The Submanifolds with Parallel Mean Curvature Vector in a Locally Symmetric and Conformally Flat Riemannian Manifold

In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If then M^n lies in a totally geodesic submanifold N^(n+1).

C-Totally Real Submanifolds with Parallel Mean Curvature Vector of Sasakian Space Form

We have discussed the C-totally real subrnanifolds with parallel mean curvature vector of Sasakian space form, obtained a formula of J.Simons type, and improved one result of S.Yamaguchi.

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.

A rigidity theorem for submanifolds in S^(n+p) with constant scalar curvature

Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE S_p^(n+p)(c) WITH CONSTANT SCALAR CURVATURE

Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of Mn,respectively.Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for Mn immersed in ~S^n+p _p(c) with parallel normalized mean curvature vector field is proved.When n≥3, the pinching constant is the best.Thus,the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math,1998,95:499-505) is corrected.Moreover,the reduction of the codimension when Mn is a complete submanifold in S^n+p _p(c) with parallel normalized mean curvature vector field is investigated.

The Pinching Theorem of Global Umbilic Submanifolds in a Riemannian Manifold with Quasi Constant Curvature

We study the global umbilic submanifolds with parallel mean curvature vector fields in a Riemannian manifold with quasi constant curvature and get a local pinching theorem about the length of the second fundamental form.

A Theorem on Infinitesimal I-isometry of Surfaces Immersed in a Space with Constant Curvature

In this paper we discuss the infinitesimal I-isometric de formations of surfaces immersed in a space with constant curvature. We obtain a sufficient condition for the de formation vector field to be zero vector field which is generalization of the results in [1] and [2].

CURVATURE COMPUTATIONS OF 2-MANIFOLDS IN IR^k

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk with k > 3.

On Totally Real Pseudo-Umbilical Submanifolds in a Complex Projective Space

Let M^n be a totally real submanifold in a complex projective space CP^(n+p).In this paper,we study the position of the parallel umbilical normal vector field of M^n in the normal bundle.By choosing a suitable frame field,we obtain a pinching theorem,in the case p>0, for the square of the length of the second fundamental form of a totally real pseudo-umbilical submanifold with parallel mean curvature vector.

A complete classification of Blaschke parallel submanifolds with vanishing Mbius form

The Blaschke tensor and the Mbius form are two of the fundamental invariants in the Mobius geometry of submanifolds;an umbilic-free immersed submanifold in real space forms is called Blaschke parallel if its Blaschke tensor is parallel.We prove a theorem which,together with the known classification result for Mobius isotropic submanifolds,successfully establishes a complete classification of the Blaschke parallel submanifolds in S^n with vanishing Mobius form.Before doing so,a broad class of new examples of general codimensions is explicitly constructed.

Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds

For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian （1 〈 p 〈 ＋∞） on M. This result can be seen as an extension of Reilly＇s bound for the first non-zero closed eigenvalue of the Laplace operator.

Proper Biharmonic Submanifolds in a Sphere

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.

On the Tangent Bundle of a Hypersurface in a Riemannian Manifold

Let （Mn, g） and （N^n＋1, G） be Riemannian manifolds. Let TMn and TN^n＋1 be the associated tangent bundles. Let f ： （M^n, g） → （N^＋1, G） be an isometrical immersion with g = f^＊G, F = （f, df） ： （TM^n,g） → （TN^n＋1, Gs） be the isometrical immersion with g= F＊Gs where （df）x ： TxM → Tf（x）N for any x∈ M is the differential map, and Gs be the Sasaki metric on TN induced from G. This paper deals with the geometry of TM^n as a submanifold of TN^n＋1 by the moving frame method. The authors firstly study the extrinsic geometry of TMn in TN^n＋1. Then the integrability of the induced almost complex structure of TM is discussed.

A Characterization of the Standard Tori in C^(2) as Compact Lagrangian ξ-Submanifolds

In this paper,the authors give a characterization theorem for the standard tori S^(1)(a)×S^(1)(b),a,b>0,as the compact Lagrangianξ-submanifolds in the two-dimensional complex Euclidean space C^(2),and obtain the best version of a former rigidity theorem for compact Lagrangianξ-submanifold in C^(2).Furthermore,their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in C^(2).

On Submanifolds of the Unit Sphere with Vanishing Mobius Form and Parallel Para-Blaschke Tensor

The para-Blaschke tensor are extended in this paper from hypersurfaces to general higher codimensional submanifolds in the unit sphere S^(n),which is invariant under the Mobius transformations on Sn.Then some typical new examples of umbilic-free submanifolds in Snwith vanishing Mobius form and a parallel para-Blaschke tensor of two distinct eigenvalues,D_(1) and D_(2),are constructed.The main theorem of this paper is a simple characterization of these newly found examples in terms of the eigenvalues D_(1) and D_(2).