2} h(\nabla_V V_i,V_i)\\
&=\lambda_2(\lambda_2-\lambda_3)\sum_{i=1}^{n_2} (\nabla h) (V,V_i,V_i)=0,
\end{align*}
as by assumption $h(K(V,W),W))= h(K(V,T),T)$. So $M_1$ is an hyperbolic
affine hypersphere. Choosing now the constant $d_1$ appropriately we may
assume that $M_1$ has mean curvature $-1$. A similar argument also holds for $M_2$.

As
\begin{align*}
\phi &=\tfrac{1}{f (\lambda_2-\lambda_3)}\phi_2 - \tfrac{1}{g (\lambda_2-\lambda_3)} \phi_3\\
&=(\tfrac{1}{d_1} e^{\tfrac{\sqrt{n_3+1}}{\sqrt{n_2+1}} t} \phi_2
 + \tfrac{1}{d_2} e^{-\tfrac{\sqrt{n_2+1}}{\sqrt{n_3+1}} t} \phi_3)(\frac{\sqrt{(n_2+1)(n_3+1)}}{n_2+n_3+2}).
\end{align*}
We note from Section 2 that we must have that $d_1^{n_2+1} d_2^{n_2+1} = 1$ and that therefore $\phi$ is given
as the Calabi product of the immersions $\phi_1$ and $\phi_2$.

\begin{remark} In case that $M$ has parallel difference tensor, i.e. if $\hat \nabla K=0$, the conditions
of Theorem 2 can be weakened. Indeed we can prove:
\begin{theorem} Let $M$ be a hyperbolic affine sphere with mean curvature $\lambda$,
where $\lambda<0$. Suppose that $\hat \nabla  K= 0$
and there exists $h$-orthonormal distributions $\mathcal D_1$ (of dimension $1$),
$\mathcal D_2$ (of dimension $n_2$)
and such that  $\mathcal D_3$ (of dimension $n_3$) such that
\begin{align*}
&K(T,T)=\lambda_1 T,\\
&K(T,V)=\lambda_2 V,\\
&K(T,W)=\lambda_3 W,
\end{align*}
where $T$ is a unit vector spanning $\mathcal D_1$ and $V \in
\mathcal D_2$, $W \in \mathcal D_3$. Moreover we suppose that
$\lambda_2 \ne \lambda_3$ and $2\lambda_2 \ne \lambda_1 \ne 2
\lambda_3$. Then $\phi:M^n \rightarrow \mathbb R^{n+1}$ can be
decomposed as the Calabi product of two hyperbolic affine sphere
immersions $\psi_1:M_1^{n_2} \rightarrow \mathbb R^{n_2+1}$ and
 $\psi_2:M_2^{n_3} \rightarrow \mathbb R^{n_3+1}$ with parallel cubic form.
\end{theorem}
\begin{proof} By applying an homothety we may choose $\lambda=-1$.
As $\hat \nabla K=0$, we also have that $\hat R. K = 0$. This means that
\begin{equation*}
\hat R(X,Y)K(Z,U)= K(\hat R(X,Y)Z,U)+K(Z, \hat R(X,Y)U).
\end{equation*}
So, taking $X=Z=U=T$ and $Y=V$, we find that
$$
\hat R(T,V)T=V -K_T K_V T +K_V K_TT=(1-\lambda_2^2+\lambda_1 \lambda_2) V.
$$
Hence we deduce that
\begin{equation*}
(\lambda_1-2\lambda_2)(-1-\lambda_1 \lambda_2 +\lambda_2^2)=0.
\end{equation*}
Similarly we have
\begin{equation*}
(\lambda_1-2\lambda_3)(-1-\lambda_1 \lambda_3 +\lambda_3^2)=0.
\end{equation*}
In view of the conditions, we must have that $\lambda_2$ and $\lambda_3$ are the two different
roots of the equation
$$-1 - \lambda_1 x +x^2=0.$$
Consequently $\lambda_2+\lambda_3 = \lambda_1$ and $\lambda_2 \lambda_3=-1$.

