Showing papers on "Scalar curvature published in 1977"

Hypersurfaces with constant scalar curvature

Abstract: Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of complete surfaces with constant curvature in R 3.

Some remarks on the existence of spacelike hypersurfaces of constant mean curvature

Abstract: Bernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

On the nature of singularities in general relativity

Abstract: It is shown that there is an upper bound to the rate of growth of the Ricci curvature near a singularity.

On the holomorphic curvature of some intrinsic metrics

Abstract: If G is a hyperbolic manifold in the sense of Kobayashi and the differential Kobayashi metric KG is of class C2, then the holomorphic curvature of KG is greater than or equal to -4. If G is Carath&odory-hyperbolic and the differential Carathxodory metric CG is of class C2, then the holomorphic curvature of CG is less than or equal to -4. With this result we obtain an intrinsic characterization of the unit ball.

Curvature invariants and space–time singularities

Abstract: This paper collects together in a general setting observer dependent curvature invariants for space–time and applies them to an analysis of curvature singularities. Observer dependent quantities, such as energy and momentum densities and tidal stresses, are dependent not only on the space–time point but also on the observer’s 4‐velocity. The properties of these invariants are discussed, and it is shown that they completely describe the behavior of curvature along timelike curves. In particular, curvature singularities can be characterized by unboundedness of these invariants.

Conformal changes of metrics and the initial-value problem of general relativity

Abstract: Conformal techniques are reviewed with respect to applications to the initial-value problem of general relativity. Invariant transverse traceless decompositions of tensors, one of its main tools, are related to representations of the group of “conformeomorphisms” acting on the space of all Riemannian metrics onM. Conformal vector fields, a kernel in the decomposition, are analyzed on compact manifolds with constant scalar curvature. The realization of arbitrary functions as scalar curvature of conformally equivalent metrics, a generalization of Yamabe's conjecture, is applied to the Hamiltonian constraint and to the issue of positive energy of gravitational fields. Various approaches to the solution of the initial-value equations produced by altering the scaling behavior of the second fundamental form are compared.

A complete Kähler metric of positive curvature on ⁿ

Abstract: A complete Kihler metric of positive curvature on C' is constructed and its importance is discussed. The purpose of this paper is to exhibit an example of a complete Kahler metric on Cn with strictly positive Riemannian sectional curvature. To accomplish this, let r2= S. 1ziz on Cn and consider metrics on Cn of the form g,-= a 2 (r2)/azij, r f(r2) E C'(R). We shall describe the conditions on f which make g,a complete metric of positive curvature, and then show that if f(x) = fx (ln(1 + T)/ T) dT, these conditions are satisfied, so for this f, gjis a C X complete Kahle1 metric of strictly positive curvature on Cn. Previously, there have been no known examples of complete noncompact Kahler manifolds of positive sectional curvature. Nevertheless several theorems have been proved regarding the structure of such manifolds. In particular, such manifolds are Stein manifolds [3], they are real diffeomorphic to R 2n [5], and they admit no nonconstant bounded holomorphic functions [6]. For these and other reasons, they have been conjectured to be biholomorphic to Cn [4]. Thus, the existence of a complete Kahler metric of positive sectional curvature on Cn is not surprising; but it is not trivial. One can easily show that C admits a complete Kahler metric of positive curvature: By the existence of isothermal coordinates [2] any complete metric on R2 of positive sectional curvature is a Hermitian metric of positive sectional curvature relative to some complex structure; the resulting complex manifold is in fact C as a consequence of the Blanc Fiala Theorem [1] or of the general result quoted on the nonexistence of nonconstant bounded holomorphic functions. If one then takes products of C with itself one obtains a complete Kahler metric of nonnegative sectional curvature and positive Ricci curvature on Cn. However, there is no obvious way to perturb this metric to obtain a complete Kahler metric of positive sectional curvature, for there is difficulty in finding a perturbation which gives a merely Hermitian metric of positive sectional curvature and, in addition, there is the difficulty of maintaining the Kahler condition, which is given, in effect, by a differential equation. Here, the problem of satisfying the Kahler condition is solved by considering only Received by the editors October 12, 1976 AMS (MOS) subject classifications (1970). Primary 53C55; Secondary 52E10.

Quantization in Spaces of Constant Curvature

Abstract: A cnllque of the conventional canonical quantization method is given. In its place an alternative quantization scheme applicable to any complete Riemannian manifold of constant curvature is presented. The scheme is based firstly on a rigorous quantization of every momentum observable associated with a one-parameter group of transformations of the mani­ fold, and secondly on a consideration of spatial symmetries and conservation laws. The formulation is coordinate independent and the troublesome generalized momenta no longer play any dominant role. Explicit examples are worked out leading to interesting results.

