Showing papers on "Scalar curvature published in 1969"

Curvature Collineations: A Fundamental Symmetry Property of the Space‐Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

Abstract: A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel‐Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.

On conformal killing tensor in a riemannian space

Abstract: where pc is a certain vector field. Because we can easily show that a conformal Killing tensor in this sense is a Killing tensor, i.e., we have pc = 0. Thus this definition of conformal Killing tensor is meaningless. In this paper we shall define a conformal Killing tensor in another way and generalize some results about a conformal Killing vector to the conformal Killing tensor. The definition which we shall adopt is suggested by the following fact. A parallel vector field in the Euclidean space E induces a

A formula of Simons' type and hypersurfaces with constant mean curvature

Abstract: In a recent work [8] J. Simons has established a formula for the Laplacian of the second fundamental form of a submanifold in a Riemannian manifold and has obtained an important application in the case of a minimal hypersurface in the sphere, for which the formula takes a rather simple form. The application is made by means of the Laplacian of the function / on the hypersurface, which is defined to be the square of the length of the second fundamental form. In the present paper, by a more direct route than Simons' we first obtain the same type of formula (see (16)) in the case of a hypersurface M immersed with constant mean curvature in a space M of constant sectional curvature, and then derive a new formula (see (18)) for the function / which involves the sectional curvature of M. Based on this new formula our main results are the determination of hypersurfaces M of non-negative sectional curvature immersed in the Euclidean space R or the sphere 5 with constant mean curvature under the additional assumption that the function / is constant. This additional assumption is automatically satisfied if M is compact. We state the general results in a global form assuming completeness of M, but they are essentially of local nature.

Curvature and the Petrov Canonical Forms

Abstract: The Petrov classification for the curvature tensor of an Einstein space M4 is related to the critical‐point theory of the sectional‐curvature function σ, regarded as a function on the manifold of nondegenerate tangent 2‐planes at each point of the space. It is shown that the Petrov type is determined by the number of critical points. Furthermore, all the invariants in the canonical form can be computed from a knowledge of the critical value and the Hessian quadratic form of σ at any single critical point.

Riemannian manifolds admitting a conformal transformation group

Abstract: Let M be a Riemannian manifold with constant scalar curvature K which admits an infinitesimal conformal transformation A necessary and sufficient condition in order that it be isometric with a sphere is obtained Inequalities giving upper and lower bounds for K are also derived

On isoperimetric and various other inequalities for a manifold of bounded curvature

Abstract: | I . F U N D A M E N T A L D E F I N I T I O N S A N D F O R M U L A T I O N O F R E S U L T S Le t F be a compac t two--dimensi0nal manifold of bounded cu rva tu re [1] with a nonempty edge L cons is t ing of a finite number of rec t i f iable curves , homeomorphlc to a disc . We also denote by F and L, r e spec t ive ly , the a r e a of the manifold F and the length of i ts boundary., Unless otherwise s tated, the a rea or length of a se t of the space (if it is meaningful to speak of them) will be denoted in the following by the s ame symbol as the se t i tself . This will not lead to confusion, since it will a lways be c l ea r f rom the context what is meant . We denote by w(E) and x(E), respec t ive ly , the cu rva tu re and the Euler cha rac t e r i s t i c of a se t E c F. The g r e a t e s t dis tance f rom points XE F to L will be cal led the internal radius of the space F, denoted by a iF). We shall define the excess of the curva ture of F with r e spec t to a rea l numoer l~. as the di f ference

Sources of curvature of a vector field

Abstract: It is known that for a vector field in three-dimensional space we can introduce the concepts of curvature and mean curvature. In the present article we derive integral formulas for these concepts; these formulas allow us to decide whether a vector field has, for example, singularities in a domain. We explain the influence of the modulus of the curvature of a vector field on the magnitude of its nonholonomity. We also consider the question of the influence of the curvature of a family of surfaces on the distortion of the enveloping space for a given size of domain. Bibliography 5 items.

Infinitely small transformations preserving the curvature of a Riemannian space

Abstract: The necessary and sufficient conditions for tensor character are obtained, for which an infinitely small transformation of the space Vn preserves its Riemannian curvature for any two-dimensional area. It is proved that for n>3 the subprojective spaces of the exceptional case, satisfying a certain condition, and only they, permit nontrivial, infinitely small conformal transformations preserving the Riemannian curvature of each two-dimensional area.

On the interior regularity of weak solutions of non-stationary Navier-Stokes equations on a riemannian manifold

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