Characterizations of umbilic hypersurfaces in warped product manifolds

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.     

Constant mean curvature surfaces in warped product manifolds

We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.     

On the rigidity of constant mean curvature complete vertical graphs in warped products

We investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriated convergence condition. In this setting, under suitable restrictions on the values of the mean curvature and the norm of the gradient of the height function, we obtain rigidity theorems concerning to such graphs. Furthermore, applications to the hyperbolic and Euclidean spaces are given.     

Scalar curvature rigidity of hyperbolic product manifolds

This paper presents a scalar curvature rigidity result of real hyperbolic product manifolds in analogy to M. Min–Oos result in [14]. In order to prove this, we consider Dirac bundles obtained from the spinor bundle, and we derive Killing equations trivializing these Dirac bundles. Moreover, an integrated Bochner–Weitzenböck formula is shown which allows the usage of the non–compact Bochner technique.Mathematics Subject Classification (2000): Primary 53C24, Sec. 53C21in final form: 11 August 2003     

Minkowski identities for hypersurfaces in constant sectional curvature manifolds

We give a new proof of the generalized Minkowski identities relating the higher degree mean curvatures of orientable closed hypersurfaces immersed in a given constant sectional curvature manifold. Our methods rely on a fundamental differential system of Riemannian geometry introduced by the author. We develop the notion of position vector field, which lies at the core of the Minkowski identities.     

Relative radial mass and rigidity of some warped product manifolds

We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a fixed point or orthogonal to a special fixed hypersurface. We assume that one of these metrics is a warped product if the dimension n is greater than or equal to 3. This formula has non-trivial geometric consequences such as a positive mass type theorem and other rigidity results. We also apply our result to some standard models.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

On constant mean curvature hypersurfaces with finite index

We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in with finite index must be minimal. Received: 30 May 2005     

Bilipschitz embeddings of negative sectional curvature in products of warped product manifolds

Generalizing results due to Brady and Farb (1998) we prove the existence of bilipschitz embedded manifolds of negative sectional curvature in Riemannian products of certain types of warped products.

     

On ricci-pseudosymmetric hypersurfaces in spaces of constant curvature

Results in Mathematics - This article studies the conditions of pseudosymmetry and Ricci-pseudosymmetry, realizedon hypersurfaces of semi-Riemannian spaces of constant curvature. In particular, we...     

On the compactness of constant mean curvature hypersurfaces with finite total curvature

In this work we consider a complete submanifold M with parallel mean curvature vector h immersed in a space form of constant sectional curvature c ￡ 0c\leq 0. If M has finite total curvature and |H|² > -c|H|^2>-c, we prove that M must be compact.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997