Showing papers on "Scalar curvature published in 1979"

On the structure of manifolds with positive scalar curvature

Abstract: Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are felt to provide a more complete picture of manifolds with positive scalar curvature: (1) let M be a compact four-dimensional manifold with positive scalar curvature. Then there exists no continuous map with non-zero degree onto a compact K(π,1). (2) Let M be n-dimensional complete manifold with non-negative scalar curvature. Then any conformed immersion of M into Sn is one to one. In particular, any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sn by a discrete subgroup of the conformal group. (3.) Let M be a compact manifold whose fundamental group is not of exponential growth. Then unless M is covered by Sn, Sn–1 x S1 or the torus, M admits no conformally flat structure.

Riemannian geometry as determined by the volumes of small geodesic balls

Abstract: (Here o~-~the volume of the unit ball in R\". The simplest expression for eo is w = (1]( 89 \"/~ where ({rn)! ---F( 89 + 1).) First we make several remarks. 1. Our method for attacking the conjecture (I) will be to use the power series expansion for Vm(r). This expansion will be considered in detail in section 3; however, the general facts about it are the following: (a) the first term in the series is corn; (b) the coefficient of r n+~ vanishes provided k is odd; (c) the coefficients of r n+~ for/r even can be expressed in terms of curvature. Unfortunately the nonzero coefficients depend on curvature in a rather complicated way, and this is what makes the resolution of the conjecture (I) an interesting problem. 2. To our knowledge the power series expansion for Vm(r) was first considered in 1848 by Bertrand-Diguet-Puiseux [6]. See also [14, p. 209]. In these papers the first two terms of the expansion for Fro(r) are computed for surfaces in RS:

Spectra of Compact Locally Symmetric Manifolds of Negative Curvature.

Abstract: Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \\ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

L^2-harmonic forms on rotationally symmetric Riemannian manifolds

Abstract: The paper contains a vanishing theorem for P harmonic forms on complete rotationally symmetric Riemannian manifolds. This theorem requires no assumptions on curvature. This paper gives necessary and sufficient conditions for existence of L2 harmonic forms on a special class of Riemannian manifolds. Manifolds of this class were called models by Greene and Wu and played a crucial part in the study of function theory on open manifolds [GW]. Throughout the paper M will denote a model of dimension n, i.e. a C X Riemannian manifold such that: (1) there exists a point o EE M for which the exponential mapping is a diffeomorphism of ToM onto M; (2) every linear isometry p: T,oM > To,M is realized as the differential of an isometry 1: M -M, i.e., ?(o) = o and 0.(o) = p. Clearly, M is complete and can be identified with To,M via exp0. In terms of geodesic polar coordinates (r, 9) E (0, oo) x S'~1 M\{o} the Riemannian metric dj2 of M can be written as S2= dr2 + f(r)2 dO2, (3) where dO2 denotes the standard metric on S'1 and the function f(r) is C on [0, oo) and satisfies f(0) = 0, f'(0) = 1, f(r) > 0 for r > 0 (4) (cf. [S, pp. 179-183]). Complete description of the spaces 9C*(M) of L2 harmonic forms is contained in the following THEOREM. Let M be a model of dimension n > 2. Then (i) 9C(M) = {0} for q # 0, n/2, n, [(0) if f(r) dr = , 0(ii) aV(M) ==d 9M, IR if If(r) l r < X0 Received by the editors July 27, 1978 and, in revised form, December 4, 1978. AMS (MOS) subject classifications (1970). Primary 58G99.

An elementary method in the study of nonnegative curvature

Abstract: A standard technique in classical analysis for the study of eontinous sub-solutions of the Dirichlet problem for second order operators may be illustrated as follows. Suppose it is to be shown that a continuous real function ](x) is convex (respectively, striely convex) at x0; then it suffices to produce a C ~ function g(x) such that g(x)<<.](x) near x 0 and g(Xo) =/(x0), and such that 9\"(xo) >/0 (respectively g\"(xo) >1 some fixed positive constant). The main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone. Now in global differential geometry, the functions that naturally arise are often continuous but not differentiable. Since much of geometric analysis reduces to second order elliptic problems, this technique then recommends itself as a natural tool for overcoming this difficulty with the lack of differentiability. In a limited way, this technique has indeed appeared in several papers in complex geometry (e.g. Ahlfors [1], Takeuchi [20], Elenewajg [7] and Greene-Wu [11]; cf. also Suzuki [19]). The main purpose of this paper is to broaden and deepen the scope of this method by making it the central point of a general study of nonnegative sectional, Ricei or bisectional curvature. The following are the principal theorems; the relevant definitions can be found in Section 1. Let M be a noncompact complete Riemannian manifold and let 0 E M be fixed. Let {Ct}tG1 be a family of closed subsets of M indexed by a subset I of R. Assume that et = d(0, C t ) ~ as t ~ , where d(p, q) will always denote the distance between p, qEM relative to the Riemannian metric. The family of functions ~t: M-~R defined by ~t(P)=

