Showing papers on "Scalar curvature published in 1982"

Spacelike hypersurfaces with prescribed boundary values and mean curvature

Abstract: We consider the boundary-value problem for the mean curvature operator in Minkowski space, and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions. Our main results hold for non-constant mean curvature, and make no assumptions about the smoothness of the boundary or boundary data.

New examples of manifolds with strictly positive curvature

Abstract: Berger [-3], Wallach [10] and Berard Bergery [2] have classified all simply connected smooth manifolds which allow a homogeneous Riemannian metric of strictly positive curvature. Besides the rank one symmetric spaces there exist five exceptional manifolds and an infinite series of 7-manifolds of distinct homotopy type which have been studied by Aloft and Wallach [1]. These are diffeomorphic to Mpq:=SU(3)/Upq, where p, q are positive integers and Upq is the one-parameter subgroup of diagonal matrices

Geometrische Methoden zur Gewinnung von A-Priori-Schranken für harmonische Abbildungen

Abstract: In this paper, we prove a-priori estimates for harmonic mappings between Riemannian manifolds which solve a Dirichlet problem. These estimates employ geometrical methods and depend only on geometric quantities, namely curvature bounds, injectivity radii, and dimensions. An essential tool is the introduction of almost linear functions on Riemannian manifolds. Furthermore, we show the existence of almost linear and harmonic coordinates on fixed (curvature controlled) balls. These coordinates possess better regularity properties than Riemannian normal coordinates.

Axial isometries of manifolds of non-positive curvature

Abstract: Let M be a complete C ~~ Riemannian manifold of non-positive sectional curvature. We say that a geodesic 9: IR~ M bounds a fiat strip of width c > 0 (a fiat half plane) if there is a totally geodesic, isometric immersion i: [0, c) x IR~M(i: [0, oo) x IR~M) such that i(0, t) = 9(0. A 9eodesic without fiat strip (without fiat half plane) is a geodesic, which does not bound a flat strip (a flat half plane). We will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of M. In fact, many of the properties of a manifold of strictly negative curvature (resp. of a visibility manifold) still remain true if one assumes only the existence of a closed geodesic without flat half plane. We will discuss the existence of free (non-Abelian) subgroups of gl(M), the existence of infinitely many closed geodesics, the density of closed geodesics, and a transitivity property of the geodesic flow. It is, therefore, interesting to give conditions which ensure the existence of a closed geodesic without flat half plane. We will prove that M has a closed geodesic without flat half plane if vol(M)< oo and if M contains a geodesic without flat half plane. Note that a geodesic is not boundary of a flat strip (and a fortiori not boundary of a flat half plane) if it passes through a point p e M such that the sectional curvature of all tangent planes at p is negative. In the proofs of our results we investigate the action of rtl(M ) as group of isometries on the universal covering space H of M. In the proofs of many of our results we do not use the fact that this action is properly discontinuous and free. We, therefore, formulate these results for arbitrary groups D of isometries of H. The paper is organized as follows: In Sect. 1 we fix some definitions and notations and quote some standard results of non-positive curvature. Section 2 is the central section of this paper. We investigate the properties of those isometries of H which correspond to closed geodesics in M. We also prove

Some curvature properties of complex surfaces

Abstract: In this paper, we are investigating curvature properties of complex two-dimensional Hermitian manifolds, particularly in the compact case. To do this, we start with the remark that the fundamental form of such a manifold is integrable, and we use the analogy with the locally conformal KAhler manifolds, which follows from this remark. Among the obtained results, we have the following: a compact Hermitian surface for which either the Riemannian curvature tensor satisfies the KAhler symmetries or the Hermitian curvature tensor satisfies the Riemannian Bianchi identity is KAhler; a compact Hermitian surface of constant sectional curvature is a flat KAhler surface; a compact Hermitian surface M with nonnegative nonidentical zero holomorphie Hermitian bisectional curvature has vanishing plurigenera, c1(M) ⩾ 0, and no exceptional curves; a compact Hermitian surface with distinguished metric, and positive integral Riemannian scalar curvature has vanishing plurigenera, etc.

