Showing papers on "Ricci-flat manifold published in 1983"

A sphere theorem for manifolds of positive Ricci curvature

Abstract: Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that RicM > 1, KM '-K 2 and the volume of M > the volume of the (r e)-ball on the unit m-sphere, m = dim M. Then for a suitable choice of e = e(m, K) every M belonging to this class is homeomorphic to

On totally umbilical submanifolds of some class riemannian manifolds

Abstract: coordinate neighbourhoods { u j x r } . We denote by |B-|;}» H v r g t t R v t and R the Chr is tof f .e l symbols, the operator of covar iant d i f f e r e n t i a t i o n , the curvature t e n s o r , the Rioci t ensor and the s ca l a r curvature of N. The ind ices r , s , t , u,v,w run over the range | l , 2 , . . . , n | , n ^ 4 . A tensor f i e l d i r s t u i s ca l l ed a general ized curvature tensor on K ( [ 1 3 ] ) , i f

Closed geodesics on Riemannian manifolds

Abstract: The Hilbert manifold of $H^1$-curves The loop space and the space of closed curves The second order neighborhood of a critical point Appendix. The $S^1$- and the $Z_2$-action on $\Lambda M$ Closed geodesics on spheres On the existence of infinitely many closed geodesics.

Geometric properties of some classes of Riemannian almost-product manifolds

Abstract: The aim of this paper is to study the geometric properties of the thirty-six classes of Riemannian almost-product manifolds that appear considering the algebraic properties of the covariant derivative of the tensor field defining the structure with respect to the Levi-Civita connexion.

New examples of strictly almost kahler manifolds

Abstract: A parametrized family of non-Kahler almost Kahler manifolds is con- structed as the product of solvable Lie groups with almost cosymplectic structures. A family of compact strictly almost Kahler manifolds whose cohomology is consistent with that of Kahler manifolds is similarly obtained. Almost Kahler manifolds are almost Hermitian manifolds which carry the sym- plectic structure given by their closed Kahler 2-form, $, but which are not neces- sarily complex (if complex, they are Kahler). Those whose almost complex structure 7 is not integrable are called strictly almost Kahler. Few examples of strictly almost Kahler manifolds are known. In fact, only the tangent and cotangent bundles (and some related tensor bundles) of nonflat Riemannian manifolds were known to possess such structures until recently. In 1976, W. Thurston (T) reported a strictly almost Kahler structure on a 2-torus bundle over a 2-torus. He verified its non-Kahler status by the oddness of its first Betti number. In (Go), S. I. Goldberg studied the conjecture that the almost complex structure on a compact Einstein almost Kahler manifold is integrable. While some progress has been made, particularly upon restricting the curvature tensor (SI, S2, S-K, Gr), the conjecture remains unresolved. Even in dimension four, where a normal form for the curvature of an Einstein manifold exists (Hi, J, S-T), it is open. One reason may be the paucity of examples of compact strictly almost Kahler manifolds. We announce here the construction of a large family of noncompact almost Kahler manifolds which are not Kahler and an interesting compact strictly almost Kahler 10-dimensional manifold which seeds the construction of (6« + 4)- dimensional spaces of like properties. We wish to thank Allen Broughton for illuminating discussions on solvable Lie groups. 1. Almost contact metric structures. Let (M2m+X, , |, tj, g) be an almost contact metric manifold (B). Define the fundamental 2-form $> in the usual way. Numerous distinct structures are definable on M in terms of $ and tj and their covariant derivatives. We are specifically interested in: (1) M is contact if $ = dr\; (2) M is normal if ( , ) + 2dri 8> £ = 0; (3) M is Sasakian if it is normal and contact; (4) M is almost cosymplectic if d$ — 0 and dri — 0; (5) M is cosymplectic if V = 0.

