Showing papers on "Scalar curvature published in 1968"

Growth of finitely generated solvable groups and curvature of Riemannian manifolds

Abstract: If a group Γ is generated by a finite subset 5, then one has the "growth function" gs, where gs(m) is the number of distinct elements of Γ expressible as words of length

The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature

Abstract: With an immersion x of a Riemannian n-manifold M into a Euclidean Nspace E there is associated the Gauss map, which assigns to a point p of M the n-plane through the origin of E and parallel to the tangent plane of x(M) at x(p), and is a map of M into the Grassmann manifold G Λ f # = O(N)/O(n)χO(N-n). An isometric immersion of M into a Euclidean iV-sphere S can be viewed as one into a Euclidean (N + l)-sρace E**, and therefore the Gauss map associated with such an immersion can be determined in the ordinary sense. However, for the Gauss map to reflect the properties of the immersion into a sphere, instead of into the Euclidean space, it seems desirable to modify the definition of the Gauss map appropriately. To this end we consider the set Q of all the great n-spheres in 5\", which is naturally identified with the Grassmann manifold of (« -f l)-planes through the center of S in E^ , since such {n + l)-planes determine unique great n-spheres and conversely. In this paper by the Gauss map of an immersion x into 5* is meant a map of M into the Grassmann manifold Gn+ltK+1 which assigns to each point p of M the great H-sphere tangent to x(M) at *(/?), or the (n + l)-plane spanned by the tangent space of x(M) at x(p) and the normal to S at x(p) in E. More generally, with an immersion x of M into a simply-connected complete N-sρace V of constant curvature there is associated a map which assigns to each point p of M the totally geodesic w-subspace tangent to x(M) at*(p). Such a map is called the (generalized) Gauss map. Thus the Gauss map in our generalized sense is a map: M —• Q, where Q stands for the space of all the totally geodesic H-subspaces in V. The purpose of the present paper will be first to obtain a relationship among the Ricci form of the immersed manifold and the second and third fundamental forms of the immersion, and then to give a geometric interpretation of the

On hypersurfaces satisfying a certain condition on the curvature tensor

Abstract: where the endomorphism R(X, Y) operates on R as a derivation of the tensor algebra at each point of M. Conversely, does this algebraic condition (•*) on the curvature tensor field R imply that M is locally symmetric (i.e. Vi? = 0) ? We conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3 . For partial and related results, see [4], p.ll, [9], Theorem 8, and [6]. The main purpose of the present paper is to give an affirmative answer in the case where M is a complete hypersurface in a Euclidean space. More precisely, we prove

Exploding spheres of dust

Abstract: In this paper, we extend the work of Oppenheimer & Snyder (1939) who treated the zero curvature case in the gravitational collapse of spheres of dust intended to represent collapsing stars. A solution is given which is valid for all space and is characterized by negative curvature of the space within a sphere of dust. This solution is obtained by matching the negative curvature interior solution (as well as, for completeness, zero and positive curvatures interior solutions) to an exterior Schwarzschild geometry. In this solution, corresponding to the case of a Newtonian system with positive total energy, the mass as seen by an observer at infinity is found to be positive definite. Also, in each case, the positive definite massm is related to the densityν and radiusr [defined as the square root of the (surface area/4π)] of the dust cloud viam=(4π/3)pr3. The methods employed here for matching interior and exterior solutions are applicable to the construction of cosmological models in which the sign of the curvature and/or expansion rate differ in two or more regions, e.g. a universe expanding in one region and contracting in another.

Divergence properties of curvatures of a vector field and a family of surfaces

Abstract: A representation in divergent form is given for the total curvature and mean curvature of a vector field.