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Negative curvature, surface of
From Encyclopedia of Mathematics
(in the direct sense)
A two-dimensional surface in three-dimensional Euclidean space that has negative Gaussian curvature $ K < 0 $ at every point. The simplest examples of this are: a one-sheet hyperboloid (Fig.1a), a hyperbolic paraboloid (Fig.1b) and a catenoid.
Figure: n066200a
Figure: n066200b
The concept of a surface of negative curvature can be generalized, for example, with respect to the dimension of the surface itself or the dimension and structure of the ambient space.
Surfaces of negative curvature locally have a saddle-like structure. This means that in a sufficiently small neighbourhood of any of its points, a surface of negative curvature resembles a saddle (see Fig.1b, not considering the behaviour of the surface outside the part of it that has been drawn). The local saddle-like character of the surface is clearly illustrated in this figure, which shows the principal sections of the surface at an arbitrary point $ O $. Let $ 1/R _ {1} $, $ 1/R _ {2} $ be their normal curvatures, i.e. the principal curvatures at the point $ O $( cf. Principal curvature). According to the classical definition, the Gaussian curvature at $ O $ is the number $ K = 1/R _ {1} R _ {2} $. Since $ K < 0 $, the principal curvatures have different signs, for which reason the principal sections are convex in opposite directions; in Fig.1bthe section $ 0 2 $ is convex in the direction of the normal $ n $, while the other is convex in the opposite direction, which fits in with the saddle-like character of the surface. The topological structure in the large ( "globally" ) of a surface of negative curvature can be very different. For example, a hyperbolic paraboloid (Fig.1b) is topologically equivalent to the plane, while a one-sheet hyperboloid (Fig.1a) is equivalent to a cylindrical tube. Let two one-sheet hyperboloids with parallel axes intersect along a hyperbola (Fig.2a). Let these hyperboloids not be transparent, and let their invisible parts (i.e. the parts of any of them which lie within another) be disregarded. The surface thus obtained has negative curvature everywhere, apart from the points of the hyperbola mentioned above. At the points of this hyperbola there is, in general, no curvature (in the classical sense), since the hyperbola is an edge of the surface. However, the surface near the edge can, in the given case, be smoothed (see [7]), such that a surface is obtained that has a curvature at all points and that is, moreover, negative (see Fig.2b). It is topologically equivalent to a torus with two punctures. The topological structure of a surface of negative curvature can be made as complex as desired by the inclusion of a larger number of one-sheet hyperboloids.
Figure: n066200c
Figure: n066200d
A surface is said to be complete if it is a complete metric space with respect to the distance given by its intrinsic geometry (cf. also Internal metric). In all examples presented above, complete surfaces were considered. The idea of constructing complete surfaces of negative curvature of various topological types arising from very simple types (as shown above) is attributed to J. Hadamard [1]. A fundamentally different example is proposed in Fig. c(see [6], [7]). This surface has negative curvature everywhere, and is completely enclosed in a sphere. It has an infinite number of branches, and a variable point of it moving along the branches can get infinitely close to the boundary of the sphere without ever touching it. The surface is constructed such that a variable point moving along the branches always follows a path of infinite length. In this way, the completeness of the surface in the sense of its interior metric is guaranteed. This example shows that an intrinsically-complete surface of negative curvature does not necessarily move to infinity in space, like those in Hadamard's examples.
Figure: n066200e
In this example, the condition $ K < 0 $ can generally be observed, allowing for a slight irregularity of the surface. This means that the surface is presumed to be smooth (everywhere of class $ C ^ {1} $), but that at certain isolated points it does not belong to $ C ^ {2} $; at these points, the curvature must have a limit value. After completing the curvature with those limit values, it is defined everywhere and is continuous. The question of the extension of the surface and its connection with the condition of regularity is complemented by the following theorems [8].
If the Gaussian curvature $ K $ of a complete surface of class $ C ^ {2} $ satisfies the inequality $ - a ^ {2} \leq K \leq 0 $, where $ a = \textrm{ const } $, then the surface cannot be contained within a sphere of radius less than $ \sqrt 2/3 /a $.
If on a complete surface of class $ C ^ {2} $, which is unbounded in the intrinsic sense, the Gaussian curvature $ K \rightarrow 0 $ at infinity, then the surface is not bounded in space (in this theorem, the sign of the curvature may change).
Particular attention has been (and still is) paid to surfaces of constant negative curvature, i.e., they have Gaussian curvature with the same negative value at all points. They are notable for the fact that their intrinsic geometry locally coincides with the geometry on the Lobachevskii plane (see Lobachevskii geometry). This means that for figures which lie on a surface of constant negative curvature, precisely the same relations occur as those which hold in the Lobachevskii planimetry (the role of straight lines is played by geodesic lines); for example, the trigonometric formulas for triangles on a surface of constant negative curvature and the corresponding formulas in non-Euclidean Lobachevskii geometry coincide exactly. Thus, on a surface of constant negative curvature, a local model of the Lobachevskii geometry is obtained. A model of this type was constructed by E. Beltrami in 1868 (cf. Beltrami interpretation), which made an essential contribution to the recognition of Lobachevskii geometry. Beltrami also proposed an important example of a surface of negative curvature, which he called a pseudo-sphere (Fig. d) and which is formed by the rotation of a tractrix around its asymptote (Fig. e).
Figure: n066200f
Figure: n066200g
A characteristic property of a tractrix is that the segment of the tangent from a point of tangency to the asymptote is constant and equal to a number $ a $. The Gaussian curvature of the pseudo-sphere is $ K = - 1/a ^ {2} $. Surfaces of rotation of negative curvature were studied even earlier than Beltrami by F. Minding in 1839. He discovered two forms of periodic surfaces of rotation of constant negative curvature (Fig. f) which, with the pseudo-sphere, exhaust all possible surfaces of rotation of constant negative curvature (for their equations, see [3], [5]).
Figure: n066200h
The construction of a global model of Lobachevskii planimetry on a surface of rotation of constant negative curvature, for example on a pseudo-sphere, is not possible for two reasons. First, there are irregular points on a pseudo-sphere; these form a cuspidal edge (Fig. d). Secondly, a pseudo-sphere is topologically equivalent to a tube, and therefore, even from the topological point of view, differs from a Lobachevskii plane. However, it is possible to construct a model of the infinite part of the Lobachevskii plane on a regular surface of rotation of constant negative curvature. To this end, an infinite (contractible) domain $ U $ is cut out of the pseudo-sphere by a plane perpendicular to the axis. The plane is so formed that it does not contain the cuspidal edge. The part of the pseudo-sphere in which there is no cuspidal edge is taken as $ U $. Thereafter, only $ U $ is considered, and, for the sake of convenience, it is also called a pseudo-sphere; it has no singular points. Let $ U ^ \prime $ be the universal covering surface of $ U $. This concept can be introduced in the following way. A countably infinite number of identical and superposed copies of the pseudo-sphere $ U $ is taken, which are distinguished using the integers: $ \dots, U _ {-} 2 , U _ {-} 1 , U _ {0} , U _ {1} ,\dots $. Then they are all cut along a common meridian, and a left and right side of the cut is marked on each of them, such that all left sides are superposed (the same applies also to the right sides). Now, for any number $ k $, $ - \infty < k < \infty $, the left side of $ U _ {k} $ is joined to the right side of $ U _ {k+} 1 $( similarly, the right side of $ U _ {k} $ is joined to the left side of $ U _ {k-} 1 $). A connected surface $ U ^ \prime $ is thus obtained, which is also the universal covering surface of $ U $, and is sometimes called the simply-connected envelope of the surface $ U $. A universal covering can be constructed for any surface, but the visual description of its construction will be far less simple than that of a pseudo-sphere. It is essential that the universal covering is always simply connected. It is also important that the local intrinsic geometry of the covered surface can be transferred to the universal covering. For this to happen, it is sufficient to ensure that every point $ p $ of the covered surface has a convex neighbourhood $ V( p) $ which is covered one-sheetedly by a neighbourhood $ V ^ \prime ( p ^ \prime ) $ in the covering surface (here $ p ^ \prime $ is a point covering the point $ p $; the correspondence $ V \rightarrow V ^ \prime $ is bijective). The geometry of $ V( p) $ is transferred to $ V ^ \prime ( p ^ \prime ) $ identically, i.e. the distance between two points $ p _ {1} ^ \prime , p _ {2} ^ \prime \in V ^ \prime $ is taken to be equal to the distance between their inverse images. Then the global geometry of the universal covering is defined, but, as a rule, it will be completely different from the global geometry of the covered surface.
On the pseudo-sphere $ U $, the meridians, when moving towards infinity, converge; moreover, they are orthogonal to the boundary of $ U $. The boundary of the universal covering is therefore a curve in the Lobachevskii plane, which is the orthogonal trajectory of a pencil of Lobachevskii-parallel (converging) straight lines. This curve is called a horocycle in the Lobachevskii geometry. Thus, from the point of view of its intrinsic geometry, the universal covering of the pseudo-sphere $ U $ is the interior domain of an horocycle in the Lobachevskii plane. The universal covering $ U ^ \prime $ of the pseudo-sphere $ U $ is more convenient for the construction of a model of the Lobachevskii geometry than the pseudo-sphere $ U $ itself. E.g., a circle of any radius can be placed on $ U ^ \prime $, but a large circle without overlappings may prove to be too large to be contained in $ U $. However, even $ U ^ \prime $ is not extensive enough to contain some object of the Lobachevskii geometry, such as, for example, a complete straight line. In other words, it is not possible to construct a model of the entire Lobachevskii plane using $ U ^ \prime $.
In 1900, D. Hilbert posed the question of whether there exists a surface whose intrinsic geometry coincides completely with the geometry of the Lobachevskii plane. Using a simple reasoning, it follows that if such a surface does exist, it must have constant negative curvature and be complete. Moreover, it must be simply connected, since the Lobachevskii plane is simply connected. It would in fact be sufficient, however, to find a complete surface of constant negative curvature (as stated above, the universal covering of such a surface is complete and simply connected). In saying this, one considers only regular surfaces (at least twice continuously differentiable at any point, i.e. of class $ C ^ {2} $).
