Showing papers on "Scalar curvature published in 1971"

A Scalar-Tensor Theory of Gravitation in a Modified Riemannian Manifold

Abstract: A new scalar‐tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance. This is in contrast to the well‐known Brans‐Dicke theory where the tensor field alone is geometrized and the scalar field is alien to the geometry. The static spherically symmetric solution of the exterior field equations is worked out in detail.

Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold.

Abstract: If M is an ra-dimensional differentiable manifold (n £ N) with the Riemannian metric g, then the tangent b ndle TM of M admits a canonical Riemannian metric Tg (see [1], [2]). In other words, a metric connection v on M induces, in a canonical way, a metric connection v on TM. Further, A. J. Ledger and K. Yano ([3], [4]) found a different construction joining to any linear connection v on M a linear connection v on TM. The basic result of [4] says that the space (M, v) is locally Symmetrie if and only if the space (T M, v) is locally Symmetrie, In the present paper we compute the curvature tensor of the Riemannian space (TM, Tg) corresponding to (If, g). We deduce, in contrast to the Yano-Ledger's theory, the following result: The space (TM, Tg) is never locally Symmetrie unless (M, g) is locally euclidean.

Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds

Abstract: Let (M,g) denote a smooth (say C) compact two-dimensional manifold, equipped with some Riemannian metric g. Then, as is well-known, M admits a metric gc of constant Gaussian curvature c in fact the metrics g and gc can be chosen to be conformally equivalent. Here, we determine sufficient conditions for a given non-simply connected manifold M to admit a Riemannian structure g (conformally equivalent to g) with arbitrarily prescribed (Holder continuous) Gaussian curvature K(x). If the Euler-Poincare characteristic χ(M) of M is negative, the sufficient condition we obtain is that K(x) < 0 over M. Note that this condition is independent of g, and this result is obtained by solving an isoperimetric variational problem for g. If K(x) is of variable sign for χ(M) < 0, or if χ(M) > 0, then the desired critical point may not be an absolute minimum and our methods do not succeed. If χ(M) = 0, our methods apply when K(x) satisfies an integral condition with respect to the given metric g (see § 3) this result is perhaps not unreasonable since, for χ(M) < 0, distinct Riemannian structures on M need not be conformally equivalent.

An existence theorem for surfaces of constant mean curvature

Abstract: Let y be an oriented Jordan curve in Euclidean space, E3, which is homeomorphic to the unit circle, u2 + v 2 = 1. Let A denote the disk u2 + n2 < 1, d its closure, and let C”(d), C?(A) be th e set of n-times continuously differentiable vector functions x : A + E3 or x : d + E3. x E Co(d) n C2(A) is said to be a conformal representation of a surface of constant mean curvature, H, and boundary y, if X(U, v) satisfies

Submanifolds with parallel mean curvature vector

Abstract: J. Simons [5] has recently proved a formula which gives the Laplacian of the square of the length of the second fundamental form, and applied the formula to the study of minimal hypersurfaces in the sphere (see also [1], [2]). K. Nomizu and B. Smyth [4] have obtained a formula of the same type for a hypersurface immersed with constant mean curvature in a space of constant sectional curvature, and derived a new formula for the Laplacian of the square of the length of the second fundamental form, in which the sectional curvature of the hypersurface appears. Using this new formula, they determined hypersurfaces of nonnegative sectional curvature and constant mean curvature immersed in the Euclidean space or in the sphere under the additional condition that the square of the length of the second fundamental form is constant. The purpose of the present paper is to generalize Nomizu-Smyth formulas to the case of general submanifolds and to use the formulas to study submanifolds, immersed in a space of constant curvature, whose normal bundle is locally parallelizable and mean curvature vector field is parallel in the normal bundle.

Mean Curvature of Riemannian Immersions

Abstract: 1. Let M and M' denote complete riemannian manifolds of dimension n and m respectively, and suppose that M is compact and oriented. For simplicity we assume that both manifolds and their metrics are smooth (i.e. of class C). In terms of local co-ordinates (x, x, ...,x") on M and local co-ordinates (y,y, •••,/") on M', the riemannian metrics are written ds = gtJ dx l dx, ds' = g'aP dy* dy * where Roman suffixes take values 1,2, ..., n and Greek suffixes take values 1,2,..., m. Let/: M ->• M' be a smooth map. Following Eells and Sampson [1], we associate with / a real number called its energy. We define an inner product on the space of 2-covariant tensors at P £ M in terms of local co-ordinates by

Curvature and differentiable structure on spheres

Abstract: L Introduction. The purpose of this note is to outline a proof of the following result: A simply connected, complete, riemannian manifold whose curvature tensor R is sufficiently close to the curvature tensor Ro of the standard sphere 5 of the same dimension is diffeomorphic to S. Traditionally, the proximity of R and Ro has been measured in terms of the sectional curvature as follows: A riemannian manifold is called 8-pinched if the sectional curvature K satisfies the condition ô

Rigidity of hypersurfaces of constant scalar curvature

Abstract: Let Mn and Mn+i be Riemannian manifolds of dimension n and n+1 respectively. Assume Mn isometrically immersed in Mn+iIf each point p of Mn is contained in an open neighborhood U This concept allowed us to show that the result of NaganoTakahashi [3] holds without any restriction, i.e. that any homogeneous hypersurface of the Euclidean space JSn+i is isometric to the Riemannian product of a ^-dimensional sphere and an n—p dimensional Euclidean space. This result is a consequence of the following

On the mean curvature of submanifolds of euclidean space

Abstract: Let ( , ) denote the scalar product of E. If there exists a function f on M such that (h(u, v), H)=f(u, v) for all tangent vector fields u, v on M, then M is called a pseudo-umbilical submanifold of E. If the covariant derivative of i f in E is tangent to x(M) everywhere, then H is said to be parallel in the normal bundle. In [2], [3], the author proved that if M is closed, then the mean curvature vector H satisfies

An estimate from below for the spatial diameter of a surface in terms of its intrinsic radius and curvature

Abstract: In this paper we prove the followingTheorem. Let be a regular simply connected surface of class in . There exist positive absolute constants and such that if , where is the Gaussian curvature and is the area element on , the estimate holds. Bibliography: 11 items.

Integral formulae on hypersurfaces in Riemannian manifolds

Abstract: The coefficients of the complete set of n fundamental forms of a hypersuface Vn−1 imbedded in an n-dimensional Riemannian space Vn, as recently introduced[(5)], are used to construct certain tensor fields over Vn−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of Vn, the coefficients of the n fundamental forms, and the rth curvatures of Vn−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on Vn−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of Vn−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever Vn is a space of constant curvature.