Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds

Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.

Second Order Parallel Tensors on Quasi-constant Curvature Manifolds

The author establishes in this paper the following results: (1) In a quasiconstant curvature manifold M a parallel tensor of type is constant multiple of the metric tensor. (2) On a quasi_constant curvature manifold there is no nonzero parallel 2_form. Unless the Ricci principal curvature corresponding to the generator of M is equal to zero.

The Manifolds with Ricci Curvature Decay to Zero

The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.

COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES(RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)

In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.

The extension for mean curvature flow with finite integral curvature in Riemannian manifolds

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be extended over time T. Moreover,we show that the condition is optimal in some sense.

An Interpolation of Hardy Inequality and Moser–Trudinger Inequality on Riemannian Manifolds with Negative Curvature

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser-Trudinger inequality on M. Furthermore, the constant we obtain is sharp.

DARBOUX EQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITH NONNEGATIVE CURVATURE IN R

The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.

ON THE SECTIONAL CURVATURE OF A RIEMANNIAN MANIFOLD

In this paper the author establishes the following1.If M^n(n≥3)is a connected Riemannian manifold,then the sectional curvatureK(p),where p is any plane in T^x(M),is a function of at most n(n-1)/2 variables.Moreprecisely,K(p)depends on at most n(n-1)/2 parameters of group SO(n).2.Lot M^n(n≥3)be a connected Riemannian manifold.If there exists a point x ∈ Msuch that the sectional curvature K(p)is independent of the plane p∈T_x(M),then M is aspace of constant curvature.This latter improves a well-known theorem of F.Schur.

WEYL CURVATURE OF A FINSLER SPACE

The Weyl curvature of a Finsler metric is investigated.This curvature constructed from Riemannain curvature.It is an important projective invariant of Finsler metrics.The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature.A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved.It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.

The Extension of the H^k Mean Curvature Flow in Riemannian Manifolds

In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").

A Pinching Theorem for Riemannian Foliations with Parallel Mean Curvature in a Local-Symmetric Riemannian Manifold

We discuss the Riemannian foliations with parallel mean curvature in a local- symmetric Riemannian manifold,and obtain a pinching theorem about it.

Manifolds of positive Ricci curvature,quadratically asymptotically nonnegative curvature,and infinite Betti numbers

In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than or equal to 6.The purpose of the present paper is to use a different technique to exhibit a family of complete I-dimensional(I≥5)Riemannian manifolds of positive Ricci curvature,quadratically asymptotically nonnegative sectional curvature,and certain infinite Betti numbers bj(2≤j≤I-2).

ON COMPLETE SPACE-LIKE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR

Let M n be a complete space-like submanifold with parallel mean curvature vector in an indefinite space form N n+p p (c).A sharp estimate for the upper bound of the norm of the second fundamental form of M n is obtained. A generalization of this result to complete space-like hypersurfaces with constant mean curvature in a Lorentz manifold is given. Moreover, harmonic Gauss maps of M n in N n+p p(c) in a generalized sense are considered.

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2^(1/m-1),2^(1/2)}, Such that λ_1≥π~2/d^2·1/(2-(11)/(2π~2))+11/2π~2e^cm、

The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold

Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M, a, b are constants and . Then there exist harmonic functions on M .

RIGIDITY THEOREMS FOR HYPERSURFACES IN RIEMAN-NIAN MANIFOLD OF CONSTANT CURVATURE

In this letter we prove several global rigidity theorems for hypersurfaces in Riemannian manifold of constant curvature, which are generalizations of some wellknown theorems for convex hypersurfaces in En+1, Sn+1 and Hn+1. Our main results are as follows:

SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR

Let Sn+p be a unit （n+p） sphere and f: M→Sn+p an isometric immersion of an n-ditmensional Riemannian manifold M into Sn+p, If the length of the mean curvature vector ξ of f（M） is constant and the vector ξ/|ξ| is parallel im the normal bundle, then f（M） is called the submanifold

Yamabe invariant for complete manifold with nonpositive scalar curvature

A sufficient condition for Yamabe invariants to be non-positive and its applications are obtained.

COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE

A natural extension of minimal submanifolds is the Riemannian submanifolds with parallel mean curvature, which were discussed a lot by Yau, S. T., Okumura, M. and the other authors. Recently, by applying the

A Quantum Representation of the Homogeneous 5D Manifold and the Perelman Mappings of 5D onto Non-Homogeneous Lorentz 4D Manifolds

The expression of the Maxwell magnetic monopole was employed to correlate the space to space projection that gives rise to the Gell-Mann standard model, and space to time projection which gives the leptons;and how does it correlate to the Perelman mappings from the homogeneous 5D manifold to the Lorentz 4D manifold, together with correlating the physical consequences caused by the breaking of the Diagonal Long Range Order [DLRO] of the monopoles quantum states affected by the motion of massive particles in the Lorentz 4D boundary of the 5D manifold, which leads to gravitons and the gravity field via the General Relativity covariant Riemannian 4D curvatures metric equation.