As early as 1901, Hilbert solved this problem (see [2]), and in the negative sense, so that no complete regular surface of constant negative curvature exists in three-dimensional Euclidean space. This theorem has attracted the attention of geometers over a number of decades, and continues to do so today. The reason for this is that a number of interesting questions are related to it and to its proof (see below). The occurrence of singular lines on surfaces of rotation of constant negative curvature (Fig. d, Fig. f) is not coincidental, but is in line with Hilbert's theorem.
The proof of Hilbert's theorem on surfaces of any topological structure is of special interest, since the concept of a universal covering surface is essential to it. This concept reduces the problem to a surface with the simplest topology, namely, a simply-connected surface. Hilbert's theorem was evidently one of the earliest mathematical statements to explain this concept. Following the publication of Hilbert's first article expounding the proof of his theorem, doubts were expressed on the validity of his proofs. The criticism related to the limitless behaviour of the asymptotic lines, which were studied on the surface in terms of the complexity of the topology of that surface. It was even suggested that Hilbert had refined his proof to make it more accurate. Nevertheless, Hilbert's proof was quite faultless throughout. These criticisms would probably not have arisen if Hilbert had stated clearly that the object of study was a complete simply-connected manifold of constant negative curvature, i.e. the Lobachevskii plane itself, assuming that it can be isometrically and regularly immersed in $ 3 $- dimensional Euclidean space.
Figure: n066200i
It is also interesting that Chebyshev nets appear unexpectedly in Hilbert's proof. These are nets in which every net quadrangle has opposite sides of equal length (Fig. g). Moreover, an asymptotic net (which is defined and non-degenerate at all points on any surface of negative curvature) is a Chebyshev net in the case of a surface of constant negative curvature. This net can be taken to be a coordinate net: $ u = \textrm{ const } $, $ v = \textrm{ const } $. The line element then takes the Chebyshev form
$$ \tag{1 } ds ^ {2} = du ^ {2} + 2 \cos \omega du dv + dv ^ {2} , $$
where $ \omega $ is the angle in the net. Assuming that the surface is regular, complete and simply connected, and taking account of the Chebyshev nature of the net, it can be proved that $ u , v $ take all values: $ - \infty < u < \infty $, $ - \infty < v < \infty $; these exhaust all of the net. In other words, an asymptotic net in the large is homeomorphic to the Cartesian net on the Euclidean coordinate plane (see [9]). On the other hand, assuming that the Gaussian curvature is constant, for example $ K = - 1 $, the equation
$$ \tag{2 } \frac{\partial ^ {2} \omega }{\partial u \partial v } = \ \sin \omega $$
is obtained from (1), and the restrictions
$$ \tag{3 } 0 < \omega < \pi $$
follow from geometric considerations. Hilbert demonstrated that equation (2) cannot have a regular solution $ \omega = \omega ( u, v) $ subject to the condition (3) on the entire coordinate plane, and the complete Lobachevskii plane can therefore not be regularly and isometrically imbedded in $ 3 $- dimensional Euclidean space. As explained above, this leads to a more general formulation of this theorem: In $ 3 $- dimensional Euclidean space there does not exist a complete regular surface of constant negative curvature (whatever its topological structure).
In various areas of mathematics it happens that one particular statement or one particular question is of great importance in the development of the whole area. In differential geometry, Hilbert's theorem has been, and still remains, one such assertion, as witnessed by the amount of publications devoted to it. Interest in this theorem stems largely from the combination of different ideas and concepts in its proof (for example, the universal covering and the Chebyshev net). A connection with physics was subsequently established.
It turned out that equation (2), if written slightly differently and without restriction (3), proves to express a condition on the potential in the so-called Josephson effect from the theory of superconductivity. Physicists have named equation (2) the "sine-Gordon" equation. Using the connection between equation (2) and the theory of surfaces of negative curvature (see [13], [14]), certain solutions of the equation have been found in closed form, which are regular in the entire plane.
S.E. Cohn-Vossen proposed (see [15]) that a far more general theorem than Hilbert's theorem holds; namely, the condition on the Gaussian curvature $ K = \textrm{ const } < 0 $ can be replaced by the condition $ K \leq \textrm{ const } < 0 $. However, up to the beginning of the 1960's, no theorem had been found which included Hilbert's theorem as a specific case. Not until 1961 (see [16]) was the correctness of Hilbert's theorem proved, i.e. a theorem was proved concerning a surface of variable negative curvature, for which Hilbert's theorem is a limiting case. In other words, the properties of a surface of constant negative curvature, which give rise to the effect of Hilbert's theorem (disruption of regularity) and the effect itself, can be taken in an approximate sense. In the proof, an equation was found in the case of variable curvature, generalizing equation (2):
$$ \tag{4 } k ^ {2} \frac \partial {\partial s _ {2} } \left ( \frac{1}{K ^ {2} } \frac{\partial \omega }{\partial s _ {1} } \right ) = \ \{ k ^ {2} + A + \alpha | \mathop{\rm grad} \omega | \} \sin \ \omega , $$
where $ \omega $ has its previous sense, $ k ^ {2} = - K $, differentiation takes place along the arcs of the asymptotic lines, and $ A $ and $ \alpha $ are variables whose absolute values allow an intrinsic estimate, i.e. an estimate depending only on the imbedded metric. When $ K = \textrm{ const } $, then $ A = 0 $, $ \alpha = 0 $ are obtained, and equation (4) becomes equation (2).
In the case of variable curvature the equation
$$ \tag{5 } L _ {s _ {2} } L _ {s _ {1} } ^ {*} ( \omega ) = M \ \sin \omega $$
has been found (see [17]), where $ L _ {s _ {2} } $ is a linear operator proportional to the differentiation operator along the arc of the asymptotic lines of the second family, and $ L _ {s _ {1} } ^ {*} $ is a quasi-linear operator of similar structure. It is essential that the function $ M $, given certain conditions on the metric, has an intrinsic estimate from below by a positive number, thanks to which, and to the structure of the left-hand side, equation (5) is consistent with equation (2). Therefore, given these conditions on the metric, arguments can be put forward which are consistent with those of Hilbert (although far more complex) and which produce a similar result. In this way, a generalization of Hilbert's theorem was first obtained (see [18]). As regards the conditions on the metric mentioned above, they are included in the requirement that the Gaussian curvature $ K $ be bounded away from zero and changes slowly. It is proposed that if $ K \leq - 1 $, then a slow change of the curvature must be understood in the sense that the first and second derivatives of $ K $ along the arc of any geodesic are sufficiently small. Sufficient estimates are made such that in the calculation the above arguments, coming from Hilbert, can be used for equation (5). These estimates (see [19]) are expressed using positive constants, denoted by $ h $ and $ \Delta $. Consequently, metrics with a slowly-changing negative curvature are called $ ( h, \Delta ) $- metrics (for their definition see [18] or [19]); moreover, metrics of constant curvature are obtained when $ \Delta = + \infty $. Thus, a complete $ ( h, \Delta ) $- metric does not permit a regular isometric immersion in three-dimensional Euclidean space.
The question of generalizing Hilbert's theorem in the form in which it was stated in the 1930's (without restrictions on the character of the behaviour of the curvature, see [12]) along the lines indicated above, i.e. using differential equations of the theory of surfaces, has so far not succeeded at all. However, a solution to this problem has been obtained along entirely different lines, namely, through detailed studies of the boundary properties of the spherical image of a surface of negative curvature. Using this method, it has been proved ([19], [20]) that a complete metric with Gaussian curvature $ K \leq \textrm{ const } < 0 $ does not allow a regular isometric immersion in $ E ^ {3} $. This answers the question in the classical formulation. At the same time, the theorem that, in $ E ^ {3} $, the equality
$$ \tag{6 } \sup K = 0 $$
holds on every complete regular surface of negative Gaussian curvature $ K $, has been proved. Here, $ C ^ {2} $ regularity is sufficient. However (see [7]), in the case of $ C ^ {1,1} $ there is a counter-example (in the form of a weakly irregular surface). See [39] for these results.
Apart from (6), other equalities and estimates have been obtained (see [21]), which are universal in the sense that they refer to all surfaces of negative curvature in $ E ^ {3} $. Moreover, the method of proving (6) makes it possible to obtain theorems which give sufficient criterions for a mapping from a plane onto a plane with a negative Jacobian to be not only a local, but also a global diffeomorphism.
Despite the results which have now been obtained, studies on $ ( h, \Delta ) $- metrics and other metrics with slowly-changing curvature (for example, $ q $- metrics; see [9]) are still relevant. The situation is that for immersions of metrics with slowly-changing negative curvature, certain effects are created which do not occur in the case of more general metrics of negative curvature (see, for example, the theorem on the simple zone in [9]). As a rough rule, one can say that the slower the change of curvature, the "harder" it is for the metric of negative curvature to be immersed in $ E ^ {3} $. This is particularly noticeable in the case of a metric of constant curvature (see the strengthened Hilbert theorem [9]). As regards differential equations which arise in connection with the question of immersibility of $ ( h, \Delta ) $- metrics, they were essentially used in studies of other questions. Indeed, a number of strong theorems on the connection between intrinsic and extrinsic properties of surfaces of negative curvature has been proved using these equations; these include theorems on the dependence of the extrinsic regularity of the curvature of a surface of negative curvature on the regularity of its metric (see [7]).
The effect of the regularity of the metric on the extrinsic regularity of a surface with negative curvature is of particular interest, since surfaces of negative curvature form part of a special class of surfaces which can be defined purely geometrically (it is interesting that a purely geometric definition is also possible for the class of convex surfaces). Surfaces of this class are called saddle-like. Saddle-like surfaces are, in a certain sense, the antipodes of convex surfaces (see [22]–[25]).