A necessary and sufficient condition for York data to specify an asymptotically flat spacetime

Abstract: This paper studies the conditions under which Cauchy data (ḡ,π,ν,T) for an asymptotically flat spacetime are determined by the freely specifiable York data (g, σ, ν, T) (τ=0), where trgσ=0, divgσ=8πν. It is shown that the space of such σ′s is infinite dimensional. Furthermore,it is shown that (g, σ, ν, T) determine conformally equivalent Cauchy data if and only if g is conformally equivalent to an asymptotically flat metric with nonnegative scalar curvature.

Anomalies, topological invariants, and the Gauss-Bonnet theorem in supergravity

Abstract: The Einstein multiplet ${W}_{a}$ is constructed and extra terms in its transformation laws, due to the index $a$, are found. Application of the tensor calculus to the Einstein, Weyl, and scalar curvature multiplets yields (i) supersymmetric off-shell complete extensions of the multiplet of anomalies, (ii) the super-Gauss-Bonnet theorem, (iii) topological invariants of supergravity. The latter coincide with those of general relativity.

Quantum effects and regular cosmological models

Abstract: Allowance for the quantum nature of matter fields and weak gravitational waves on the background of the classical metric of a cosmological model leads to two main effects: vacuum polarization and particle production. The first of these effects can be taken into account qualitatively by the introduction into the Lagrangian density of the gravitational field of corrections of the type A+BR/sup 2/+CR/sup 2/lnvertical-barR/R/sub 0/vertical-bar; the second, by the specification of a local rate of production of particles (gravitons) proportional to the square of the scalar curvature R/sup 2/. It is shown that simultaneous allowance for the influence of these effects on the evolution of a homogeneous anisotropic metric of the first Bianchi type eliminates the Einstein singularities. Asymptotic approach to the classical model, however, is attained only if additional assumptions are made. In the contraction stage the solution is close to the anisotropic vacuum Kasner solution; in the expansion stage it tends to the isotropic Friedmann solution, in which matter is produced by the gravitational field.

Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity.

Abstract: We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in the affirmative.

Path integrals for a particle in curved space

Abstract: We consider a particle obeying the Schr\"odinger equation in a general curved $n$-dimensional space, with arbitrary linear coupling to the scalar curvature of the space. We give the Feynman path-integral expressions for the probability amplitude, $〈x,s|{x}^{\ensuremath{'}},0〉$, for the particle to go from ${x}^{\ensuremath{'}}$ to $x$ in time $s$. This generalizes results of DeWitt, Cheng, and Hartle and Hawking. We show in particular, that there is a one-parameter family of covariant representations of the path integral corresponding to a given amplitude. These representations are different in that the covariant expressions for the incremental amplitudes, $〈{x}_{l+1},{s}_{l}+\ensuremath{\epsilon}|{x}_{l},{s}_{l}〉$, appearing in the definition of the path integral, differ even to first order in $\ensuremath{\epsilon}$ (after dropping common factors). Finally, using the proper-time representation, we give the corresponding generally covariant expressions for the propagator of a scalar field with arbitrary linear coupling to the scalar curvature of the spacetime.

Perturbation technique for quantum fields in curved space

Abstract: We derive a concrete expression for the vacuum expectation value of the stress tensor for a massless, nonconformal scalar field propagating in a background spatially flat Robertson-Walker spacetime, up to second order in perturbation theory in the conformal-breaking parameter. The result, which is valid for an arbitrary Robertson-Walker scale factor (subject only to vanishing scalar curvature at some moment) is manifestly nonlocal, yet can still be written as an integral expression in closed form. The method should be extendible to the massive field and anisotropic spacetime cases.

Relativistic Kepler problem. II. Asymptotic behavior of the field in the infinite past

Abstract: The standard weak-field, slow-motion approximation to Einstein's relativistic theory of gravitation is used to express the curvature tensor, up to order ${r}^{\ensuremath{-}5}$ on a flat background space-time, as a functional of the motion of the source of this curvature. The behavior, in the distant past, of the orbit of two particles weakly interacting gravitationally, with radiation reaction taken into account, is then used to compute the asymptotic behavior of the corresponding curvature tensor along past-directed null straight lines in the flat background. It is found, on the one hand, that the falloff of the curvature is fast enough to guarantee satisfaction of a condition to exclude incoming gravitational radiation. On the other hand, the falloff is slower than would have been expected if the conformally rescaled curvature tensor had been regular on the hypersurface at past null infinity of the flat background.