Showing papers on "Scalar curvature published in 1976"

Homogeneous Manifolds With Negative Curvature

Abstract: This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra < satisfying these conditions and any member of an easily constructed family of inner products on i, a metric deforma- tion argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group. 1. Introduction. This paper was motivated by the following problem: Which connected Lie groups admit a left invariant Riemannian metric with nega- tive (sectional) curvature? We emphasize that throughout the paper, we under- stand "negative" to mean "less than or equal to zero". Since the property in question is not sensitive to groups linked by a local isomorphism, we deal primar- ily with simply connected groups. Results of J. A. Wolf (13) and E. Heintze (4) show that the above problem is closely linked with the classification of connected, homogeneous Riemannian manifolds with negative curvature. Indeed, if M is such a manifold and if M is simply connected, then M is isometric to a solvable Lie group endowed with a left-invariant metric. In this paper, we give a complete solution to our original problem by show- ing that a necessary and sufficient condition for a group to have the property in question is that its Lie algebra be what we call an "NC algebra". Roughly speak- ing, the crucial properties of an NC algebra $ are that in addition to being solv- able, e must contain an abelian subalgebra a complementary to the derived

Riemannian Structures and Triangulations of Manifold

Abstract: Let X be a C ∞ closed manifold of dimension N. Two additional structures on X have been extensively studied. One is the Riemannian structure giving rise to Riemannian geometry and the other is the triangulation of X giving rise to combinatorial or polyhedral topology.

The Scalar Curvature

Abstract: We shall deal with some problems concerning the scalar curvature of compact riemannian manifolds. In particular we shall deal with the problem of Yamabe: Does there exist a conformal metric for which the scalar curvature is constant? And also problems posed by Chern, Nirenberg and others. All these problems are almost entirely solved, however there remain some open questions (see the conjectures).

Isotropic transport process on a Riemannian manifold

Abstract: We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class C0. The special case of a manifold of negative curvature is considered as an illustration.

Submanifolds of a Riemannian manifold with semisymmetric metric connections

Abstract: We derive the Gauss curvature equation and the CodazziMainardi equation with respect to a semisymmetric metric connection on a Riemannian manifold and the induced one on a submanifold. We then generalize the theorema egregium of Gauss.

Harmonic analysis on riemannian symmetric spaces of negative curvature and scattering theory

Abstract: Harmonic analysis on a Riemannian symmetric space can be connected with the study of a nonstationary system of equations that has been constructed with respect to the ring of Laplace operators. The scattering theory for this system generalizes the scattering theory for hyperbolic equations constructed by Lax and Phillips. The paper contains a series of new spectral theorems generalizing the Harish-Chandra theorem and a formulation of a causality principle for scattering operators.Bibliography: 23 items.

Geometrical Interpretations of Scalar Curvature and Regularity of Conformal Homeomorphisms

Abstract: To complete the solution I gave [5] of A. Lichnerowicz’s conjecture concerning the conformal group of compact1 riemannian manifolds, one had still to prove that every conformal homeomorphism of C ∞ manifolds is of C ∞ class. Up to now, it seems that this result has been proved only in the euclidean case by Y. G. Resetnyak [7] and F. W. Gehring [3]. We plan in this paper to solve the general case by a method analogous to the one of Resetnyak.

Geometrical interpretations of scalar curvature and regularity of conformal

Abstract: To complete the solution I gave [5] of A. Lichnerowicz's conjecture concerning the conformal group of compad riemannian manifolds, one had still to prove that every conformal homeomorphism of C~ manifolds is of C~ class. Up to now, it seems that this result has been proved only in the euclidean case by Y. G. Resetnyak [7] and F. W. Gehring [3]. We plan in this paper to solve the general case by a method analogous to the one of Resetnyak. To understand the difficulty of this regularity problem, it is enough to give the history of the euclidean case: in 1850 Liouville proved that any conformal diffeomorphism of class C 3 of open sets of En is a product of inversions, thus of class C'. But one had apparently to wait for the works of P. Hartman [4] to extend this result to the class C' and it was only in 1960-61 that Y. G. Resetnyak and F. W. Gehring were able to handle the case of conformal homeomorphisms. Reset­ nyak's proof is based on the isoperimetrical inequality of En, Geh­ ring's proof uses the technique of quasi-conformal maps; both reduce, like Hartman's proof, to the regularity of the solutions of an elliptic equation. The case of riemannian manifolds raises new difficulties for several reasons: on one hand, no extremals for the conformal invariant fM Igrad ul n dr are known a priori and the technique of 'conformal capacities' or of 'extremal lengths' gives less accurate results, which seems to exclude an extension of Gehring's method. On the other hand no simple isoperimetrical inequality is known in riemannian geometry, and one has no regularizing kernel which commutes with the Laplacian, so that Resetnyak's method does not extend straight­ forwardly. Anyhow the problem is cleared up - if not made easier - by the appearance of the scalar curvature (see § 1 and 2) and our method will be based on 'approximate' isoperimetrical inequalities (§3 and 4) due to the fact that a manifold is 'approximately euclidean' in the neighbourhood of each point.

