Showing papers on "Riemann curvature tensor published in 1990"

On complete manifolds with nonnegative Ricci curvature

Abstract: Complete open Riemannian manifolds (Mn, g) with nonnegative sectional curvature are well understood. The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CG1]. The Splitting Theorem has been extended to manifolds of nonnegative Ricci curvature [CG2]. On the other hand, the Soul Theorem does not extend even topologically, according to recent examples in [GM2]. A different method to construct manifolds which carry a metric with Ric > 0, but no metric with nonnegative sectional curvature, has been given by L. Berard Bergery [BB]. This leads to the question (cf. also [Y1]): Is there any finiteness result for complete Riemannian manifolds with Ric > 0 ? The answer is certainly affirmative in the low-dimensional special cases n = 2, where all notions of curvature coincide, and n = 3, where nonnegative Ricci curvature has been studied by means of stable minimal surfaces [MSY, SY]. On the other hand, J. P. Sha and D. G. Yang [ShY] have constructed complete manifolds with strictly positive Ricci curvature in higher dimensions. For example they can choose the underlying space to be R4 x S3 with infinitely many copies of S3 x CP 2 attached to it by surgery; cf. also [ShY 1]. It is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions. The purpose of this paper is to establish the following main result.

On Rotationally Symmetric Hamilton’s Equation for Kähler–Einstein Metrics

Abstract: Publisher Summary This chapter focuses on rotationally symmetric Hamilton's equation for Kahler–Einstein metric. Any Riemannian metric g0 with positive Ricci curvature on a compact three-dimensional manifold is deformed to an Einstein metric along the equation , where rt denotes the Ricci tensor of gt and the mean value of the scalar curvature. The chapter explains how the solution of an equation converges to a Kahler–Einstein metric if it exists, even on a compact Kahler manifold with positive first Chern class.

Legendre transformation and dynamical structure of higher derivative gravity

Abstract: Among the so-called 'non-linear' (purely metric) Lagrangians for the gravitational field, those which depend in a quadratic way on the components of the Riemann tensor have been given particular consideration by many authors. In this paper, the authors deal with the most general quadratic Lagrangian depending on the full Riemann tensor, in arbitrary dimension; instead of considering the corresponding fourth-order Euler-Lagrange equations, they investigate an equivalent set of second-order quasilinear equations which are obtained by (a suitably generalised) Legendre transformation. In this framework, they compare this class of theories with those depending on the Ricci tensor only, showing that the Weyl tensor dependence breaks the equivalence with general relativity, but the new auxiliary field arising in this case has no dynamical term. The degeneracy occurring for a suitable choice of the parameters in the Lagrangian is widely discussed, and some effects of a non-minimal coupling with an external scalar field are also described.

Solution of the Dirichlet Problem for Curvature Equations of Order m

Abstract: Solvability conditions for curvature equations of order which are sufficient, and almost necessary, are obtained, and theorems concerning the existence of solutions in , , , are proved. The first-order curvature equation coincides with the curvature equation of order , and the curvature equation of order with the Monge-Ampere equation.Bibliography: 18 titles.

Conformal symmetries and fixed points in space‐time

Abstract: A general study is made of conformal vector fields on a four‐dimensional Lorentz manifold with particular emphasis being laid on the structure of the zeros (critical points) of such vectors fields. The implications for general relativity are investigated and a discussion of conformal vector fields in generalized plane wave space‐times is given. An attempt is made to clarify the well‐known theorem of Bilyalov and Defrise‐Carter.

Gravitational instantons as su(2) gauge fields

Abstract: Self-dual solutions to the complex Einstein equation with cosmological constant are characterised as SL(2, C) gauge fields satisfying an equation quadratic in the curvature. For a solution, a tetrad is constructed with respect to which the connection is the left handed spin connection. For real Euclidean signature, the connection reduces to an SU(2) connection.

Twistor spaces with hermitian ricci tensor

Abstract: The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics ht compatible with the almost-complex structures J, and J2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds M for which the Ricci tensor of (Z , ht) is ./"-Hermitian, n = 1 or 2. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kahler manifold with Hermitian Ricci tensor is Kahlerian.

