Showing papers on "Riemann curvature tensor published in 1982"

New examples of manifolds with strictly positive curvature

Abstract: Berger [-3], Wallach [10] and Berard Bergery [2] have classified all simply connected smooth manifolds which allow a homogeneous Riemannian metric of strictly positive curvature. Besides the rank one symmetric spaces there exist five exceptional manifolds and an infinite series of 7-manifolds of distinct homotopy type which have been studied by Aloft and Wallach [1]. These are diffeomorphic to Mpq:=SU(3)/Upq, where p, q are positive integers and Upq is the one-parameter subgroup of diagonal matrices

Axial isometries of manifolds of non-positive curvature

Abstract: Let M be a complete C ~~ Riemannian manifold of non-positive sectional curvature. We say that a geodesic 9: IR~ M bounds a fiat strip of width c > 0 (a fiat half plane) if there is a totally geodesic, isometric immersion i: [0, c) x IR~M(i: [0, oo) x IR~M) such that i(0, t) = 9(0. A 9eodesic without fiat strip (without fiat half plane) is a geodesic, which does not bound a flat strip (a flat half plane). We will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of M. In fact, many of the properties of a manifold of strictly negative curvature (resp. of a visibility manifold) still remain true if one assumes only the existence of a closed geodesic without flat half plane. We will discuss the existence of free (non-Abelian) subgroups of gl(M), the existence of infinitely many closed geodesics, the density of closed geodesics, and a transitivity property of the geodesic flow. It is, therefore, interesting to give conditions which ensure the existence of a closed geodesic without flat half plane. We will prove that M has a closed geodesic without flat half plane if vol(M)< oo and if M contains a geodesic without flat half plane. Note that a geodesic is not boundary of a flat strip (and a fortiori not boundary of a flat half plane) if it passes through a point p e M such that the sectional curvature of all tangent planes at p is negative. In the proofs of our results we investigate the action of rtl(M ) as group of isometries on the universal covering space H of M. In the proofs of many of our results we do not use the fact that this action is properly discontinuous and free. We, therefore, formulate these results for arbitrary groups D of isometries of H. The paper is organized as follows: In Sect. 1 we fix some definitions and notations and quote some standard results of non-positive curvature. Section 2 is the central section of this paper. We investigate the properties of those isometries of H which correspond to closed geodesics in M. We also prove

Some curvature properties of complex surfaces

Abstract: In this paper, we are investigating curvature properties of complex two-dimensional Hermitian manifolds, particularly in the compact case. To do this, we start with the remark that the fundamental form of such a manifold is integrable, and we use the analogy with the locally conformal KAhler manifolds, which follows from this remark. Among the obtained results, we have the following: a compact Hermitian surface for which either the Riemannian curvature tensor satisfies the KAhler symmetries or the Hermitian curvature tensor satisfies the Riemannian Bianchi identity is KAhler; a compact Hermitian surface of constant sectional curvature is a flat KAhler surface; a compact Hermitian surface M with nonnegative nonidentical zero holomorphie Hermitian bisectional curvature has vanishing plurigenera, c1(M) ⩾ 0, and no exceptional curves; a compact Hermitian surface with distinguished metric, and positive integral Riemannian scalar curvature has vanishing plurigenera, etc.

A note on a local effect at the Schwarzschild sphere

Abstract: Passage of the Schwarzschild radius is shown to be locally measurable by a sign change in a certain scalar. In the Kerr solution this scalar changes sign at the stationary limit. This is an example of the use of a coordinate-invariant method, based on the curvature tensor and a finite number of its covariant derivatives, for investigating gravitational fields.

Kahler manifolds and quantum gravity

Abstract: The geometry of four-dimensional Kahler manifolds is discussed, and it is shown that the existence of a certain constant spinor enables one to obtain relations between the spectra of wave operators of different spins in an Einstein-Kahler space. This result can be regarded as a generalisation of one discovered recently by Hawking and Pope (1978) in spaces with half-flat Riemann tensor.

The Riemann tensor, the metric tensor, and curvature collineations in general relativity

Abstract: The equation xμνRμ λαβ+xμλRμ ναβ = 0, where xμν and Rμ ναβ are the components of an arbitrary symmetric tensor and of the Riemann tensor formed from the metric tensor gμν, is trivially satisfied by xμν = φgμν. Nontrivial solutions are important in various areas of general relativity such as in the study of curvature collineations, and also in the study of algebraic methods given by Hlavatý and Ihrig for the determination of gμν, from a given set of Rμ ναβ. We have found all Rμ ναβ for which there exist nontrivial solutions of the above equation, and we have given the form of the xμν in each case. Various examples of space–times for explicit nontrivial solutions are discussed.

