Showing papers on "Curvature of Riemannian manifolds published in 1985"

Conformal deformations of complete manifolds with negative curvature

Abstract: A basic problem in Riemannian geometry is that of studying the set of curvature functions that a manifold possesses. In this generality there has been such a great deal of work that we cannot here record the different contributions. (For a fairly complete account, see [23].) However, in this paper we shall be concerned with the special case of (\"pointwise\") conformal deformations of metrics which we shall call problem (K):

Structure of space-time curvature

Abstract: The structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined. Applications to Einstein's theory of gravitation are given.

Curvature Tensor for {Kaluza-Klein} Theories With Homogeneous Fibers

Abstract: We give explicit formulas for all components of the Riemannian connection and curvature tensor for a class of metrics which describe low-energy deformations in Kaluza-Klein theories with homogeneous fibers.

Minimal disks and compact hypersurfaces in euclidean space

Abstract: Let M" be a smooth connected compact hypersurface in (n + 1)- dimensional Euclidean space E" + l, let A"+l be the unbounded component of £-»+1 _ M"t an(j let K( , De tne principal curvatures of M" with respect to the unit normal pointing into A" + 1. It is proven that if k2 + • • ■ + ic" < 0, then A " + ' is simply connected. 1. Introduction. Recently the theory of minimal surfaces has yielded many striking results relating topology to curvature of Riemannian manifolds. We are interested in applying minimal surfaces to extrinsic problems which relate topology to curvature of submanifolds of low codimension in Euclidean space. The simplest case is that of a smooth connected compact hypersurface M" lying in (n + l)-dimensional Euclidean space E" + l. Such a hypersurface divides E" + l into an unbounded connected open region A"+l and a bounded region B"+1. In this case, the basic local invariants are the principal curvatures k,

Riemannian manifolds with harmonic curvature

Abstract: The present paper is a survey of results on manifolds with harmonic curvature, i.e., on those Riemannian manifolds for which the divergence of the curvature tensor vanishes identically. The curvatures of such manifolds occur as a special case of Yang-Mills fields. These manifolds also form a natural generalization of Einstein spaces and of conformally flat manifolds with constant scalar curvature. After describing the known examples of compact manifolds with harmonic curvature, we give, in Sect. 5, a review of theorems concerning such manifolds. Most of their proofs are either omitted or only briefly sketched. For a complete presentation of the results mentioned in this paper (except for Sect. 3, Sect. 7 and 4.4) the reader is referred to the forthcoming book [5], where one of the chapters deals with generalizations of Einstein spaces. The preparation of the present article was begun under the program SFB 40 in the Max Planck Institute of Mathematics in Bonn, and completed at the Mathematical Sciences Research Institute in Berkeley. The author is obliged to these institutions for their hospitality and assistance. He also wishes to thank Jerry Kazdan for helpful remarks concerning the topics discussed in 4.4.

On the number of diffeomorphism classes in a certain class of Riemannian manifolds

Abstract: The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

Metrical geometry of the classical gauge fields

Abstract: The usual geometric interpretation of the theory of the gauge fields is implemented and generalized, utilizing concepts of the metrical Riemannian geometry. A classic theorem—due to Riemann and Vermeil—allows us to give an elementary solution to the problem of the passage from the field strength (curvature) to the potential (connection).

Examples of compact Einstein Kahler manifolds with positive Ricci tensor

Abstract: On donne des exemples de varietes d'Einstein-Kahler compactes avec premiere classe de Chein positive qui ne sont pas homogenes. On donne une condition necessaire et suffisante pour l'existence de metriques d'Einstein-Kahler sur des P 1 (C)-fibres sur des espaces symetriques hermitiens de type compact

On the Structure of Complete Manifolds with Positive Scalar Curvature

Abstract: One of the greatest contributions of Rauch in differential geometry is his famous work on manifolds with positive curvature. His comparison theorems, which are needed for his proof of the pinching theorem, are fundamental for later developments in Riemannian geometry. His work initiated a systematic research developed by Klingenberg, Berger, Gromoll, Meyer, Cheeger, Gromov, Ruh, Shio-hama, Karcher, etc. This work depends heavily on how a length-minimizing geodesic behaves under the influence of the curvature. Since geodesic is one-dimensional, the information we need from the curvature tensor is the curvature of the two planes which are tangential to the geodesic. This means that we need to know the behavior of the sectional curvature or the Ricci curvature of the manifold. Therefore, it seems very unlikely that arguments based only on length-minimizing geodesics can be used to deal with problems related to scalar curvature. The problem of scalar curvature, however, has drawn a lot of attention of the differential geometers in the late sixties and the seventies, partly because of its interest in general relativity.