COMPLETE $ l$-DIMENSIONAL SURFACES OF NONPOSITIVE EXTRINSIC CURVATURE IN A RIEMANNIAN SPACE

Abstract: This article studies complete -dimensional surfaces of nonpositive extrinsic 2-dimensional sectional curvature and nonpositive -dimensional curvature (for even) in Euclidean space , in the sphere , in the complex projective space , and in a Riemannian space . If the embedding codimension is sufficiently small, then a compact surface in or is a totally geodesic great sphere or complex projective space, respectively. If is a compact surface of negative extrinsic 2-dimensional curvature in a Riemannian space , then there are restrictions on the topological type of the surface. It is shown that a compact Riemannian manifold of nonpositive -dimensional curvature cannot be isometrically immersed as a surface of small codimension. The order of growth of the volume of complete noncompact surfaces of nonpositive -dimensional curvature in Euclidean space is estimated; it is determined when such surfaces are cylinders. A question about surfaces in which are homeomorphic to a sphere and which have nonpositive extrinsic curvature is looked at.Bibliography: 25 titles.

Geometry in visual space-some method-dependent (arti)facts.

Abstract: The measure of scalar curvature K varies with the method used to determine it. Overconstancy is found only with simultaneous comparison, nearly perfect constancy results from successive judgement, while absolute judgement produces underconstancy.

Spectral geometry of symmetric spaces

Abstract: Let M be a compact Riemannian manifold without boundary. Let D be a differential operator on M. Let spec (D, M) denote the eigenvalues of D repeated according to multiplicity. Several authors have studied the extent to which the geometry of M is reflected by spec (D, M) for certain natural operators D. We consider operators D which are convex combinations of the ordinary Laplacian and the Bochner or reduced Laplacian acting on the space of smooth functions and the space of smooth one forms. We prove that is is possible to determine if M is a local symmetric space from its spectrum. If the Rcci tensor is parallel transported, the eigenvalues of the Ricci tensor are spectral invariants of M. Introduction. Let M be a compact connected Riemannian manifold without boundary and let Do = d* d be the Laplacian acting on the space of smooth functions. Let spec (DO, M) denote the set of eigenvalues 0 < XI < ?2 ...; each eigenvalue is repeated according to the multiplicity. The basic question we will be considering is to what extent the geometry of the manifold M is reflected by spec (Do, M) and by the spectra of certain other natural differential operators acting on M. Sakai [5] has proved THEOREM (SAKAI). Let M and M' be Einstein manifolds of dimension 6. Suppose that M and M' have the same Euler characteristic and that spec (Do, M) = spec (Do, M'). Then if M is a local symmetric space, so is M'. If we enlarge the class of differential operators which we are willing to consider, other geometrical properties of the manifold M are reflected by the spectrum. Let D = d* d + dd* be the Laplacian acting on the space of smooth p-forms; let spec (4D M) denote the eigenvalues of 4 repeated according to multiplicity. Patodi [4] has proved THEOREM (PATODI). Suppose that spec (D4, M) = spec (Dp, M') for p = 0, 1, 2. Then: (a) if M has constant scalar curvature c, so does M'; (b) if M is Einstein, so is M'; Received by the editors January 30, 1976. AMS (MOS) subject classifications (1970). Primary 35P20; Secondary 53C35, 35J05, 35K05. (1) Research partially supported by NSF grant MPS72-04357 and by a grant from the Sloan foundation. @) American Mathematical Society 1977 341 This content downloaded from 157.55.39.224 on Wed, 11 May 2016 05:46:31 UTC All use subject to http://about.jstor.org/terms

An optimal poincaré inequality for convex domains of non-negative curvature

Abstract: In this note we estimate from below the first non-zero eigenvalue of the Neumann problem of two dimensional convex domains of non-negative Gaussian curvature. The Neumann problem is described as follows: Let M be an oriented manifold with fixed complete Riemannian metric, and let A be the Laplace-Beltrami operator associated with this metric, acting on realvalued functions on M. Also let f2 denote a fixed compact connected submanifold of M whose boundary c?f2 is continuous and piecewise of class C ~, and whose interior f2 ~ has the same dimension as M. Our task is to study the eigenvalues problem

A generalization of the concept of constant mean curvature and canonical time

Abstract: Some compact spaces of achronal hypersurfaces are constructed in various types of space-time A variational principle is introduced on these spaces, smooth extremals of which are spacelike hypersurfaces of constant mean curvature The integrand of the variational principle is shown to be upper semicontinuous and the direct methods of the calculus of variations are applied to obtain aC0 extremal, which is defined to be a spacelike hypersurface of generalized constant mean curvature The family of such hypersurfaces generated by altering the value of the mean curvature is discussed and the mean curvature itself is shown to have many of the properties of a canonical time coordinate

On minimal immersions with parallel normal curvature tensor

Abstract: In this paper we first prove the fo11owing theorem on reduction of codimension of minima1 immersions: Theorem 1 - Let x: Mn→X be a minima1 immersion of an n-dimensiona1 connected manifold Mn into an (n+l)-dimensiona1 space X of constant curvature. Assume that the curvature tensor of the norma1 connexion is paral1e1 in the norma1 bundle and the first norma1 space of the immersion has constant dimension k.

Curvature on holomorphic plane curves. II

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