The theorem on the non-existence in Euclidean space of a surface of complete negative curvature $ K \leq \textrm{ const } < 0 $ has a particularly strengthened form in the case of a surface which can be projected injectively onto a plane, i.e. which can be represented as the graph of a function $ z = f( x, y) $. In this case, universal estimates for the extension of the surface exist, and its completeness becomes immaterial. The following theorems have been proved (by N.V. Efimov, see [26]):
1) if a surface with Gaussian curvature $ K \leq - \alpha ^ {2} = \textrm{ const } < 0 $ can be regularly projected onto a rectangle with sides $ a, b $, then the inequality $ \sqrt \alpha \geq C _ {1} + \epsilon $, $ \epsilon > 0 $, implies $ a \sqrt \alpha \leq C _ {1} + C _ {2} / \epsilon $, where $ C _ {1} , C _ {2} $ are universal constants (for example, $ C _ {1} = 4 \pi $, $ C _ {2} = 12 \pi $);
2) if $ a, b $ are as above and (for the sake of simplicity) $ \alpha = 1 $, then $ a \leq 18.9 $.
Let a metric $ ds ^ {2} $ with negative Gaussian curvature $ K $ be given, and let $ k = \sqrt - K $. Let the metric $ ds ^ {2} $ be immersed in $ E ^ {3} $, and let $ l, m, n $ be the reduced coefficients of the second fundamental form (reduced means: satisfying the equation $ l n - m ^ {2} = - K $). In the immersion problem, $ l $, $ m $ and $ n $ are unknown functions. The Riemann invariants are introduced as new unknown functions:
$$ r = m+ \frac{k}{n} ,\ \ s = m- \frac{k}{n} . $$
The system of fundamental equations of the theory of surfaces will then take a symmetric form
$$ \tag{7 } \left . \begin{array}{c} r _ {x} + sr _ {y} = \ A _ {0} + A _ {1} r + A _ {2} s + A _ {3} r ^ {2} + A _ {4} rs + A _ {5} r ^ {2} s , \\ s _ {x} + rs _ {y} = \ A _ {0} + A _ {1} s + A _ {2} r + A _ {3} s ^ {2} + A _ {4} rs + A _ {5} s ^ {2} r, \end{array} \right \} $$
where $ A _ {0} \dots A _ {5} $ are expressed in terms of only the metric $ ds ^ {2} $.
It has already been shown that a complete surface of negative curvature with $ K \leq \textrm{ const } < 0 $ does not permit an isometric regular immersion in $ E ^ {3} $. The results obtained on the possibility of an isometric immersion of a two-dimensional manifold of negative curvature in $ E ^ {3} $ can be restricted to parts of complete manifolds. The manifold itself is taken to be simply connected. Thus, it has been proved (see [29], [30]) that every geodesic disc on an arbitrary regular manifold of negative curvature $ K \leq \textrm{ const } < 0 $ can be regularly isometrically immersed in $ E ^ {3} $. This theorem is obtained as a corollary of a more general theorem: Given certain natural conditions on the regularity of the metric of a complete surface of negative curvature with $ K \leq \textrm{ const } < 0 $, every infinite equidistant strip on this manifold can be isometrically and regularly immersed in $ E ^ {3} $. This theorem in turn is obtained as a result of the existence theorem of solutions of hyperbolic systems of quasi-linear equations of the form (7), where $ r \neq s $.
An example of a simply-connected manifold of sign-alternating curvature has been constructed on which there is a geodesic disc which cannot be isometrically immersed in $ E ^ {3} $( see [10]). It has been proved [31], given certain restrictions on the complete metric of a simply-connected surface of negative curvature $ K \leq \textrm{ const } < 0 $, that the interior of an arbitrary horocycle of such a manifold can be isometrically immersed in $ E ^ {3} $. Thus, for simply-connected manifolds of negative curvature $ K \leq \textrm{ const } < 0 $ it has been proved that isometric immersion is possible for all their compact and non-compact parts which are compatible with parts of the Lobachevskii plane, the isometric immersions of which are known through the works of Minding and Beltrami.
Immersions of a metric with Gaussian curvature $ K = \textrm{ const } < 0 $ have been studied, i.e. immersions of parts of the Lobachevskii plane. Thus, it has been proved that all polygons, even those of infinite extension, of the Lobachevskii plane can be isometrically immersed in $ E ^ {3} $; for new results on this kind of immersions see [40]–[42].
The variety of topological types of complete surfaces of negative curvature in $ E ^ {3} $ makes it necessary to consider different subclasses of this too large class, satisfying some supplementary conditions. The subclass of complete surfaces of negative curvature which have a one-to-one spherical ( "single-sheeted" ) mapping is particularly natural (see ). If a surface $ F $ of this class does not have self-intersections, then it is simply or doubly connected. If such a surface $ F $ has a horn, then, using the unboundedness of the saddle horn [34], a number of external geometric properties of this surface can be obtained: $ F $ can be defined by an equation $ z = f( x, y) $ over a domain obtained from the $ ( x, y) $- plane by eliminating a compact convex set; $ F $ has a bounding cone $ A( F ) $ consisting of a ray corresponding to the horn and a convex cone corresponding to the cup of the surface; the closure $ F ^ { * } $ of the spherical image of $ F $ has as its completion to a sphere an open semi-sphere and an open convex set (Fig. h–Fig. j).
Figure: n066200j
Figure: n066200k
Figure: n066200l
A number of similar questions has been studied for other types of complete surfaces of negative curvature with a one-to-one spherical mapping.
Another class which is of interest because of its properties and the properties of its spherical mapping is the class of contractible surfaces of negative curvature (see , [37]). An example of this type of surface is the surface defined by an equation $ x ^ {2} y ^ {2} + y ^ {2} z ^ {2} + z ^ {2} x ^ {2} = a ^ {2} $. In a number of its properties, a contractible surface of negative curvature resembles a closed surface. Its spherical image can be viewed as a Riemann surface with boundary, each component of which is a spherical broken line lying on one of the large circles.
The theory of surfaces of negative curvature in a pseudo-Euclidean space $ E _ {2,1} ^ {3} $ is viewed differently. In this space, surfaces of negative curvature are convex; here the curvature is understood in the usual way, as the curvature of the metric induced by the ambient space. Namely, it is supposed to be negative.
Figure: n066200m
The unit sphere of the space $ E _ {2,1} ^ {3} $ consists of three connected components, $ L _ {1} , L _ {2} , L _ {3} $( see Fig. k). The components $ L _ {1} $ and $ L _ {2} $ are convex surfaces on which a positive-definite metric of constant negative curvature is induced; these surfaces provide a well-known interpretation of the Lobachevskii geometry. The surface $ L _ {3} $ is saddle-like, and an indefinite metric of constant positive curvature is induced on it. The metric of this surface is known as the two-dimensional de Sitter metric.
Surfaces of negative curvature with a definite metric form a natural broad class of surfaces in $ E _ {2,1} ^ {3} $ generalizing the properties of the surfaces $ L _ {1} $ and $ L _ {2} $, just as surfaces of positive curvature form in Euclidean space a class generalizing the sphere (see [38]).
The properties of surfaces with a definite metric and negative curvature, the total curvature of which is finite, have been studied exhaustively. The properties of these surfaces are analogous to the properties of convex surfaces with curvature less than $ 2 \pi $ in Euclidean space (the class of Olovyanishnikov surfaces). Namely, a complete definite metric of negative curvature and finite total curvature $ \widetilde{K} $ and a geodesic ray $ \gamma $ on a manifold carrying this metric uniquely determine a convex surface with the given limiting cone of complete curvature $ \widetilde{K} $, without isotropic generators and with a given generator which is limiting for the ray $ \gamma $. The regularity of the surfaces obtained is determined by the regularity of the metric.
Complete surfaces of negative curvature with an indefinite metric are also convex. However, their properties are partially compatible with the properties of saddle-like surfaces in Euclidean space. In particular, the estimate $ \sup K = 0 $ holds for them. These surfaces would have to correspond to the component $ L _ {3} $ of the unit sphere of the space $ E _ {2,1} ^ {3} $, which is not convex, however. Here, both the sign of the Gaussian curvature and the sign of the determinant of the metric form of the surface play a determining role.
