Showing papers on "Scalar curvature published in 1973"

Spectral Asymmetry and Riemannian Geometry

Abstract: This has an analytic continuation to the whole s-plane as a meromorphic function of 5 and s = 0 is not a pole: moreover CA(o) can be computed as an explicit integral over the manifold [9]. In this note we shall introduce a refinement of this invariant when A is no longer positive and we shall study its geometrical significance for an important class of operators (first order systems) arising from Riemannian geometry. A full exposition will be given elsewhere. Suppose therefore that A is self-adjoint and elliptic but no longer positive. The eigenvalues are now real but can be positive or negative. We define, for Re(s) large,

Surfaces of constant mean curvature in manifolds of constant curvature

Abstract: An immersed surface in a three-dimensional Euclidean space E has constant (scalar) mean curvature if the length of the mean curvature vector H is constant. An arbitrary isometric immersion M<=—>M of Riemannian manifolds is said to have constant mean curvature if H is parallel in the normal bundle of the immersion (for definitions see § 1). This condition is stronger than the requirement \\H\\ = constant c. In the case of immersions of surfaces into manifolds of constant curvature we generalize many known facts and theorems about surfaces of constant (scalar) mean curvature in E. The main theorems of this paper were announced in Hoffman [7], and we refer the reader there for a more lengthy introduction and statement of results. What follows is a brief sketch of the principal results. To a surface of constant mean curvature given in conformal coordinates we associate an analytic function φ constructed out of the second fundamental form in the mean curvature direction (Lemma 2.1). This was first done for surfaces in E by Heinz Hopf [8]. Under certain additional assumptions, the same procedure works for other normal directions. These functions have direct geometrical meaning which is discussed in § 2. In particular they are used to prove Theorem 2.2(b): The only genus zero surfaces of constant mean curvature in E or the standard 4-sρhere S are the standard 2-sρheres. Theorem 3.1 gives a local characterization of constant (nonzero) mean curvature immersions which have constant Gauss curvature they are shown to be pieces of 2-spheres or products of 1-spheres, S(r) x S(ρ), 0 < r < oo, 0 < p < oo. Theorem 4.1 classifies complete surf aces of constant mean curvature in E and S\\ whose Gauss curvature does not change sign they must be minimal surfaces, 2-sρheres or S^r) x S^p), 0 < r < o o , 0 < / ? < oo. In § 5 we use the analytic functions of Lemma 2.1 to construct local examples of surfaces of constant mean curvature in 4-dimensional manifolds of constant curvature (Theorem 5.1). In these examples for the case of immersions into E or S\\ the surfaces do not lie minimally in hyperspheres of £ 4 or S

Curvature tensors and covariant derivatives

Abstract: The problems considered here are of two types.(i) What are implications of vanishing k-th covariant derivatives of curvature tensors?(ii) Under what conditions on curvature tensors, does the k-th covariant derivative ∇kT=0 for a tensor T mean ∇T=0?

Isometric immersions of two-dimensional riemannian metrics in euclidean space

Abstract: In this paper we consider the problem of global isometric immersions of two-dimensional Riemannian metrics in Euclidean space. The dimension of the ambient space depends on the character of the Riemannian metric in question.

On a Variational Problem on Hypersurfaces

Abstract: \«\dV £ cm, (1) M where cm denotes the area of a unit m-sphere. The equality sign of (1) holds when and only when M is a hypersphere in E. In order to know whether the inequality (1) can be improved for some given hypersurfaces in E, it is important to know the S-hypersurfaces in E, i.e., the stable hypersurfaces in E with respect to the integral ot!"dV. The main purpose of this paper is to study the S-hypersurfaces in E. In §1, we obtain a necessary and sufficient condition for S-hypersurfaces in terms of mean curvature and scalar curvature. In §2, we obtain two applications.