Finally we take $Z=U=T$, $X=V$ and $Y=W$. Then we find that
\begin{align*}
\lambda_1 \hat R(V,W)T&= 2 K(\hat R(V,W)T,T)=-2  K(K_V K_W T,T)+2
K(K_W K_V T,T)\\
&=-2(\lambda_3-\lambda_2)K_T K_VW.
\end{align*}
Hence
\begin{align*}
\lambda_1 (\lambda_2-\lambda_3) K_V W&= 2 K(\hat R(V,W)T,T)=-2 K(K_V
K_W T,T)+2 K(K_W K_V T,T)\\
&=-2(\lambda_3-\lambda_2)K_T K_VW.
\end{align*}
This implies that $K_V W$ is an eigenvector of $K_T$ with eigenvalue $\tfrac{1}{2} \lambda_1$. Given the form
of $K_T$ we deduce that $K(V,W) = 0$. We are now in a position to apply Theorem 2 and deduce that
$M$ can be obtained as the Calabi product of the hyperbolic affine spheres.
\end{proof}

\end{remark}

\section{Characterisation of the Calabi product of a hyperbolic affine sphere and a point and the proof of Theorem 1

Throughout this section we will assume that $\phi:
M^n\longrightarrow\mathbb R^{n+1} $ is a hyperbolic affine
hypersphere with mean curvature $-1$ and we will prove Theorem 1.
Therefore, we shall also assume that $M$ admits two mutually
orthogonal differential distributions $\mathcal D_1$ and $\mathcal
D_2$ of dimension $1$ and $n_2> 0$, respectively, with $1+n_2=n$,
and, for unit vector $T\in \mathcal D_1$ and all vectors $V\in
\mathcal D_2$,
\begin{gather*}
K(T,T) = \lambda_1 T,\;\;\;\;\;\;
K(T,V) = \lambda_2 V.\\
\end{gather*}
By the apolarity condition we must have that
\begin{equation}
\lambda_1 +n_2 \lambda_2= 0,
\end{equation}
Moreover, we will assume that
\begin{equation}
1+\lambda_1 \lambda_2-\lambda_2^2=0.
\end{equation}

The above conditions imply that $\lambda_1$ and $\lambda_2$ are constant and
can be determined explicitly in terms of the dimension $n$.
Indeed, if necessary by replacing $T$ with $-T$, we have that
\begin{align*}
&\lambda_2= \tfrac{1}{\sqrt{n}},\\
&\lambda_1=-\tfrac{n-1}{\sqrt{n}}.
\end{align*}

We now proceed as in the previous case. Using the fact that
$\hat \nabla K$ is totally symmetric it follows that
\begin{lemma} \label{lemme3} We have
\begin{enumerate}
\item $\hat \nabla_{T} T=0$,
\item $\hat \nabla_{V} T=0$,
\item $h(\hat \nabla_V \tilde V,T)=0$.
\end{enumerate}
\end{lemma}

The  previous lemmma tells us that the distributions determined by
$\mathcal D_1$ and $\mathcal D_2$ are totally geodesic.
This is sufficient to conclude that locally $(M,h)$ is isometric with
$I \times M_1$ where $T$ is tangent to $I$ and $\mathcal D_2$ is tangent
to $M_1$.

The product structure of $M$ implies the existence of local
coordinates $(t,p$ for $M$ based on an open subset containing
the origin of $\mathbb R \times \mathbb R^{n_2}$,
 such that $\mathcal D_1$ is given by $dp=0$ and
$\mathcal D_2$ is given by $dt=$. We may also assume that $T = \tfrac{\partial}{\partial
t}$. We now put

\begin{equation}\label{eqphi2}
\quad \phi_2= f  \tfrac{1}{\lambda_2} \phi + f T, \quad \phi_3 =
 g \lambda_2 \phi - g T,
\end{equation}
where the functions $f$ and  $g$, which depend only on the variable $t$,  are determined by
\begin{align*}
f'=-f \lambda_2= -\tfrac{1}{\sqrt{n}} ,\\
g'=g(\lambda_2 - \lambda_1)=\sqrt{n}.
\end{align*}
It is clear that solutions are given by
\begin{equation*}
f(t) = d_1 e^{-\tfrac{1}{\sqrt{n}}t} \qquad \text{and} \qquad g(t)
= d_2 e^{\sqrt{n} t}.
\end{equation*}