Dynamical symmetry breaking due to radiative corrections in cosmology

Abstract: Dynamical symmetry breaking in massless $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory in curved spacetime with non-conformal coupling to the curvature is investigated It is shown that in general the one-loop self-energy for such a field will involve a term of the form $R\mathrm{ln}|R{\ensuremath{\mu}}^{\ensuremath{-}2}|$, where $R$ is the scalar curvature, and $\ensuremath{\mu}$ is a mass This term can give rise to symmetry breaking Two models, the Einstein universe and a spatially flat Robertson-Walker universe with a power-law expansion, are considered where this term is the sole contribution to the one-loop self-energy In both cases a phase transition will occur at a critical value of the curvature The form of the two-loop corrections to the self-energy and the limits of validity of the one-loop approximation are discussed

Curvature, geodesics and the Brownian motion on a Riemannian manifold. I. Recurrence properties

Abstract: Let M be an n-dimensional, complete, connected and locally compact Riemannian manifold and g be its metric. Denote by ΔM the Laplacian on M.

Splitting theorems of riemannian manifolds

Abstract: © Foundation Compositio Mathematica, 1982, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the compatibility of relativistic wave equations in Riemann spaces. II

Abstract: For pt.I see Nouvo Cimento vol.45, p.486 (1962). Some details of a previous paper relating to the internal consistency in a V4 of spin S equations (S>or=3/2) are re-examined and a special case is brought to light. This occurs when one of the field spinors has indices of only one kind. It is argued that the prescription of minimal gravitational coupling is to be abandoned. In the special case just referred to, the minimally coupled equations are modified by the addition to one of them of a certain term which contains the Weyl tensor as a factor, so that consistency in an arbitrary V4 then obtains.

Smooth approximation of polyhedral surfaces regarding curvatures

Abstract: In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.

Curvature, geodesics and the Brownian motion on a Riemannian manifold. II. Explosion properties

Abstract: Let M be an n-dimensional, complete, connected and non compact Riemannian manifold and g be its metric. ΔM denotes the Laplacian on M.

Birkhoff's theorem for ghost-free, tachyon-freeR + R 2 +Q 2 theories with torsion

Abstract: We investigate the conditions under which the class of ghost-free, tachyon-freeR + R2 +Q2 theories with torsion satisfy Birkhoff's theorem. We prove a weakened Birkhoff theorem requiring an additional assumption of parity invariance for two Lagrangians one of which contains torsion squared terms in addition to curvature squared terms. For another Lagrangian, also containing torsion squared terms, a weakened Birkhoff theorem requiring the additional assumptions of parity invariance and constant scalar curvature is proven. A special case of this Lagrangian is shown to satisfy a weakened Birkhoff theorem requiring only the additional assumption of constant scalar curvature. In addition the explicit dependence of torsion on parity noninvariant quantities is displayed.

Compactness criteria for Riemannian manifolds

Abstract: Ambrose, Calabi and others have obtained Ricci curvature conditions (weaker than Myers' condition) which ensure the compactness of a complete Riemamnan manifold. Using standard index form techniques we relate the problem of finding such Ricci curvature criteria to that of establishing the conjugacy of the scalar Jacobi equation. Using this relationship we obtain a Ricci curvature condition for compactness which is weaker than that of Ambrose and, in fact, which is best among a certain class of conditions. One of the most well-known results relating the curvature and topology of a complete Riemannian manifold M is the classical theorem of Myers [8] which states that if the Ricci curvature with respect to unit vectors on M has a positive lower bound then M is compact. (Myers also gives a diameter estimate in terms of this bound.) In 1957 Ambrose [1] published an interesting generalization of Myers' theorem. He proved that if there is a point q M in such that along each geodesic y: [0, oo) -M emanating from q (and parameterized by arc length t) the Ricci curvature satisfies f Ric(dY dtY ) dt = +oo then M is compact. One of the important features of this result is that the Ricci curvature is not required to be everywhere nonnegative. The author, together with T. Frankel, has applied Ambrose's theorem to certain problems in general relativity (see [4]). In this paper we present a general technique for establishing compactness criteria for complete Riemannian manifolds. As an application of this technique we obtain a generalization of Ambrose's theorem, which, with respect to a certain class of curvature conditions, is best. This generalization can be used to improve some of the results in [4]. (See, especially, Theorem 5 and Corollary 6 of that paper.) We take a moment to introduce some notation and terminology. Throughout, let M denote a smooth complete Riemannian manifold of dimension n > 2. Let be the Riemannian metric on M and let V be the associated Levi-Civita connection. If t -y(t) is a curve in M, let D/dt be the covariant derivative operator on vector fields along y induced by the connection V. For vector fields X and Y let R(X, Y) be the Riemann curvature transformation, i.e. R(X, Y)Z = VxV yZ-V yVxZ -V[XyZ Received by the editors October 3, 1980. 1980 Mathematics Subject Classificatio. Primary 53C20. o 1982 American Mathematical Society 0002-9939/82/0000-0025/$02.25