Scalar curvature, inequality and submanifold

Abstract: Using an inequality relation between scalar curvature and length of second fundamental form, we may conclude that a submanifold must have nonnegative (or positive) sectional curvatures. An application to compact submanifolds in obtained. 1. Statement of results.' Let M be an n-dimensional submanifold of an (n+p)-dimensional Riemannian manifold N of constant sectional curvature c, and let h and H be the second fundamental form and the mean curvature vector field respectively. Let M., i, j=1,* , n, O= n.+ 1,* , n+p, be the coefficients of the second fundamental form h with respect to a local field of orthonormal frame el, * *, en, en+l, *' *, en+p. Then the square of length of second fundamental form, S, and the scalar curvature, R, of M are given respectively by n+p n (1) S = E E j)2 a=n+l i, j=l (2) R =n2H H-S + n(n-l)c, where dot "" denotes the scalar product of vectors. A normal vector field q is said to be parallel if Dr=O identically, where D denotes the connection of the normal bundle. The purpose of this paper is to show the following THEOREM 1. Let M be an n-dimensional submanifold of a Riemannian manifold N of constant curvature c. If the scalar curvature R satisfies 3 R _ (n 2)S + (n 2)(n l)c (3) (resp. R > (n 2)S (n 2)(n 1)c) at a point p E M, then the sectional curvatures of M are nonnegative (resp. positive) at p. Received by the editors May 2, 1972 and, in revised form, July 31, 1972. AMS (MOS) subject classifications (1970). Primary 53B25, 53C40.

Isometric embedding of a compact Riemannian manifold into Euclidean space

Abstract: An isometric immersion of an «-dimensional compact Riemannian manifold with sectional curvature always less than A~2 into Euclidean space of dimension In — 1 can never be contained in a ball of radius X. This generalizes and includes results of Tompkins and Chern and Kuiper. Tompkins (4) proved that the flat «-dimensional torus could not be isometrically embedded into E, the Euclidean space of dimension 2«—1. Chern and Kuiper (1) and Otsuki (3) generalized this and showed that a compact «-dimensional manifold whose sectional curvatures are every- where nonpositive also cannot be isometrically embedded in E. Apparently these results also hold for immersions. The purpose of this note is to point out that a standard proof of this nonembeddability result essentially establishes a more general quantitative version. Theorem. Let E be Euclidean space of dimension 2«—1 and M a compact n-dimensional Riemannian manifold whose sectional curvatures are everywhere less than some constant X~2. Then no isometric immersion of M into E is contained in a ball of radius X. To prove the theorem, we adapt the proof for nonpositive curvature given in Kobayashi and Nomizu (2, pp. 26-29). Note the theorem in- cludes the results mentioned above. Let/:Af—>-£ be an isometric embedding. Identify the tangent space TXM for x e M with its realization as a linear subspace of E. Denote the length and inner product of vectors in £ by \X\ and (X, Y). To prove the theorem, let us assume such an isometric immersion did exist. We can assume \f(x)\^X while \f(x0)\ = X for some x0 and all x in M. Thus (f(x0), A'>=0forallArin TxiiM. Let L(X, Y)denote thesecondfundamental form at x0 of M in E. For X and Y in Tx M, L(X, Y)is a vector in Forthog- onal to M at x0. The sectional curvature of the two-plane spanned by

The exterior diameter of an immersed riemannian manifold

Abstract: In this paper we establish a general lower bound for the radius of the sphere containing a complete regular Riemannian manifold in Euclidean space n$ SRC=http://ej.iop.org/images/0025-5734/21/3/A07/tex_sm_2027_img2.gif/>.Bibliography: 10 titles.

On three-dimensional Riemannian spaces with curvature bounded above

Abstract: For three-dimensional closed simply-connected Riemannian spaces one gets a lower bound for the radius of injectivity of the exponential map from a lower bound of the sectional and an upper bound of the Ricci curvatures.

Estimates for solutions of quasilinear elliptic equations connected with problems of geometry "in the large"

Abstract: The following questions are presented in this paper: A geometric method for obtaining two-sided estimates for general quasilinear elliptic equations and its applications to problems of the calculus of variations and the problem of recovering a hypersurface from its mean curvature in spaces of constant curvature. Estimates of the modulus of the gradient for a hypersurface with boundary in a Riemannian space by means of its mean curvature and the metric tensor of the space. Estimates of the modulus of the gradient of a hypersurface depending on the distance of a point from the boundary and its mean curvature in Euclidean space. Estimates of these three types are of independent interest and play a fundamental role in the proofs of existence theorems for a hypersurface with prescribed mean curvature in Riemannian spaces. Bibliography: 3 items.

Families of negatively curved Hermitian manifolds

Abstract: A complex analytic family of compact hermitian manifolds has negative holomorphic sectional curvature in a neighborhood of any fibre having negative holomorphic sectional curvature.

Generalizing Riemannian geometry

Abstract: The properties of Riemannian geometry necessary to relativity have been used as a basis to derive a more general geometry. Emphasis is placed on a natural development with the result of considerable generalization. Several examples are discussed including the Brans‐Dicke field equation which are but one special case of the new manifolds. The scalar field is not introduced ad hoc but is a natural geometrical quantity.