A straightforward computation, now shows that
\begin{align*}
D_T (\phi_2)&=D_T(f \sqrt{n} \phi + f T)\\
&=-f(\phi+ \tfrac{1}{\sqrt{n}}T)+ f ( \sqrt{n} T +(K(T,T)+\phi))\\
&=f T (- \tfrac{1}{\sqrt{n}}+ \sqrt{n}-\tfrac{n-1}{\sqrt{n}})\\
&=0.
\end{align*}
Similarly
\begin{align*}
&D_T (\phi_3)=0,\\
&D_V (\phi_3)=0.
\end{align*}
The above implies that $\phi_2$ reduces to a map of $M_1$ in $\mathbb R^n$ whereas
$\phi_3$ is a constant vector in $\mathbb R^n$. As we have
that
\begin{equation*}
d\phi_2(V)=D_V (\phi_2)=f(\sqrt{n} V
+K(V,T))=f(\sqrt{n}+\tfrac{1}{\sqrt{n}})V,
\end{equation*}
the map $\phi_2$ is actually immersions. Moreover, denoting by $\nabla^1$ the
$\mathcal D_2$ component of $\nabla$, we find that
\begin{align*}
D_{V}d\phi_2(\tilde V)&=f(\sqrt{n}+\tfrac{1}{\sqrt{n}})D_{V}\tilde V\\
&= f(\sqrt{n}+\tfrac{1}{\sqrt{n}})\nabla_{V}\tilde V+ f(\sqrt{n}+\tfrac{1}{\sqrt{n}})h(V,\tilde V)\phi \\
&= f(\sqrt{n}+\tfrac{1}{\sqrt{n}})\nabla^1_{V}\tilde V+f(\sqrt{n}+\tfrac{1}{\sqrt{n}})(h(K(V,\tilde V),T)T+h(V,\tilde V)\phi)\\
&=d\phi_2(\nabla^1_{V}  \tilde V)+ f(\sqrt{n}+\tfrac{1}{\sqrt{n}})h(V,\tilde V)(\tfrac{1}{\sqrt{n}} T +\phi)\\
&=d\phi_2(\nabla^1_{V} \tilde V)+\tfrac{n+1}{n}
h(V,\tilde V)\phi_2.
\end{align*}
The above formulas imply that $\phi_2$ can be interpreted as a
centroaffine immersion contained in an $n_2+1$-dimensional vector
subspace of $\mathbb R^{n+1}$ with induced connection $\nabla^1$ and
affine metric $h_1 = \tfrac{n+1}{n} h$. Of course as the vector $\phi_3$ is transversal
to the immersion $\phi_2$, we may assume by a linear transformation that the
$\phi_2$ lies in the space spanned by the first $n$
coordinates of $\mathbb R^{n+1}$ whereas the constant vector lies in the direction of the last coordinate,
and by choosing $d_2$ appropriately we may assume that $\phi_2= (0,\dots,0,1)$.

As before we get that $M_1$ satisfies the apolarity condition and hence is a hyperbolic
affine hypersphere. Choosing now the constant $d_1$ appropriately we may
assume that $M_1$ has mean curvature $-1$.

As
\begin{equation*}
\phi
=(\tfrac{1}{d_1} e^{\tfrac{1}{\sqrt{n}} t} \phi_2
 + \tfrac{1}{d_2} e^{-\sqrt{n} t} \phi_3)(\frac{\sqrt{(n)(n_3+1)}}{n+1}).
\end{equation*}
We note from Section 2 that we must have that $d_1^{n_2+1} d_2 = 1$ and that therefore $\phi$ is given
as the Calabi product of the immersions $\phi_1$ and a point.

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Parapallene hospitalis är en havsspindelart som beskrevs av Loman, J.C.C. 1908. Parapallene hospitalis ingår i släktet Parapallene och familjen Callipallenidae. Inga underarter finns listade i Catalogue of Life.

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