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[2] D. Hilbert, "Ueber Flächen von konstanter Gausscher Krümmung" , Gesammelte Abhandlungen , 2 , Springer (1933) pp. 437–448
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[10] E.G. Poznyak, "Isometric immersions of two-dimensional Riemannian metrics in Euclidean space" Russian Math. Surveys , 28 : 4 (1973) pp. 47–77 Uspekhi Mat. Nauk , 28 : 4 (1973) pp. 47–76 Zbl 0289.53004
[11] E.G. Poznyak, E.V. Shikin, "Surfaces of negative curvature" J. Soviet Math. , 5 : 6 (1976) pp. 865–887 Itogi Nauk. i Tekhn. Ser. Algebra. Topol. Geom. , 12 (1974) pp. 171–207 Zbl 0107.15502
[12] E.G. Poznyak, "Geometric investigations connected with " J. Soviet Math. , 13 : 5 (1980) pp. 667–686 Itogi Nauk. i Tekhn. Probl. Geom. , 8 (1977) pp. 225–241 Zbl 0439.53006
[13] I.V. Gribkov, "Construction of some regular solutions of the "sine-Gordon" equation by means of surfaces of constant negative curvature" Vestnik Mosk. Univ. Ser. Mat. Mekh. , 4 (1977) pp. 78–83 (In Russian) (English abstract) MR0467036 Zbl 0409.53017 Zbl 0359.53003
[14] I.V. Gribkov, "Some solutions of the sin-Gordon equation obtained by means of a Bäcklund transformation" Russian Math. Surveys , 33 : 2 (1978) pp. 231–232 Uspekhi Mat. Nauk , 33 : 2 (1978) pp. 191–192
[15] S.E. Cohn-Vossen, "Global deformation of surfaces" Uspekhi Mat. Nauk , 1 (1936) pp. 33–76
[16] N.V. Efimov, "The validity of Hilbert's theorem on surfaces of constant negative curvature" Soviet Math. Dokl. , 2 : 1 (1961) pp. 189–191 Dokl. Akad. Nauk SSSR , 136 : 6 (1961) pp. 1283–1286 MR137036
[17] N.V. Efimov, E.G. Poznyak, "Some transformations of the fundamental equations of the theory of surfaces" Soviet Math. Dokl. , 2 : 2 (1961) pp. 225–227 Dokl. Akad. Nauk SSSR , 137 : 1 (1961) pp. 25–27 MR130619 Zbl 0108.33901
[18] N.V. Efimov, E.G. Poznyak, "Generalization of Hilbert's theorem on surfaces of constant negative curvature" Soviet Math. Dokl. , 2 : 2 (1961) pp. 300–303 Dokl. Akad. Nauk SSSR , 137 : 3 (1961) pp. 509–512 MR130661
[19] N.V. Efmov, "Imposibility of a complete regular surface in euclidean 3-space whose Gaussian curvature has a negative upper bound" Soviet Math. Dokl. , 4 : 3 (1963) pp. 843–846 Dokl. Akad. Nauk SSSR , 150 : 6 (1963) pp. 1206–1209
[20] N.V. Efimov, "Generation of singularities on surfaces of negative curvature" Mat. Sb. , 64 : 2 (1964) pp. 286–320 (In Russian)
[21] N.V. Efimov, "Differential criteria for homeomorphism of certain mappings with applications to the theory of surfaces" Math. USSR Sb. , 5 : 4 (1968) pp. 475–488 Mat. Sb. , 76 : 4 (1968) pp. 499–512 Zbl 0182.54702
[22] S.Z. Shefel', "Studies on the geometry of saddle-like surfaces" , Novosibirsk (1963) (In Russian)
[23] S.Z. Shefel', "On the intrinsic geometry of saddle surfaces" Sib. Mat. Zh. , 5 : 6 (1964) pp. 1382–1396 (In Russian)
[24] S.Z. Shefel', "A condition of compactness of a family of saddle surfaces" Siberian Math. J. , 8 : 3 (1967) pp. 528–535 Sib. Mat. Zh. , 8 : 3 (1967) pp. 705–714 Zbl 0161.41801
[25] S.Z. Shefel', "Saddle surfaces bounded by a rectifiable curve" Soviet Math. Dokl. , 6 : 3 (1965) pp. 684–687 Dokl. Akad. Nauk SSSR , 162 : 2 (1965) pp. 294–296 Zbl 0128.40601
[26] N.V. Efimov, "Estimates for the dimensions of the domains of regularity of solutions of certain Monge–Ampère equations" Math. USSR Sb. , 29 : 3 (1976) pp. 319–326 Mat. Sb. , 100 : 3 (1976) pp. 356–363 MR425786 Zbl 0376.35037
[27] N.V. Efimov, "Study of a single-valued projection of a surface of negative curvature" Dokl. Akad. Nauk SSSR , 93 : 4 (1953) pp. 609–611 (In Russian)
[28] E. Heinz, "Ueber Flächen mit eindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind" Math. Ann. , 129 : 5 (1955) pp. 451–454 Zbl 0065.37201
[29] E.G. Poznyak, "On the regular realization in the large of two-dimensional metrics of negative curvature" Soviet Math. Dokl. , 7 : 5 (1966) pp. 1288–1291 Dokl. Akad. Nauk SSSR , 170 : 4 (1966) pp. 786–789 Zbl 0168.19501
[30] E.G. Poznyak, "Regular global realization of two-dimensional metrics with negative curvature" Ukr. Geom. Sb. , 3 (1966) pp. 78–92 (In Russian)
[31] E.V. Shikin, "On the global isometric imbedding in of some metrics of nonpositive curvature" Soviet Math. Dokl. , 15 : 2 (1974) pp. 448–451 Dokl. Akad. Nauk SSSR , 215 : 1 (1974) pp. 61–63
[32] B.L. Rozhdestvenskii, "A system of quasi-linear equations in the theory of surfaces" Dokl. Akad. Nauk SSSR , 143 : 1 (1962) pp. 50–52 (In Russian)
[33a] A.L. Verner, "On the extrinsic geometry of elementary complete surfaces with nonpositive curvature I" Mat. USSR Sb. , 3 : 2 (1967) pp. 205–224 Mat. Sb. , 74 : 2 (1967) pp. 218–240 Zbl 0172.22702
[33b] A.L. Verner, "On the extrinsic geometry of elementary complete surfaces with nonpositive curvature II" Math. USSR Sb. , 4 : 1 (1968) pp. 99–123 Mat. Sb. , 75 : 1 (1967) pp. 112–139 Zbl 0176.18804
[33c] A.L. Verner, "Correction to "On the extrinsic geometry of elementary complete surfaces with nonpositive curvature" " Mat. Sb. , 77 : 1 (1968) pp. 136 (In Russian) MR0232323
[34] A.L. Verner, "Tapering saddle surfaces" Siberian Math. J. , 11 : 4 (1970) pp. 567–581 Sib. Mat. Zh. , 11 : 4 (1970) pp. 750–769 Zbl 0219.53052
[35] A.L. Verner, "The finiteness of the set of branch points of a spherical mapping of a narrowing saddle surface" Mat. Notes , 12 : 3 (1972) pp. 603–605 Mat. Zametki , 12 : 3 (1972) pp. 281–286 Zbl 0258.53009
[36a] D.D. Sokolov, "The structure of a limiting cone of a convex surface in Euclidean space" Uspekhi Mat. Nauk , 30 : 1 (1975) pp. 261–262 (In Russian)
[36b] D.D. Sokolov, "On convex surfaces with an indefinite metric" Russian Math. Surveys , 33 : 4 (1978) pp. 265–266 Uspekhi Mat. Nauk , 33 : 4 (1978) pp. 227–228 Zbl 0414.53042
[36c] D.D. Sokolov, "Convex surfaces having bounded total curvature in pseudo-Euclidean space" Russian Math. Surveys , 34 : 3 (1979) pp. 222–223 Uspekhi Mat. Nauk , 34 : 3 (1979) pp. 213–214 Zbl 0437.53054
[37] A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)
[38] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian) MR0346714 MR0244909 Zbl 0311.53067
[39] T. Klotz Milnor, "Efimov's theorem about complete immersed surfaces of negative curvature" Adv. in Math. , 8 : 3 (1972) pp. 474–543 MR301679
[40] D.V. Tunitskii, "On the regular isometric immersion in of unbounded domains of negative curvature" Mat. USSR Sb. , 62 : 1 (1989) pp. 121–138 Mat. Sb. , 134 : 1 (1987) pp. 119–134 MR912415
[41] Zh. Kaidasov, E.V. Shikin, "Isometric immersion in of a convex domain of the Lobachevskii plane containing two horocycles" Math. Notes , 39 : 4 (1986) pp. 335–340 Mat. Zametki , 39 : 4 (1986) pp. 612–617
[42] Zh. Kaidasov, E.V. Shikin, "Isometric immersion in of extended strips on manifolds of type " Math. Notes , 42 : 6 (1987) pp. 962–968 Mat. Zametki , 42 : 6 (1987) pp. 842–851
The Hilbert result that a surface of constant negative curvature cannot be imbedded isometrically in $ E ^ {3} $ needs $ C ^ {2} $ regularity. The general method of $ C ^ {1} $ isometric imbedding, [a1], [a2], applies also here.
[a1] N.H. Kuiper, "On -isometric imbeddings I" Indag. Math. , 17 : 4 (1955) pp. 545–556 MR75640
[a2] N.H. Kuiper, "On -isometric imbeddings II" Indag. Math. , 17 : 5 (1955) pp. 683–689 MR75640
[a3] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) MR0917479 Zbl 0629.53001
[a4] W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German) MR0474045 Zbl 0366.53001
[a5] D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1961) MR0939369 MR1528072 MR0036551 Zbl 0105.14707
[a6] A.V. Pogorelov, "Intrinsic geometry of surfaces" , Amer. Math. Soc. (1973) (Translated from Russian)
[a7] M. Spivak, "A comprehensive introduction to differential geometry" , 3; 5 , Publish or Perish (1975) pp. 1–5 MR0394453 MR0394452 MR0372756 Zbl 0306.53003 Zbl 0306.53002 Zbl 0306.53001
[a8] M. Gromov, "Partial differential relations" , Springer (1986) (Translated from Russian) MR0864505 Zbl 0651.53001
[a9] M. Gromov, "Embeddings and immersions in Riemannian geometry" Russian Math. Surveys , 25 : 5 (1970) pp. 1–58 Uspekhi Mat. Nauk , 25 : 5 (1970) pp. 3–62 Zbl 0222.53053 Zbl 0202.21004
How to Cite This Entry:
Negative curvature, surface of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_curvature,_surface_of&oldid=47952
This article was adapted from an original article by N.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Negative_curvature,_surface_of&oldid=47952"
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Kiersten Woodward, 16, of Martinsville, passed away as a result of injuries sustained from the accident on Monday.
MOORESVILLE, Ind. – The Mooresville community is grieving after a 16-year-old high school junior died Monday in a car accident, Fox59 reported.
Kiersten Woodward, 16, of Martinsville, passed away as a result of injuries sustained from the accident.
The crash happened Monday afternoon on Ind. 144 between Mooresville and Waverly.
Kiersten attended Waverly Elementary, Paul Hadley Middle School, and Herron High School. She transferred to Mooresville High School for her junior year.
Friends and family will gather to remember and celebrate the life of Kiersten Woodward Monday, December 10, 2018, 3 to 7 p.m., at the Tube Factory Art Space, located at 1125 S. Cruft Street, Indianapolis, IN 46203.
A Celebration of Kiersten's life will begin at 7 p.m. In lieu of flowers, memorial contributions may be made to the Peyton Manning Children's Hospital.
The story originally appeared at Fox59.com.


Started the practice of daswandh (tithe) - contributing one tenth of one's earnings for community purposes.
Author of Sukhmani Sahib bani - the Prayer for Peace.
Guru Hargobind put on two swords - one signifying miri (secular power) and other piri (spiritual power).
Defended the integrity of the Guru Granth Sahib by refusing to modify it's words.
Gurdwara Bangla Sahib in New Delhi was constructed in the Guru's memory. This is where the Guru stayed during his visit to Delhi.