On two theorems of I. M. Singer about homogenous spaces

Abstract: A theorem of I. M. Singer [9] states that a Riemannian manifold is locally homogeneous if and only if the Riemannian curvature tensor and its covariant derivatives are the same at each point up to some orderkM + 1.

Inconsistency of spin 4-spin 2 gauge field couplings

Abstract: The authors invert the usual problem of coupling higher spin gauge fields to gravity by treating (linearised) gravity as the 'matter field' source of spin 4 gauge theory. This is motivated by the existence of the conserved gravitational four-index Bel-Robinson tensor as a possible current for the spin 4 field. They first derive this tensor as a Noether current, thereby linking it to a novel invariance of spin 2. It is then shown that, as usual for higher spins, consistency does not survive beyond lowest-order cubic coupling.

Positive scalar curvature and periodic fundamental groups

Abstract: is formed by double contraction of the riemannian curvature tensor of g (compare [He, pages 74-75]). Geometrically speaking, the scalar curvature function measures the difference between the volumes of the riemannian and euclidean geodesic disks. The existence of a riemannian metric with positive scalar curvature on a smooth manifold turns out to be of interest in many contexts. For example, by results of J. Kazdan and F. Warner [KW] the entire question of realizing a smooth function as a scalar curvature reduces to the existence of such a metric, and results of R. Schoen [Schn] on the Yamabe problem show that a metric with positive scalar curvature can be conformally deformed to one with

Counter-Example to the "Second Singer's Theorem"

Abstract: We give an explicit example showing that a theorem by I. M. Singer announced in [3] (about the existence of a Riemannian homogeneous space with the prescribed curvature tensor and some of its covariant derivatives) cannot hold without an additional topological condition of closeness. All references in this short note concern Chapter 3 of the paper by L. Nicolodi and F. Tricerri [2] published in the same volume. We shall use freely the concepts and formulas from there. Consider the infinitesimal model (V,T,K) given as follows: LetV be a 5-dimensional

Singular Spaces of Non-Positive Curvature

Abstract: In [Gr5] Gromov explains the definition of upper curvature bounds for singular spaces, a concept which goes back to A.D. Aleksandrov, cf. [ABN]. Below is a discussion of this material. The main application is a criterion for the hyperbolicity of certain simply connected polyhedra.

Transversal affine connection and quantization of constrained systems

Abstract: The Dirac quantization of a finite‐dimensional relativistic system with a quadratic super‐Hamiltonian and linear supermomenta is investigated. In a previous work, the operator constraints were consistently factor‐ordered in such a way that the resulting quantum theory was invariant under all relevant transformations of the classical theory. The method was based on a special choice of coordinates and gauge. Here, coordinate‐independent methods are worked out and a quite general gauge is used. A new mathematical concept, the so‐called ‘‘transversal affine connection,’’ is introduced. This connection is not a linear connection and is associated with a degenerate metric. The corresponding curvature tensor is defined and its components are calculated. The formalism is used to reconstruct the operator constraints, clarify their geometric meaning, and calculate their commutators.

On real hypersurfaces of a complex space form with ^-parallel ricci tensor

Abstract: Let Mn(c) denote an n-dimensional complex space form with constant holomorphic sectional curvature c. It is well known that a complete and simplyconnected complex space form consists of a complex projective space CPn, a complex Euclidean space Cn or a complex hyperbolic space CHn, according as c>0, c=0 or c<0. In this paper we consider a real hypersurface M of CPn or CHn. The study of real hypersurfaces of CPn was initiated by Takagi [10], who proved that all homogeneous hypersurfaces of CPn could be divided into six types which are said to be of type Au A2, B, C, D and E. Moreover, he showed that if a real hypersurface M of CPn has two or three distinctconstant principal curvatures, then M is locally congruent to one of the homogeneous ones of type Au A2 and B ([11]). Recently, to give another charac terization of homogeneous hypersurfaces of type Alt A2 and B in CPn Kimura and Maeda [6] introduced the notion of an ^-parallel second fundamental form, which was defined by g((FxA)Y, Z)=0 for any vector fields X, Y and Z orthogonal to the structure vector field£,where A means the second fundamental form of M in CPn, and g and V denote the induced Riemannian metric and the induced Riemannian connection, respectively. On the other hand, real hypersurfaces of CHn have also been investigated by many authors (Berndt [1], Montiel [8], Montiel and Romero [9]). Using some results about focal sets, Berndt [1] proved the following.