The topology of certain Riemannian manifolds with positive Ricci curvature

Abstract: 1. Let M be a complete connected Riemannian manifold of dimension n, and let Ric denote its Ricci curvature. Understanding the Ricci curvature is one of the important problems in today's geometry. In these notes, we assume that Ric > n — 1. The classical theorem of Myers then asserts that M is compact and has diameter dM < TΓ. R. Bishop showed that the volume of M also satisfied volM < vol5«, where S n is the unit Euclidean sphere in R + , and that the equality holds only if M is isometric to S. In [3], S. Y. Cheng proves Theorem A. If' dM — π, then M is isometric to S . It is interesting to ask to what extent these theorems can be perturbed. Our main result is Main Theorem. Given any upper bound K for the sectional curvature of M, there exists a constant v > 0, depending only on n and K, such that whenever vol M > (1 — υ)vols«, then M has the homotopy type of S . By using some of the same methods, we can also show Theorem B. There is a constant p > 0, depending only on n, such that if M has the injectivity radius iM> π — p, then M is homeomorphic to S . In §2 of these notes, we describe the main tools which can be used to prove these theorems. In §§3 and 4, we outline the proofs of Theorem B and Main Theorem. In §5, we describe a new geometric proof for Theorem A. Finally, we discuss some remarks and open question in §6. Details and additional applications will appear in [10]. The author would like to express gratitude to D. Gromoll for many helpful discussions.

Chronoprojective Cartan Structures on Four-Dimensional Manifolds

Abstract: As indicated by its denomination Cartan structures have been derived from Cartan's works [1] initiating the projective and conformal geometries. In the fifties a precise description of this notion in a modern mathematical language has been given by using the fibre bundle of second order frames [2, 3]. The starting point can be viewed as a generalization of the Klein's Erlangen program. Indeed Cartan considered various spaces at each point of which an homogeneous space of the same dimension is tangentially associated, with the possibility of connecting these tangent spaces at different neighbouring points of the base space. Moreover these spaces were endowed with a \"normal\" connection which allows to develop the base space on the tangent homogeneous space along a curve. In a geometrical language the above depicted situation is described by using the notions of Cartan connection and Cartan structure. The classical geometries i.e. the projective [4] and conformal [5] geometries are the standard examples of Cartan structures; they correspond to the case where the bigger concerned Lie group is semisimple and its Lie algebra is jlj-graded. A general study of this case can be found in the literature [6]. On the contrary the geometrical structures considered in this paper, do not enter this scheme. They deal with a group which is not semi-simple and whose Lie algebra is |2|-graded, the so-called chronopro-

The local homology of cut loci in Riemannian manifolds

Abstract: point q-and the position of q in the simplicial decomposition. Cut points of order one are extreme points of the simplicial complex; those of order two are interior to an edge; and those of order k•†3 are vertices where k edges meet. In short, the topology of a neighborhood of a point in the cut locus is determined by the order of the cut point. Recently, Ozols [12] and Buchner [3] have shown that the cut locus of a point p in a real analytic Riemannian manifold admits a simplicial decomposition. Moreover, Ozols [12] describes the structure of the cut locus near a non-conjugate cut point q as a finite (depending on the order of q) intersection of hyperspaces and half-planes, while Buchner [4] completely classifies the local structure of generic cut loci in low dimensional manifolds. In general, however, the relation between the set, called the link, of minimal geodesics connecting p to a cut point q and the structure of the cut locus near q remains obscure. This paper establishes, as a consequence of Poincare duality, a duality between the Cech cohomology of the link and the local homology groups of cut loci in smooth Riemannian manifolds, thereby weakly generalizing the result of Myers on the order and local topology of cut loci in real analytic surfaces. Using standard arguments from algebraic topology, we show that certain local homology groups of cut loci are torsion free. Finally, we prove interconnections between the dimension of the cut locus and the vanishing of high dimensional local homology which lead up to a generalization of a theorem of Bishop [2] on the decomposition of cut loci.

The Gauss-Bonnet integrand for a class of Riemannian manifolds admitting two orthogonal complementary foliations

Abstract: Abstract With the general assumption that the manifold admits two orthogonal complementary foliations, one of which is totally geodesic, we study the components of the curvature tensor field of the characteristic connection. In the case where the manifold is compact, orientable of dimension 6 or 8 and the dimension of the totally geodesic foliation is 4, we relate the sign of the Euler characteristic of the manifold and that of the sectional curvature of the leaves of both foliations.