Sacrificed his life upholding the "right to freedom of religion". Guru Tegh Bahadur was responsible for saving Kashmiri Hindu pandits who being persecuted by the Mughals, but had to lay down his own life to protect their freedom of religion.
Gurdwara Rakab Ganj Sahib in New Delhi is located where the Guru's body was cremated.
In 1699, Guru Gobind Singh baptized the Sikhs and created the Khalsa (the "Pure"). The Khalsa consists of Sikhs who have been baptized and dedicate themselves to living by the high standards of the Sikh Gurus at all times.
Instructed the Sikhs to keep the five K's.
All four of Guru Gobind Singh's sons were martyred by the Mughals.
Guru Amar Das not only preached the equality of people but also tried to foster the idea of women's equality. He tried to liberate women from the practices of purdah (wearing a veil) and preached strongly against the practice of sati (Hindu wife burning on her husband's funeral pyre).


The Emek Executive Administrative Staff, a committed and energetic group of dynamic educators, are committed to each child's development. They work together as a team to provide the backbone of the Emek educational platform. Guiding the faculty and staff, Rabbi Shifman and his team, Principal Rabbi Levi Solomon and Vice-Principals Mrs. Michelle Andron and Mrs. Esther Markel, work diligently every day to help map out a plan for success, child by child, balancing all aspects of development through a holistic approach. This group has been instrumental in the implementation and maximization of the "Cap-it Reading Program," Balanced Literacy, 21st Century Learning, CIJE and Project Kavod curriculum components. These cutting edge educational tools are impacting the entire Emek Community as they raise the bar for our students and for Jewish schools across the country.


Just last month, we reported on a project by Matt Stevens, a graphic designer with an addiction to Air Max 1. His MAX100 Book Project now has 24 days to go on KICKSTARTER and with 226 backers, the project looks very likely to reach its goal of $30,000 pledge. Graphically inclined sneakerheads should take a look, if you have not done so already, as the production of the book is dependent on each person who pledges to back this project to make it a reality. Here is an update of some new artworks that will be part of the pages in the book and expect 40 or so new illustrations to take up the rest of the book before printing.


text:
Bruce Most
Bruce has been a writer his entire adult life, spending most of his days in a small office in front of a computer (a typewriter before that, just to date himself). Not exactly dramatic, though he hopes the stories he creates are exciting to readers. He made a living as a freelance writer, writing fiction on the side. His articles appeared in Parade, TV Guide, American Way, Popular Mechanics, and other magazines. He also illustrated some of the articles with his own photography. He ghosted a self-help book and wrote over 1,000 articles on financial planning topics for the Financial Planning Association.
Catch the author at these sites
Website: https://www.brucewmost.com/
Amazon: https://www.amazon.com/dp/B07VT9Q76K?ref_=pe_3052080_276849420
Facebook: https://www.facebook.com/brucemostbooks
Twitter handle: @BruceWMost
Amazon Author Central: https://authorcentral.amazon.com/gp/profile
SP: What does your typical writing day look like? How do you work in your writing and promoting?
BM: I devote some of my mornings promoting my fiction (such as answering this Q&A) and afternoons researching and writing my fiction. Brainstorming occurs anytime I'm awake—and sometimes when I'm not awake.
SP: Tell me how and why you got into writing fiction? How long have you been doing it? And how do you go about creating a story? Elaborate here.
BM: I can point to no epiphany inspiring me to purse writing fiction, other than I loved stories from the time I was a kid and wanted to tell my own. I began writing bad short stories (and a never-finished novel) in my late teens and I have a box of colorful rejection slips to prove it. (This was back in pre-email days.)
As for how I create my novels and short stories, that's a big question. Ideas come from everywhere: things I read and see, research, personal experiences, even hobbies. For example, my first two published Ruby Dark mystery novels grew out of a local article on women bail bond agents. Another mystery (Rope Burn), set in contemporary Wyoming involving cattle rustling and murder, grew from having in-laws who work ranches and my having written magazine articles about modern-day cattle rustling (yep, it still goes on). I recently published two short crime stories based on a famous 1930s crime photographer (Weegee) due to my interest in photography.
As for my writing process, I fall between being a pantser and a plotter. Starting with the germ of an idea, I do enough research and brainstorming (and continue both as I go along) so I have at least a rudimentary idea of plot and characters. But I don't exhaustively plot every scene and character in advance. I don't like to lock myself into an outline because I prefer to discover characters or plot elements in the act of drafting. It feels more life-like. In fact, in two of my mysteries, minor characters walked into a scene and completely changed the plot of the books.
SP: Tell me about the protagonist and antagonist of your latest book. What are their goals in the story?
BM: My latest mystery is The Big Dive, the sequel to my award-winning Murder on the Tracks (published by Black Opal Books). The protagonist is Joe Stryker, a street cop in 1951 Denver. For the second time in Joe's headline-grabbing police career, a partner is brutally slain in the line of duty, almost in front of Joe's eyes. How did the killer pull off such a brazen murder and escape? And why was Joe's "choir boy" partner murdered while burglarizing a pawnshop? As in his debut novel, Joe investigates clandestinely outside of official police oversight. Which creates plenty of antagonists beyond the killer. To protect his dead partner's reputation and save his own career while trying to solve a series of murders, Joe conceals his investigation from his new partner, his superiors, even his wife, who hates his being a cop. His investigation leads him to a ring of dirty cops—and a dark secret that goes back to the shameful internment of Japanese-American citizens in relocation camps during World War II. All while dodging a homicide detective hell-bent to pin the crimes on Joe.
SP: Which do you prefer, character driven stories or plot driven stories and why should readers care? Which do you write?
BM: To me, this is not be an either/or question. A book that is not both character and plot driven is likely a boring book. Granted, my books are murder mysteries. Genres generally lean heavily toward plot. I stack up bodies and tense plot moments, to be sure. But my protagonists are amateur sleuths sucked into the mystery, often reluctantly, for deeply personal reasons. (My street cop is not a homicide detective and thus operates well out of his assigned patrol duties.) Consequently, my protagonists' motives and decisions drive the plot. As for whether character or plot comes first in the creative process, it varies from book to book. I've started with the germ of a character (a woman bail bond agent) and with the germ of a plot (cattle rustling). Regardless, in the end character and plot both serve the story.
SP: Tell me about your story point of view preferences, tense preferences, and any other thing of interest to you along these lines as we delve further into the mechanics of creating a story.
BM: My books are either in first-person POV or close third, depending on what feels right for that character, the tone I want to convey, and plot demands. Occasionally, out of plot necessity, I'll do a scene or two in third-person POV from a secondary character. I always write in past tense.
SP: What are your all-time favorite books – top three? Why? Don't skimp here. Explore it with us? Tell us what genre they are. Who is your favorite author, if different from your favorite book?
BM: Tough question. While I've long read mysteries, I've been most inspired by one of the founders of hard-boiled detective fiction, Raymond Chandler, particularly Farewell, My Lovely. I love his imagery, characterization, and tone of writing. My Joe Stryker novels are homages to him. J.R.R. Tolkein's Lord of the Rings trilogy (actually written as one book) is my favorite fantasy book. It remains a powerful story about good, evil, and how we as humans are caught between them. As for looking into the darkest elements of our soul, Joseph Conrad's Heart of Darkness has stuck with me since I read it in college.
SP: What do you find is the most difficult thing about writing fiction? The easiest thing? Elaborate.
BM: Not sure I would ever use any word resembling "easy" when it comes to writing fiction. The most frustrating is that I'm a slow writer. It's rare I crank out page after page in an hour or even an afternoon. I'm not a perfectionist in early drafts, but it's difficult to just splash down words from beginning to end without looking back and fiddling as I go along. I need to get off to a good start or the rest of the story will veer off into no-man's land, never to be seen again. I'm an obsessive rewriter as well. I believe my books are better for it, but the process can be painful. If anything does come easy, it's writing dialogue. Writing description is agony. Despite the challenges, I look forward every day to writing.
SP: Tell me about your travels. Where is an interesting place you've been? Do you travel to research your stories, or is research important to you for your stories?
BM: I've traveled across much of the world, from Asia to Europe, mostly for pleasure. I even met my future wife while traveling. Because most of my novels are set in Denver, Colorado, my research is mostly local. For Rope Burn I traveled to Wyoming ranch country. I've done research in New York City for my Weegee stories. I have an idea for a Vietnam War mystery and would travel to Vietnam if I get to the book (remember that slow writer challenge?). Research is vital to my stories, not only for authenticity but as inspiration for plot and characters.
SP: What kind of stories do you refuse to read?
BM: Romance, though I must shamefully confess my wife coerced me into reading The Bridges of Madison County (I could feel the steam coming off the pages).
SP: Finally, the most important question – DC or Marvel? Or does that even matter to you? What is your favorite comic book character ever? If none, just make something up!
BM: Marvel, for producing better movies than DC. Batman is my favorite comic book character. I love his dark side.
The Big Dive
"Benedict," I said loudly.
No sounds. Even the mice weren't moving.
I held my flashlight out at arm's length and flicked it on. No shots rang out.
I was in a backroom. A desk with ledgers. Pawn tickets pinned to a wall along with a pinup calendar two years out of date. Shelves cluttered with hocked items not ready for sale. A small workbench for repairing pawn. A wall clock ticked 10:44. A whiff of cleaning solvents and scorched coffee mixed with the smell of my own sweat.
I crept to the open doorway between the backroom and the showroom. An exterior neon sign at the front of the pawnshop flickered red and green off two scratched glass cases of jewelry running down the center of the room. Guns, musical instruments, army surplus, toasters, sewing machines, radios and record players, brass table lamps, clocks, leather jackets, and a mish-mash of other pawn crammed the narrow aisles and cluttered the walls. A car passed on Curtis, headlights momentarily brightening the interior, highlighting a shattered side case.
"Benedict!" Only my raw breathing answered.
The cold heaviness of my gun filled my hand.
My flashlight combed the shadows for the slightest movement. My ears alert for the faintest of sounds, for breathing, for any hint Benedict was okay—or someone lying in wait.
