Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

On some class of hypersurfaces with three distinct principal curvatures

Abstract: ON SOME CLASS OF HYPERSURFACES WITH THREE DISTINCT PRINCIPAL CURVATURES KATARZYNA SAWICZ Institute of E onometri s and Computer S ien e Te hni al University of Cz sto howa Armii Krajowej 19B, 42-200 Cz sto howa, Poland E-mail: ksawi z zim.p z. zest.pl Abstra t. We investigate hypersurfa es M in spa es of onstant urvature with some spe ial minimal polynomial of the se ond fundamental tensor H of third degree. We present a urvature hara terization of pseudosymmetry type for su h hypersurfa es. We also prove that if su h a hypersurfa e is a manifold with pseudosymmetri Weyl tensor then it must be pseudosymmetri . 1. Introdu tion. Let M , n = dimM ≥ 3, be a onne ted hypersurfa e in a semiRiemannian manifold (N, g ). We denote by g the metri tensor indu ed on M from g . Further, let H, resp. A, be the se ond fundamental tensor, resp. the shape operator, of (M, g) in (N, g ). It is well known that H(X,Y ) = g(AX,Y ), for any ve tor elds X and Y tangent to M . We de ne the (0, 2)-tensor Hk, k ≥ 1, by H(X,Y ) = g(AX,Y ), where H = H and A = A. In Se tions 3 and 4 we present further basi fa ts relating to hypersurfa es. A hypersurfa e M , n ≥ 3, in (N, g ) is said to be quasi-umbili al at x ∈M if at this point we have H = αg + βu⊗ u, u ∈ T ∗ xM, α, β ∈ R. (1) If α = 0 (resp., β = 0 or α = β = 0) at x then M is alled ylindri al (resp., umbili al or geodesi ) at x. If (1) is ful lled at every point of M then it is alled a quasi-umbili al hypersurfa e. A hypersurfa e M , n ≥ 4, in (N, g ) is said to be 2-quasi-umbili al at x ∈M (see [16℄ and referen es therein) if at this point we have H = αg + βu⊗ u+ γv ⊗ v, u, v ∈ T ∗ xM, α, β, γ ∈ R, (2) 2000 Mathemati s Subje t Classi ation: Primary 53B20, 53B25; Se ondary 53C25.

Principal null directions of the Curzon metric

Abstract: The Curzon metric has an invariantly defined surface, given by the vanishing of the cubic invariant of the Weyl tensor. On a spacelike slice, this surface has the topology of a 2-sphere, and surrounds the singularity of the metric. In Weyl coordinates, this surface is defined by R=m where R= square root ( rho 2+z2). The two points where the z-axis, the axis of symmetry, cuts this surface, z=m and -m, have the interesting property that, at both of them, the Riemann tensor vanishes. The Weyl tensor is of Petrov type D at all non-singular points of the z-axis, rho =0, except at the two points where it is zero. Off the axis, the Weyl tensor is of type I(M+), and the metric asymptotically tends to flat spacetime away from the source. The principal null directions of the Weyl tensor are shown to be everywhere independent of the angular basis vector delta / delta phi , and their projections into a t=constant, phi =constant plane are presented graphically.

A Characterization of the Dependence of the Riemannian Metric on the Curvature Tensor by Young Symmetrizers

Abstract: In differential geometry several differential equation systems are known which allow the determination of the Riemannian metric from the curvature tensor in normal coordinates. We consider two of such differential equation systems. The first system used by Gunther [8] yields a power series of the metric the coefficients of which depend on the covariant derivatives of the curvature tensor symmetrized in a certain manner. The second system, the so-called llcrglotz relations [9], leads to a power series of the metric depending on symmetrized partial derivatives of the curvature tensor. We determine a left ideal of the group ring C[81+4[ of the symmetric group Sr+4 which is associated with the partial derivatives 0' ) R of the curvature tensor H of order rand construct a decomposition of this left ideal into three minimal left ideals using Young symmetrizers and the Littlewood-Richardson rule. Exactly one of these minimal left ideals characterizes the so-called essential part of (9( ' ) R on which the metric really depends via the Herglotz relations. We give examples of metrics with and without a non-essential part of (') H. Applying our results to the covariant derivatives of the curvature tensor we can show that the algebra of tensor polynomials 1?. generated by V ( , . . . Vi , ) R. 1 and the algebra 1' generated by V ( ,, . . . V, RIkIj,+1+2), fulfil R. =

Bounds for eigenvalues of nonsingular H-tensor

Abstract: The bounds for the Z-spectral radius of nonsingular H -tensor, the upper and lower bounds for the minimum H-eigenvalue of nonsingular (strong) M -tensor are studied in this paper. The sharper bounds are obtained. Numerical examples illustrate that our bounds give tighter bounds.

On stability of Ricci ∞ows based on bounded curvatures

Abstract: Recognizing the deflciency that C. Guenther's arguments can not solve the stability of Ricci ∞ows because of the Ricci ∞ow equation being not strictly parabolic, our previous paper flrst studied the stability of Ricci ∞ows based on Killing conditions. In this paper, we consider the stability of Ricci ∞ows, and of quasi-Ricci ∞ows based on bounded curvature conditions, and also obtain some interesting results. @ @t g = i2Rc(g); g(0) = g0 A fundamental and di-cult problem in difierential geometry is to flnd a standard metric satisfying some prescribed conditions over a Riemannian manifold. For in- stance, concerning the celebrated Yamabe problem (20), it is essential to flnd a metric with a constant scalar curvature; and for the constant Ricci curvature, one needs to solve an Einstein equation. The study of Ricci ∞ows, in general, is exactly to flnd a standard metric satisfying the given conditions, and to solve Ricci equation. The typical problem related to Ricci ∞ows is the following short-time existence theorem: Given a compact and smooth Riemannian manifold (M n ;g0), there exists a unique smooth solution g(t) deflned on a short-time-interval such that g(0) = g0. It is natural to ask that in which case the long-time existence theorem of Ricci ∞ows is tenable and the solution converges to a constant curvature metric. The