A note on the conformal quasi-invariance of the Laplacian on a pseudo-Riemannian manifold

Abstract: We consider the Laplacian on a pseudo-Riemannian manifold with constant scalar curvature (e.g. Euclidian space with an arbitrary signed inner product or its conformal compactification and coverings of this) and show that for this minus a constant we have quasi-invariance with respect to an action of the conformal group on functions.

Differential geometry on Grassmann algebras

Abstract: H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus ‘curvature’ is represented by a tensor of the third rank and ‘Killing's equations’ now involve this ‘curvature tensor’. The immediate reason for this is that otherwise familiar terms appear with the wrong sign. We have found that these unaesthetic features disappear, and formal elegance is marvellously restored, when the manifold is replaced by a Grassmann algebra. The connection with supersymmetry is explained but applications are not reported here.

K-spaces of constant holomorphic sectional curvature

Abstract: In this note we prove the equivalence of the pointwise constancy and the global constancy of the holomorphic sectional curvature of a K-space. A criterion for the constancy of the holomorphic sectional curvature of a K-space is found. It is proved that every proper K-space of constant holomorphic sectional curvature is a six-dimensional orientable Riemannian manifold of constant positive curvature, which is isometric with the six-dimensional sphere in the case of completeness and connectedness.

ON THE INSTABILITY OF A MINIMAL SURFACE IN AN $ n$-DIMENSIONAL RIEMANNIAN SPACE OF POSITIVE CURVATURE

Abstract: In this paper the question of minimal two-dimensional surfaces in an n-dimensional Riemannian space (surfaces with zero mean curvature vector) is investigated.Bibliography: 24 titles.

Conformally flat manifolds and a pinching problem on the Ricci tensor

Abstract: There is a formal similarity between the theory of hypersurfaces and conformally flat d-dimensional spaces of constant scalar curvature provided d > 3. For, then, the symmetric linear transformation field Q defined by the Ricci tensor satisfies Codazzi's equation (Vx Q)Y = (Vy Q)X. This observation leads to a pinching theorem on the length of the Ricci tensor. 1. Statement of results. Recently, one of the authors [1] obtained THEOREM G. Let M be a d-dimensional compact conformallyflat manifold with definite Ricci curvature. If the scalar curvature r is constant and if the square of the length of the Ricci tensor is not greater than r2/(d 1), d > 3, then M is a space of constant curvature. Note that the square length of the Ricci tensor is greater than or equal to r2ld, so the Ricci tensor has been "pinched". In the present paper the following two theorems are proved, the first of which generalizes Theorem G. THEOREM 1. Let M be a d-dimensional compact conformally flat manifold with constant scalar curvature r. If the length of the Ricci tensor is less than r/ d -1, d > 3, then M is a space of constant curvature. THEOREM 2. In a d-dimensional compact conformally flat manifold M, if the length of the Ricci tensor is constant and less than r/ d -1, then M is a space of constant curvature. 2. Conformally flat manifolds. Let M be a Riemannian manifold of dimension d > 3. We cover M by a system of local coordinate neighborhoods (U, xh), and denote by gji, V>, Rk_h, and R1i the Riemannian metric, the operator of covariant differentiation in terms of the Riemannian connection, the curvature tensor and the Ricci tensor, respectively. We say that M is conformally flat if its Riemannian metric is conformally related to a locally Euclidean metric. In a conformally flat manifold, Received by the editors June 9, 1975 and, in revised form, October 15, 1975. AMS (MOS) subject classifications (1970). Primary 53A30, 53B20, 53C20.

Isometry groups of manifolds of negative curvature

Abstract: Solvable subgroups of the isometry groups of a simply-connect- ed manifold of negative curvature are characterized and this characterization is used to show that the isometry group of the universal Riemannian covering of a compact manifold of negative curvature is either discrete or semisimple. 0. Introduction. A number of recent papers have related the geometry of manifolds of negative curvature to the algebra of various groups of isometries (for example (4), (8)). In this paper we study various groups of isometries of a simply-connected manifold M of negative curvature. In Theorem 5 we use results of Bishop and O'Neill (2) to show that a solvable group of isometries either leave a single geodesic invariant, permute a class of asymptotic geodesies, or else have a nonempty fixed point set. If the total isometry group I(M) does not satisfy either of two former conditions, we show in Theorem 7 that there is a compact normal subgroup K such that I(M)/K is semisimple and acts effectively on a closed, connected, totally convex submanifold of M. Using these results we show in Theorem 9 that if M is the universal Riemannian covering of a compact manifold of negative curvature, then the isometry group 7(Af) is either discrete or semisimple. This may be viewed as an extension of the classical situation where the compact manifold may be considered as a double coset space T\G/7<" of a connected semisimple Lie group G and where the symmetric space G/K can be given an invariant metric of nonpositive curvature so that G is isomorphic to the identity component of the isometry group (5).

Differential Geometry of Totally Real Submanifolds

Abstract: Publisher Summary This chapter describes the differential geometry of totally real sub-manifolds It discusses the properties of totally real submanifolds for cases in which the submanifolds are totally geodesic The chapter highlights preliminaries on the real space form, the Weyl conformal curvature tensor, the complex space form, and the Bochner curvature tensor It presents fundamental formulas for a totally real submanifold M n of a Kaehlerian manifold M 2m The chapter discusses the case in which the ambient space M 2m is a complex space form and the case in which the Bochner curvature tensor of the ambient space M 2m vanishes