I sucked in another deep breath and stepped into the room, ducking behind the end of one of the center cases. I panned my flashlight across the side case with the broken glass. The beam caught a small pile of watches scattered on the linoleum floor, spotted with a dark liquid.
My flashlight moved, then froze. The sole of a heavy shoe stared at me several feet beyond the watches. I inched the light. A second sole, this one worn unevenly along one side of the shoe. My hand shook as I panned the light up the legs. The body lay sprawled on its back. A uniformed body. Something long and wicked protruded from the middle of the chest.
"Benedict!"
I scrambled to his side and touched his neck for a pulse. A sticky wetness clung to my fingers. I jerked my hand back and shined my flashlight on his neck.
Someone—someone hateful and ugly—had slashed my partner's throat.
I focused my flashlight on his chest. The killer had buried a large, black-handled knife in it. I pressed a wrist with my fingertips, holding no hope of finding a pulse. Blood drained in all directions, even under the glass case. I put my ear to his mouth and my hand on his chest, avoiding the knife sticking out of it, straining to hear a sound, feel a beat.
Nothing but my own desperate breathing.
I felt his dime-store notebook in his breast pocket, the one he'd scribbled in earlier in the evening. I tugged it out. The knife had narrowly missed piercing it. Blood stained the edges of the pages. I slipped it into my pocket next to my own notebook, and slumped against the glass case. The backroom clock ticked distantly, slightly out of rhythm.
It was happening again.
Dear god, it was happening again!

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Sorenson Media hired a new CEO, Marcus Liassides, with big plans to change Sorenson from a sleepy encoding company to a media powerhouse. I intervewed Marcus on Google Hangouts, which you can watch below. Or, you can read a summary of the interview on streamingmedia.com.
The second session of my afternoon workshop covered closed captioning. Here's the description; click over to the main article to download the handout.


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Cant say if it was the starfall or the creatures.. or maybe this is another one of those early thurst marks in the series lol… For crack doors, you have to crack the whole.dont care if i delete it or not i want it:..; 26 singles; 16 doubles; no triple-doubles.
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Note: Each year participants in the Masters, U.S. Open, and the PGA Championship are given the option to "opt out" of competing in certain events. Players who opt out of the Masters will not be eligible to win that tournament, except for winners of the competition's "last opportunity qualifier".
Finished as the runner-up at the 2006 PGA Championship, a major.
Finalist of the 2004 Masters Tournament, a major.
Won the 2002 Kraft Nabisco Championship, a major.
Won the 2001 Sony Open in both strokes-made and scoring-average categories.
Won the 2000 Sony Open in scoring-average and winner of the last opportunity qualifier.
Won the 2000 PGA Grand Slam of Golf (Masters, U.S. Open, and the PGA Championship) in consecutive seasons.
Won the 1997 BellSouth Classic in a three-way playoff.
Won the 1996 Bob Hope Chrysler Classic.
Won the 1995 Ben Hogan Greater Hartford Open.
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The Nacho Monreal Appreciation Society
New Author 14th January 2018 Arsenal, Exclusive, fans, tactics, wengerTagged defender, malaga, monreal, nacho, spanishLeave a Comment on The Nacho Monreal Appreciation Society
Whisper it, but sometimes, just sometimes, karma does dish out its own form of retribution. Occasionally, the people who perform above and beyond the call of duty receive their just desserts.
Step forward Nacho Monreal.
With the recent news that the Spanish full-back has been awarded a contract extension, it would seem that if you need evidence that people get what they deserve, this is it.
Nacho Monreal during this season has been the epitome of consistency. He has been at the fulcrum of excellence, the veritable cream that has risen to the top. The defender who joined in 2013 for a paltry £7m has simply taken on all obstacles and grown game by game. He is the poster boy for what can be achieved with a wholly committed attitude.
It wasn't always this way though. Nacho wasn't always so accomplished.
Monreal's first season was always going to be difficult. Any player who joins a new club in the middle of the season – Nacho joined in the January window of the 2013/14 season – will struggle to force himself into proceedings. If you also consider that this was a player who had spent his entire career in the technically sound but physically undemanding La Liga, then his beginnings were always going to err towards a toilsome time.
Monreal gamely played through the strain, but the stamina levels and awareness towards his defensive needs weren't always so sharp. Indeed, his first season was such hardship that teams were targeting his flank as Nacho floundered with the Premier League and its own set of rigours.
So many times, he was caught out on the attack. Even when hemmed in defence, he more often than not had to contend with two players on the rampage. Word had spread that this new Spanish defender who resembled Peter Crouch's shorter brother was easy pickings. Kieran Gibbs – his positional rival – took the reins for the majority and the season's showpiece – the F.A Cup Final versus Hull City – would be enjoyed from the bench by Nacho.
The summer break could not come quickly enough for Monreal. Surely, after such a tough welcoming to the Premiership, the Spaniard must have been doubting whether a move away from Spain was the right move?
The 2014/15 season was Nacho's rebirth.
A crisis is either the chance a person takes or they let it consume them. Nacho hails from the former category. A lack of defensive bodies saw first Mathieu Debuchy and then Monreal fill in at centre-back as Arsene Wenger utilised all the players at his disposal. Bodies were thin on the line. Surely the waif-like figure of Monreal would be bait to the majority of strikers? This would be the death knell of Arsene Wenger some thought.
I, along with the majority of Gooners, wondered whether Monreal was up to it. He was just starting to find his feet in his preferred role, and now he was expected to fill in at centre-back?
We should never have let it trouble us. He was an excellent partner for Mertesacker and his assured displays showed his defensive nous. He had everything necessary for a top-flight defender. He just had to acclimatise.
Since his short time as Mertesacker's partner in the centre, Nacho has been sensational. In fact, it would be fair to say he now could claim to be the Premier League's best left-back. Of course, this article is biased – but what player comes remotely close?
His postioning is exemplary, and he adjusts this from game to game dependant on who his partner on the left is. His all-round awareness is fantastic. When faced with a winger intent on going past, the sight of Monreal first tackling the opposition and then with his head up looking to effectively distribute is becoming a regularity.
He is difficult to pass. He is one of the best outlets when under pressure. He can even compete aerially. Quite frankly, Nacho Monreal ranks as the best left-back Arsenal have had since Ashley Cole.
In fact it wouldn't be a huge leap to suggest that our Nacho is among the best in his position in Europe.
He more than deserves his contract extension. Such is his stock currently, he could quite easily have sounded out some major European heavyweight team for a cash-packed move. The fact he is more than happy to stay says everything for the team spirit and Monreal's own happiness.
Long may it continue. The man who has suffered the most during Monreal's ascendance – Kieran Gibbs – could do worse than take some notes when Nacho is training. If he could pick up the habits that the Spanish defender has bred since joining, then Gibbs would be in an excellent position to take up the mantle when Monreal inevitably looks to home as his body yearns for a slower pace in a few years.
As it is, our club can boast of a defender that could hold the key to keeping quiet the worlds deadliest strikeforce – Messi, Suarez and Neymar. With his consistently high level, Nacho has made a rod for his own back. These high demands are all his own doing.
Arsenal 0-0 Hull City – A Needless Exercise In Frustration
Arsenal Vs Barcelona – Champions League Match Preview


Miami University is pleased to offer merit scholarships to academically accomplished first-year, full-time undergraduate students who enroll on the Oxford campus for Fall 2019. The application for admission is used to determine merit scholarship eligibility — no additional form is required. Priority consideration for admission, merit scholarships, and selection for the University Academic Scholars Program, the University Honors Program, and other competitive admission programs will be given to students completing all application requirements by December 1, 2018. Full consideration will be given to students who apply by February 1, 2019.
Admission to select programs in the College of Creative Arts is available fall semester only.
This binding option is intended for students who are willing to commit themselves to Miami University as their first college choice. Students may apply to only one college as an Early Decision candidate. Applicants contractually agree to withdraw all applications to other colleges and attend Miami if accepted.
Unlike Early Decision, this option is non-binding. Early Action candidates may apply to as many colleges as they wish.
As with Early Action, the Regular Decision option is non-binding. Candidates may apply to as many colleges as they wish.
* Students applying Early Decision are strongly encouraged to submit the FAFSA by December 1, 2018.
** This option is similar to Regular Decision but with an earlier deadline (November 1) and notification date (December 15).
*** This option is similar to Regular Decision but with an earlier deadline (December 1) and notification date (February 1).
Students offered admission have the right to wait until May 1 to respond to an offer of admission and/or financial aid. For more information regarding the national confirmation deadline and other policies, please visit the NACAC website.


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The Joshua Tree Boulder House by Garett Carlson
Garett Carlson, a landscape architect in Southern California, has designed the Joshua Tree Boulder House located on 2.5 acres in Joshua Tree, California.
Description of the Joshua Tree Boulder House by Garett Carlson
The Joshua Tree Boulder House proves imagination is still alive and well. Every inch of this 1700 square foot, 2-bed / 2-bath home is intentional, and every design element is deliberate. This magical creation sits on 2.5 acres in Joshua Tree, California.
The house is the brainchild of W. Garett Carlson, ASLA, a licensed landscape architect for more than 30 years (www.wgarettcarlson.com). His landscape designs grace Hollywood's recently remodelled Sunset Marquis Hotel, as well as the gardens of celebrities such as Jack Nicholson, Goldie Hawn, Johnny Depp, Blake Edwards, and more.
Having studied architecture as well as landscape architecture, designing a house has been a long-held ambition for Garett Carlson. His concept for the Boulder House came from 'imagining what the house would look like if it emerged from the ground.' The house is at one with the land in such a way that it is almost invisible as you approach it.
With Joshua Tree National Park on the doorstep, inspiration for the Boulder House is obvious. "Joshua Tree has some of the most fascinating boulder shapes anywhere in the world," explains Carlson. Painstakingly built formations of faux boulders appear to rise from the desert floor embracing and sheltering the west side of the house, and providing welcome shade from the late day heat.