Mapping Noether Identities into Bianchi Identities in General Relativistic Theories of Gravity and in the Field Theory of Static Lattice Defects

Abstract: Noether identities resulting from external symmetries represent “conservation” laws in relativistic field theories and balance laws in 3-dimensional continuum statics, respectively In a suitably selected 4-dimensional non-Euclidean space-time (3-dimensional stress space), the momentum currents (stresses) entering the conservation (balance) laws can be mapped such that the Noether identities become Bianchi identities, or irreducible pieces thereof Using a metric-affine space with independent metricg αβ and connection Γ , we derive the following types of mapping prescriptions: momentum current → (contraction of) curvature; spin current → torsion; shear current → trace-free nonmetricity; dilation current → Weyl 1-form The last two mappings constitute the main result The mapping of the dilation current turns out to be exceptional, since it does not yield a nontrivial Bianchi identity

On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection

Abstract: We give a complete decomposition of the space of curvature tensors with the symmetry properties as the curvature tensor associated with a symmetric connection of Riemannian manifold We solve the problem under the action ofS0(n) The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined Several applications for Riemannian manifolds with symmetric connection are given The group of projective transformations of a Riemannian manifold and its subgroups are considered

Codimension one tori in manifolds of nonpositive curvature

Abstract: A complete Riemannian manifold of nonpositive sectional curvature and finite volume contains a totally geodesic flat torus of codimension one provided it contains a codimension one flat.

Structure of visual perception.

Abstract: The response properties of a class of motion detectors (Reichardt detectors) are investigated extensively here. Since the outputs of the detectors, responding to an image undergoing two-dimensional rigid translation, are dependent on both the image velocity and the image intensity distribution, they are nonuniform across the entire image, even though the object is moving rigidly as a whole. To achieve perceptual "oneness" in the rigid motion, we are led to contend that visual perception must take place in a space that is non-Euclidean in nature. We then derive the affine connection and the metric of this perceptual space. The Riemann curvature tensor is identically zero, which means that the perceptual space is intrinsically flat. A geodesic in this space is composed of points of constant image intensity gradient along a certain direction. The deviation of geodesics (which are perceptually "straight") from physically straight lines may offer an explanation to the perceptual distortion of angular relationships such as the Hering illusion.

On the curvature description of gravitational fields

Abstract: The local geometry of a Riemannian manifold is described completely by the curvature tensor and a finite number of its covariant derivatives. The authors derive necessary and sufficient conditions for a line element to exist that gives rise to these quantities. The resulting system of equations can be written as a set of integrable differential equations along with a set of algebraic equations. This gives a technique for searching for solutions to Einstein's equations with special properties. It also makes it possible to perform perturbative calculations entirely in terms of invariant quantities. They illustrate the methods on homogeneous rotationally symmetric spacetimes.

Connection metrics of nonnegative curvature on vector bundles

Abstract: It is shown that a vector bundleE over a compact positively curved manifold admits complete Riemannian metrics of nonnegative sectional curvature, provided there exists a Riemannian connection onE with almost parallel curvature tensor.

On a computational method for the second fundamental tensor and its application to bifurcation problems

Abstract: An algorithm is presented for the computation of the second fundamental tensorV of a Riemannian submanifoldM ofR n . FromV the riemann curvature tensor ofM is easily obtained. Moreover,V has a close relation to the second derivative of certain functionals onM which, in turn, provides a powerful new tool for the computational determination of multiple bifurcation directions. Frequently, in applications, thed-dimensional manifoldM is defined implicitly as the zero set of a submersionF onR n . In this case, the principal cost of the algorithm for computingV(p) at a given pointp?M involves only the decomposition of the JacobianDF(p) ofF atp and the projection ofd(d+1) neighboring points ontoM by means of a local iterative process usingDF(p). Several numerical examples are given which show the efficiency and dependability of the method.