Killing vector fields on complete Riemannian manifolds

Abstract: We discuss Killing vector fields with finite global norms on complete Riemannian manifolds whose Ricci curvatures are nonpositive or negative. 1. It is well known that if a compact Riemannian manifold has nonpositive Ricci curvature then every Killing vector field is a parallel vector field (cf. [3]). In this note, we discuss Killing vector fields with finite global norms on complete Riemannian manifolds. One of our results is that if M is a complete Riemannian manifold with nonpositive Ricci curvature then every Killing vector field on M with finite global norm is a parallel vector field. This is a generalization of the above well-known result. We also discuss the volume of a complete noncompact Riemannian manifold with nonpositive Ricci curvature. Our ideas are based on those of the papers of A. Andreotti and E. Vesentini [1] and, especially, H. Kitahara [2]. We shall be in the C '-category. The manifolds considered are connected and orientable. The indices h, i, j, k,... run over the range { 1, 2, . .. , n} and the Einstein summation convention will be used. 2. Let M be an n-dimensional complete Riemannian manifold and g (resp. V) its Riemannian metric tensor field (resp. its Levi-Civita connection). Let { U: (x', ... ., x )} denote a local coordinate system on M. gij denotes the components of g and (g11) denotes the inverse matrix of the matrix (gij). We set Vi = V/xi and v V1. Let As(M) be the space of all s-forms on M and AQ(M) the subspace of As(M) composed of forms with compact supports. -q E As(M) may be expressed locally as n = (I /s!)qi . . . i dx1l A... Adx . Let denote the local scalar product; the global scalar product is defined by

Relativistic segnificance of curvature tensors

Abstract: In thi paper new curvature tensors have been defined on the lines of Weyl's projective curvature tensor and it has been shown that the “distribution” (order in which the vectors in question are arranged before being acted upon by the tensor in question) of vector field over the metric potentials and matter tensors plays an important role in shaping the various physical and geometrical properties of a tensor viz the formulation of gravitational waves, reduction of electromagnetic field to a purely electric field, vanishing of the contracted tensor in an Einstein Space and the cyclic property.

Spacetimes admitting a vector field whose inner product with the Riemann tensor is zero

Abstract: The equation vμRμναβ = 0 arises in various places in general relativity, in particular as the integrability conditions of the equations Lvg = 2φg, where φ is a constant. These are the equations of a homothetic vector field v, with a zero homothetic bivector (dv = 0) in some space‐time with metric g, and Rμναβ are the components of the Riemann tensor of that metric in some frame. In this paper the equation vμ Rμναβ = 0 is examined and the components of the Riemann tensor for the spacetimes which admit nonzero solutions vμ of this equation are given. The Petrov types of the Weyl tensors of these spacetimes are listed and, as a result, a correction is then made to a theorem in a paper by Collinson and Fugere about the Petrov types of spacetimes, which admit the type of separation that they require of the Hamilton–Jacobi equation for these spacetimes.

Mappings of empty space-times leaving the curvature tensor invariant

Abstract: It is shown that if in some local coordinate system the componentsRijkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorRijkl invariant are found for each class of these space-times.

Reduction of the codimension of regular isometric immersions

Abstract: (1.2) Theorem. Let x: M"~--~Q'~ +l be a minimal immersion. Suppose that the curvature tensor R • of the normal connection V • is parallel with respect to the normal connection and that the first normal space has constant dimension k. Then there exists a totally geodesic submanifold Q,,+k of Q~+t of dimension n+k, such that x(M") c Q,,+k. The authors point out that this result means that the curvature tensor of the normal connection is intimately related to the problem of reducing the codimension of the immersion. They also conjecture an analogous result for mregular minimal immersions (m-regular means that the k th normal space of the immersion N k satisfies: dim Nk=Constant for k = 1 . . . . , m; for further definitions, see Section 2) which satisfy (17•177 Finally they raise the possibility of replacing the condition of minimality H = 0 by other weaker geometric condition. In Section 3 the following result essentially answers these questions.

On a class of exact solutions to the Fokker–Planck equations

Abstract: In this paper we study under which circumstances there exists a general change of gross variables that transforms any Fokker–Planck equation into another of the Ornstein–Uhlenbeck class that, therefore, has an exact solution. We find that any Fokker–Planck equation will be exactly solvable by means of a change of gross variables if and only if the curvature tensor and the torsion tensor associated with the diffusion is zero and the transformed drift is linear. We apply our criteria to the Kubo and Gompertz models.

Surfaces of nonpositive extrinsic curvature in spaces of constant curvature

Abstract: This paper investigates surfaces of nonpositive extrinsic curvature in a pseudo- Riemannian space of curvature 1, Kahlerian submanifolds of complex projective space , and saddle surfaces in spherical space . It is determined under what conditions a surface is a totally geodesic submanifold.Bibliography: 14 titles.

Almost Complex Structures on Tensor Bundles

Abstract: Publisher Summary This chapter investigates a problem for tensor bundles T r s M. If a Riemannian manifold M admits an almost complex structure then so does T r s M provided r + s is odd. If r + s is even a further condition is required on M. The proofs depend on some generalizations of the notions of lifting vector fields and derivations on M, which are defined only for tangent bundles and cotangent bundles. It is shown how vector fields on T r s M can be induced from vector fields, tensor fields of type (r, s), and derivations on M. The main problem—that is, to determine a class of tensor bundles that admit almost complex structures, is considered. For this purpose, it is sufficient to consider contravariant tensor bundles because a Riemannian metric tensor field induces a fibre preserving diffeomorphism of T r s M→T r+S M.