The house was built with sustainability in mind, using an intriguing palette of metal, wood, concrete and glass to create a peaceful modern living space where traditional boundaries are dismissed. Ten-foot high, custom-made floor-to-ceiling pocket glass doors open to create a 40 feet wide expanse that connects the living area to the landscape. A fire pit to one side of the patio, and a water feature on the other side are both set within boulders to further enhance the experience. This is indoor/outdoor living at its best.
Visit Garett Carlson's website – here.
The house is currently for sale for $975,000, visit the website here for more info.


"Use of superoxide as an electron shuttle for iron acquisition by the m" by Andrew L. Rose, Tim P. Salmon et al.
Rose, AL, Salmon, TP, Lukondeh, T, Neilan, BA & Waite, TD 2005, 'Use of superoxide as an electron shuttle for iron acquisition by the marine cyanobacterium Lyngbya majuscula', Environmental Science & Technology, vol. 39, no. 10, pp. 3708-3715.
Reduction of iron from the ferric state to the ferrous state is one strategy employed by microorganisms in near-neutral environments to increase its biological availability. In recent years, the existence of mobile reducing agents produced by microorganisms to promote iron reduction, known as electron shuttles, has been demonstrated. Production of electron shuttles has been shown for several organisms, employing a variety of mostly organic molecules as the electron carrier. Here we show that the coastal cyanobacterium Lyngbya majuscula produces iron-reducing superoxide radicals (O2•-) and that this facilitates increased iron uptake. We suggest that superoxide is a useful electron shuttle because it reacts rapidly and almost indiscriminately with Fe(III)−organic complexes and its precursor, dissolved oxygen, is ubiquitous in the photic zone. We further suggest that, for these reasons, the generation of superoxide by marine oxygenic photosynthetic microorganisms and its use in facilitating iron uptake may be a reasonably widespread process.


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Home > About Ford > Media > Newsroom > 2018 > The All New Compact Utility Vehicle from Ford Debuts in India
Ford Freestyle: The All-New Compact Utility Vehicle from Ford Debuts in India
Ford's latest global product combines SUV-like design, benchmark performance, innovative in-car technology & outstanding fuel efficiency
The all-new Freestyle strengthens Ford's fun-to-drive credentials with an all-new 1.2L three-cylinder petrol engine, offering best-in-class 96PS peak power and 120 Nm torque
Clever and connected vehicle features Ford's award-winning SYNC 3 system, with a high-resolution 6.5-inch touchscreen
Strengthens Ford's diverse product portfolio in India to seize opportunities generated by fast-growing demand for SUV-like vehicles; Freestyle to be sold in India starting Q2-2018
NEW DELHI, India, January 31, 2018 – Ford today unveiled its latest global product offering – The All-New Ford Freestyle, a cool, capable & connected, compact utility vehicle (CUV) for Indian consumers.
The all-new product from Ford combines SUV-like robust design, benchmark performance, innovative technologies and outstanding fuel efficiency – for greater choice to consumers in India and markets around the world. India will be amongst the first across the globe where the all-new Freestyle will go on sale in the second quarter of 2018. "The global unveil of the Ford Freestyle strengthens Ford's continued commitment to India to bring products that consumers want and value," said Anurag Mehrotra, president and managing director, Ford India. "With the cool, capable and connected Ford Freestyle, we are adding yet another great product to our diverse portfolio that will help us expand our family of happy Ford owners in India." The all-new Freestyle will position Ford to address growing consumers demand for SUV-like vehicles. Such vehicles are being preferred owing to their commanding seating position & better ground clearance to tackle congested urban environment. Compact SUVs alone, a segment that Ford pioneered with the EcoSport, have consistently outpaced overall automotive sales in India, growing more than three-fold between 2013 and 2016. In 2017 alone, compact SUV sales in India grew 37% compared to 9% in total vehicle sales. "Just like we did with the EcoSport, we see a lot of potential in offering this compact utility vehicle to meet the SUV owning aspirations of first-time buyers or young achievers who are looking to upgrade their small cars," Mehrotra added.
The all-new Ford Freestyle embodies the aspirations of free-spirited buyers who are uninhibited and love to live life to the fullest. These consumers aren't afraid to take their chances and are driven by passion.
Their motto in life is to 'Do Everything, & Regret Nothing'.
"Freestyle is more than just an expression. This powerful word immediately encourages you to push the boundaries and do more, just like the product" said Rahul Gautam, vice president for Marketing at Ford India, said.
Cool & Compact
Featuring a striking-look, and a commanding stance, the Ford Freestyle uses design elements common to more expensive mini SUVs.
And it all begins with a unique grille with a dynamic, three-dimensional mesh.
Sweeping straight lines moving from front to the rear give the Ford Freestyle robust, muscular, & SUV looks. The front bumper of the vehicle has an integrated skid plate and sculpted fascia sections, that create a protective full-width beam element, accentuating Freestyle's SUV looks and credentials.
The sculpted rear fascia conveys Freestyle's athletic stance while precision detailing in the tail lamps ensure every colour is noticed distinctly.
The blacked-out high contrast headlamps in Freestyle, complemented by fog lamp bezel, create an assertive and commanding expression. While an inclined windshield gives the vehicle a sleek, fast profile that communicates a high-end image.
Capable & Safe
The Freestyle will come paired with Ford's all-new, three-cylinder 1.2L TiVCT petrol engine that is small, light, strong and delivers outstanding fuel efficiency. The naturally aspirated engine will generate best-in-class 96 PS of peak power and 120 Nm of torque to deliver Ford's fun-to-drive DNA. Consumers will continue to have a choice to opt for Ford's trusted 1.5L TDCi diesel engine that generates class-leading 100 PS peak power and 215 Nm of torque.
Freestyle will deliver sportier performance in all driving conditions and will feature a uniquely tuned suspension with increased track-width, high ground clearance, TCS, ABS with EBD and uniquely tuned EPAS among others.
The Freestyle will continue Ford's leadership on Safety, offering up to six airbags.
With the introduction of Active Rollover Prevention (ARP), Freestyle is designed to perform safely in extreme situations, without compromising on the fun-to-drive elements. Combined with Electronic Stability Control, the ARP system will be able to automatically detect potential rollover situations by applying brakes to individual tyres and decrease engine torque.
Clever & Connected
The Ford Freestyle will also feature Ford's in-car infotainment system, SYNC 3 with a 6.5- inch touchscreen which allows drivers to control their entertainment and their connected smartphone with conversational voice commands.
SYNC 3 system is also Apple CarPlay and Android Auto compatible.
"Freestyle represents our vision of a compact utility vehicle that delivers more on every count and is all set to surpass expectations of consumers who want and expect more," said Vinay Raina, executive director of Marketing, Sales and Service at Ford India. "With Freestyle, we are sure to delight consumers with a combination of compelling and capable features."
Kapil Sharma Nikhil Gulati
ksharma6@ford.com gnikhil1@ford.com


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Historical Reincarnation
Researchers Specializing in Reincarnation
Karma and Traumas
Traumas, Epigenetics, and Reincarnation
Group of Souls, Recognition Memories
Soul-Entity Evolution
AUMtherapy
Past-Life Therapy
Past-Life Based Coaching
A Brief Introduction to Kiera's Story
In the course of my research, I contacted various renowned researchers in the field of reincarnation in the United States, such as Walter Semkiw (2003), who worked with the psychic Kevin Ryerson, and the late Paul Von Ward (2008), who based their work on Ian Stevenson's.
Ian Stevenson and Child Case of Reincarnation Types (CORT)
Ian Stevenson (1918-2017) was a psychiatrist who founded the Division of Perceptual Studies at the University of Virginia and worked intensively on reincarnation. He based his work on investigators like Frederic W. H. Myers (1903) and their methods, which he then improved upon. Stevenson created the Cases of Reincarnation Type (CORT) based on this investigation. His intention was to give evidence of the survival of the soul, which he called the psychophore, to separate it from religious terminology. He "apparently visualized the memories and DNA patterns revealed by his research being transferred between lifetimes in such a container" (Von Ward, 2008, p. 71). Paul Von Ward (1939-2017) an independent scholar and researcher who worked on cosmology and reincarnation (2008), preferred to use the word psychoplasm to describe both the container and its contents.
Stevenson (1986) developed a methodology regarding the collection of CORT data and observations of unexpected behaviors concordant with the previous life, but not with the current one. He organized his data into tables, but did not really study the psychology of the children recalling previous lives. Rather,
Stevenson (2001) described the typical features of the cases as being: statements from the point of view of a past life; behavior related to past-life memories, including phobias (Stevenson, 1990a) and philias; recognitions of people, places or objects; and birthmarks or birth defects related to wounds experienced by the previous personality (Mills & Tucker, 2015, p. 314).
Stevenson reported that around two thirds of his CORTs were "solved" because the information that the children gave regarding their previous names and/or where they lived was verified and confirmed. However, he "compared solved and unsolved cases, noting that a violent cause of death was even more prevalent in unsolved cases, but that the cause of death is violent or premature in a high proportion of solved cases as well" (Mills & Tucker, 2015, p. 318).
According to Stevenson (1986), the identification to the previous life is one of the strongest signs of reincarnation (p. 28). For him, "Such a personification, accompanied by extremely sensitive reactions, is more valuable than simply listing information given by the child about another person who has lived previously" (Stevenson, 1986, p. 28).
Walter Semkiw and Kevin Ryerson
Walter Semkiw (2003) is a retired physician who wrote several books, and eventually built a major website focusing on reincarnation, on which he shares all his research: https://www.reincarnationresearch.com. He emphasizes Stevenson's work with the similarities of facial and physical features and personality, innate talents, and even linguistic writing style that a soul-entity recalls and hopefully improves from lifetime to lifetime. Semkiw uses these three criteria "to establish past-life matches: facial architecture, personality traits such as demeanor or habits, and a membership in a soul cohort. He later added a fourth: confirmation by a being known as Ahtun Re accessed by trance-channel Kevin Ryerson" (Von Ward, 2008, p. 77) whom he works with.
Paul Von Ward and The Soul Genome
According to Von Ward (2008), "If nature has developed in a cell the capacity to pass into the future the results of its experience, then it must also have evolved the same cumulative capacity in the cell's human host" (p. 69).
