Showing papers on "Scalar curvature published in 1984"

Eigenvalues in Riemannian geometry

Abstract: Preface. The Laplacian. The Basic Examples. Curvature. Isoperimetric Inequalities. Eigenvalues and Kinematic Measure. The Heat Kernel for Compact Manifolds. The Dirichlet Heat Kernel for Regular Domains. The Heat Kernel for Noncompact Manifolds. Topological Perturbations with Negligible Effect. Surfaces of Constant Negative Curvature. The Selberg Trace Formula. Miscellanea. Laplacian on Forms. Bibliography. Index.

Spaces of Constant Curvature

Abstract: This sixth edition illustrates the high degree of interplay between group theory and geometry The reader will benefit from the very concise treatments of riemannian and pseudo-riemannian manifolds and their curvatures, of the representation theory of finite groups, and of indications of recent progress in discrete subgroups of Lie groups

Conformal deformation of a Riemannian metric to constant scalar curvature

Abstract: A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M > 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of S X S~. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation

On the curvature of piecewise flat spaces

Abstract: We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.

On the estimate of the first eigenvalue of a compact riemannian manifold

Abstract: The main theorem proved in this work is: Let M be a compact Riemannian manifold withnon-negative Recci curvature, then the first eigenvalue -λ1, of the Laplace operator of Msatisfies λ1≥π2, where d denotes the diameter of the M. This estimate improves the recentresults due to S. T. Yan and P. Li and gives the best estimate for this kind of manifold

Renormalization Group Analysis of Grand Unified Theories in Curved Space-time

Abstract: We consider a class of grand unified theories (GUT's) based on the Georgi-Glashow model in curved spacetime. We are particularly concerned with the coupling constants involving the curvature. These include the cosmological and gravitational constants, as well as coupling constants appearing in terms quadratic in the curvature and in terms which link the Higgs bosons to the scalar curvature. For asymptotically free theories, we use the renormalization group to obtain expressions for these effective coupling constants at high curvature (between the GUT and Planck scales). We discuss the role of the effective coupling constants in the gravitational field equations. These results may be of importance for cosmology.

Hypersurfaces in IRn+1 with prescribed gaussian curvature and related equations of monge—ampére type

Abstract: (1984). Hypersurfaces in IRn+1 with prescribed gaussian curvature and related equations of monge—ampere type. Communications in Partial Differential Equations: Vol. 9, No. 8, pp. 807-838.

An elementary proof of the Cheeger-Gromoll splitting theorem

Abstract: We give a short proof of the Cheeger-Gromoll Splitting Theorem which says that a line in a complete manifold of nonnegative Ricci curvature splits off isometrically. Our proof avoids the existence and regularity theory of elliptic PDE's.

Variational principle for a prototype Rastall theory of gravitation

Abstract: A prototype of Rastall’s theory of gravity, in which the divergence of the energy-momentum tensor is proportional to the gradient of the scalar curvature, is shown to be derivable from a variational principle. Both the proportionality factor and the unrenormalized gravitational constant are found to be covariantly constant, but not necessarily constant. The prototype theory is, therefore, a gravitational theory with variable gravitational constant.

Length and curvature in the geometry of thermodynamics

Abstract: The relationship between Weinhold's metric geometry and the Riemannian geometry of thermodynamics is presented.

Uniqueness and non-existence of metrics with prescribed Ricci curvature

Abstract: We investigate whether the Ricci tensor uniquely determines the Riemannian structure, and we give conditions that a doubly covariant tensor has to satisfy in order to be the Ricci tensor for some Riemannian structure.

Self-duality of Kähler surfaces

Abstract: Soit (M,g) une surface de Kahler. Si elle est autoduale par rapport a l'orientation canonique et est Einsten, alors elle est de courbure constante holomorphe

Divergent parts of quantum fluctuation in curved space from the Adler-Zee formulas

Abstract: By using the Adler-Zee formulas, the first three terms of the adiabatic expansion of quantum fluctuation in curved space are calculated in the massive free scalar theory with arbitrary coupling to the scalar curvature. We use dimensional regularization and Pauli-Villars regularization. The former requires extension of the formulas to n dimensions. The results coincide with those given by the DeWitt-Schwinger proper-time formalism. We use two energy-momentum tensors, deltaS/deltag/sub munu/Vertical Barfixed and deltaS/deltag/sub munu/Vertical Barfixed (phi-tildeequivalent(-g)/sup 1/4/phi). Both give the same results if the extra term in the formula for the Newtonian constant is properly taken into account.

A Metric on the Manifold of Immersions and Its Riemannian Curvature.

Abstract: E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n−1) dimensional manifold into ℝn, and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.

Sectional curvature and the determination of the metric in space-time

Abstract: It is shown that if the sectional (Riemannian) curvature function is given at all spacetime points, then, apart from a very special class of space-times, the metric is uniquely determined.