Like Semkiw, Von Ward's analysis also suggested that physical features, specific memories, cognitive styles, interpersonal traits, emotional profiles, and talents carry forward from one lifetime to another through the psychoplasm. According to Von Ward (2008), "to support the psychoplasm hypothesis, a researcher must reasonably demonstrate that the correspondences in proposed past-life matches cannot be attributed to simple chance or other self-evident causes" (p. 82). In order to verify and document these similarities, Von Ward's project designed procedures to measure and validate all these areas of evidence and determine if their coincidence in two lifetimes could be attributed to chance or something more solid and real. For Von Ward (2008), reincarnation phenomenon was simple: "if reincarnation is real, it involves your real life" (p. 83).
After reading Von Ward's book The Soul Genome: Science and Reincarnation just after its release in 2008, I used his scientific protocols based on five scientific factors to assess the evidence I had accumulated in my research and any new data that was required. Von Ward agreed to help me evaluate its validity and reliability after he realized the scope of my research and its meticulous documentation.
The factors Von Ward used to measure the strength of alleged reincarnation cases are the following:
Genotype / Phenotype that corresponds to the physical characteristics of the individual.
Cognitive Cerebrotypes that represent the mental styles and capacities of the individual.
Emotional Egotypes speaks for itself and is about emotions. It "deals with strength and orientation of the basic ego structure" (p. 96).
Social Personatype determines how the individual relates to others.
Creative and Behavioral Perfomatypes talk about interests, hobbies and behaviors.
According to Von Ward (2008), "Alone [similarities] do not prove a past-life match, the the sheer volume of them makes the linear reincarnation case more persuasive" (p. 186). This confirmation helped Kiera to integrate even more her past-life and reach another level in herself and in her consciousness.
Edgar Cayce and the Akashic Records
The Akashic Records correspond to a celestial ether containing all the memories of human beings. They are accessible to anyone under certain circumstances, but particularly so to a person who has advanced psychic abilities. Edgar Cayce was one such man. Very early in his life, long before he was convinced about the fact of reincarnation, he was able obtain the great majority of his information about past lives by reading the Akashic Records. After many confirmations of the existence of reincarnation, when Cayce was asked what would convince him, he answered "an experience" (Cayce, 2006, p. 23). For him indeed, reincarnation had practical application and showed great continuity between lifetimes (Kirkpatrick, 2000, p. 288).
The experiences of past and present lives that have generated suffering, as well as all the thoughts, emotions and desires, are also contained in the individual's personal subconscious, where nothing is forgotten. As Cayce scholar, Kevin Todeschi (2013) puts it: "Since each individual is the sum total of all of his or her previous experiences, fragments of past-life memories frequently come to the surface in everyday life" (p. 56).
Matlock and Signs of Reincarnation (SOR)
James Matlock, an anthropologist and reincarnation researcher who also based his work on Stevenson's, wrote the book Signs of Reincarnation in 2019. He suggests that psi, "a term that embraces both extrasensory perception (ESP) and psychokinesis (PK)" (Matlock, 2019, p. 300), allows for the transmission of physical and behavioral traits through the action of the soul or reincarnating mind. These include the persistence after death of personality, emotion, memory, cognitive ability and the exercise of free will (Matlock, 2019, p. 246).
For Matlock, the principal reincarnation signs may be grouped into three classes:
autobiographical memories of previous lives, along with related recognition memories and spatial memories;
behavioral signs, including emotional memories and procedural memories;
physical signs like birthmarks, birth defects, and other congenital abnormalities (p. 303).
The co-occurrence of these reincarnational signs and the coherence of past-and present-life stories are significant and show the strength of a reincarnation case if there is sufficient evidence. Kiera's case was one of these, as it showed numerous signs of reincarnation as developed by Matlock (2019, p. 303):
significant impact of her previous life on her present life,
behavioral and physical signs that belonged to her previous life,
identification with her previous life and spontaneously talking about it in the first person while trying to avoid it,
spontaneous detailed memories,
recognition of people and locations
phobia of darkness related to the death in her previous life,
continuity between the two lives,
soul evolution.
However, Matlock has a different perspective on karma than mine, as he is not convinced in his research that there is such a thing as "juridical karma" – the idea that where we are reborn, and what happens to us in a subsequent life, is determined by an external, lawful force that is the product of the moral qualities of our actions and intentions in this and earlier lives. Matlock argues that the only type of karma that has been demonstrated is "processual karma," which is the carry-over from life-to-life of memory, emotions, behavior dispositions, and other attributes that inform personality.
Nevertheless, if Cayce's life readings were at all accurate, as I believe they were, there is without question such a thing as juridical karma. I believe that the reason for Matlock's skepticism about juridical karma is that his reincarnation research deals overwhelmingly with the past life claims of young children, who could not reasonably be expected to even mention karma, let alone be knowledgeable about that concept. It is remarkable enough that some young children would recall their most recent lifetime, but it seems highly illogical that they would also know why they experienced what they did from the point of view of cause and effect.
Cayce, E. (2006). Reincarnation & Karma. VA, Virginia Beach: A.R.E. Press.
Kirkpatrick, S. D. (2000). An American prophet. Rivehead Books.
Matlock, J. G. (2019). Signs of reincarnation: Exploring beliefs, cases, and theory. The Rowman & Littlefield Publishing Group.
Mills A. & Tucker, J. B. (2015). Reincarnation. In Cardena E., Palmer, J. & Marcusson-Clavertz, Parapsychology. A Handbook for the 21st Century. (pp. 314-326). McFarland & Company, Inc. Publishers.
Todeschi, K. J. (2011). Edgar Cayce on reincarnation and family karma. A.R.E. Press
Stevenson, I. (1985). 20 cas suggérant le phénomène de reincarnation. J'ai Lu. (translation)
Todeschi, K. J. (2013). Edgar Cayce on the Akashic Records. A.R.E. Press.
Von Ward, P. (2008). The soul genome: Science and reincarnation. Fenestra Books.


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HKUHKU PortalAlumni
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Overview of Group Research Activities
The Public Health and Healthy Ageing Research Group is committed to protecting and improving oral health for all people by performing and synthesising clinical, community-level, and educational research, and by applying public health sciences.
Members of the group come from various disciplines and specialties, including comprehensive dental care, dental education and e-learning, dental public health, dental materials science, oral biosciences, oral diagnosis and polyclinics, oral rehabilitation, orthodontics, paediatric dentistry, and periodontology. A number of the group's members are internationally renowned.
The Public Health and Healthy Ageing Research Group is actively involved with The University of Hong Kong's Strategic Research Area of "The Community" and its constituent themes of "Public Health" and "Sciences of Learning". A key operating principle of the group is to encourage research collaboration locally, regionally, and internationally. We have strong links with the HKU Li Ka Shing Faculty of Medicine, HKU Faculty of Education, and HKU Faculty of Social Sciences. We also extensively collaborate with the School of Public Health of the HKU Li Ka Shing Faculty of Medicine.
Our principal research Focuses
Clinical epidemiology (population-level features) of oral diseases and conditions
Clinical trials on innovative methods of preventing and treating oral diseases and conditions (including the use of traditional Chinese medicine)
Clinical communication and oral health literacy
Dental education
Oral health–related quality of life and subjective assessments of oral health
Oral health care services, including issues related to human resources and utilisation of services
Special care dentistry (dentistry for special-needs groups)
To receive further information or to enquire about collaborative research, please do not hesitate to contact the research group convenor, Professor M.C.M. Wong (e-mail: mcmwong@hkucc.hku.hk). For a selection of recent research published by the Public Health and Healthy Ageing Research Group, see our "Research Synopses".
Members of the Research Group
Group Convenor
Professor M.C.M. Wong
Primary Members
Ms. R.T.M. Chan
Dr D. Duangthip
Dr X.L. Gao
Dr S.M.L. Lai
Dr A.H.C. Lee
Dr G.H.M. Lee
Dr K.C.M. Leung
Ms J.L.K. Leung
Dr C.S.Y. Liu
Dr P. Liu
Professor E.C.M. Lo
Professor C.P. McGrath
Dr S. Peng
Dr A. Perfecto
Dr E.H.N. Pow
Dr E.Y.C. Tsang
Dr G.H.M. Wong
Professor M. Bornstein
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Hello, Re-fashion Co-Op! This is my first post on this site and I'm really excited to be sharing my refashions with so many other awesome refashionistas :) I've been blogging at Accidental Seamstress for 2 years now and I'm officially hooked on DIY fashion. My name is Laura and I'd like to share my "Memorial Day Dress" with you first.
I got this awesomely ugly 90s dress for only .99 cents. That's my "rule." I never go over $1 when I'm scouting out refashions.
This just needed a few tweeks to make it "cookout" worthy for the holiday. I put it on my homemade dress form and took it in. I also raised the shoulders a little bit, and I seam ripped off those horrible side ties and decorative buttons.
I finished that raw bottom hem with my favorite hem of all time: A rolled hem.
Easy peasy. And, this one is easily a new favorite :) If you'd like to see a more detailed description of what I did, feel free to see my original post.
Welcome to the Refashion Co-op! I look forward to more of your refashions! I'm so jealous that you're able to find great finds for $1 or less -- I have a hard time finding great steals likes that. Great refashion -- I would wear the dress all the time.
Looks great, well done! I can only join Jennifer in being jealous about $1.00 finds. Looking forward to more of your work!
Love your ''new'' dress. I have ugly 90s dress as well , even little similar, you gave me nice idea, thank you!


If you remember, the DLC was revealed earlier in the year and brings with it tons of content and customization items!
Nintendo also has confirmed that their unique team-based ink shooter will receive support and content until December 2018 and have shown off a new 3 pack of amiibo!
What do you think about the Octo Expansion? Is it releasing earlier than you thought? What about those amiibos? Comment down below and remember to follow The Loot Gaming for all your gaming news and reviews and on everything from E3! And I'll see you on the Inkling battlefield.
P.S. for those on the fence on whether or not to pick up this rad colorful shooter, check out The Loot Gaming's review!
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