共查询到20条相似文献,搜索用时 31 毫秒
1.
The aim of this work is to show that a star-shaped hypersurface of constant mean curvature into the Euclidean sphere Sn+1 must be a geodesic sphere. This result extends the one obtained by Jellett in 1853 for such type of surfaces in the Euclidean space R3. In order to do that we will compute a useful formula for the Laplacian of a new support function defined over a hypersurface M of a Riemannian manifold . 相似文献
2.
Henrique F. de Lima 《Differential Geometry and its Applications》2008,26(4):445-455
In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n+1)-dimensional Lorentz-Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1. 相似文献
3.
We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces. 相似文献
4.
We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article. 相似文献
5.
Marc Arcostanzo 《Differential Geometry and its Applications》2009,27(3):368-377
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. 相似文献
6.
Guanghan Li 《Journal of Geometry》2004,80(1-2):154-165
On a hypersurface of a unit sphere without umbilical points, we know that three Möbius invariants can be defined and analogous to Euclidean case, we have the concepts of Möbius isoparametric and isotropic hypersurfaces. In this paper, we study the relationship between Euclidean geometry and Möbius geometry, and prove that a hypersurface in a sphere with constant length of the second fundamental form is Euclidean isoparametric if and only if it is Möbius isoparametric. When restricting to the case of three distinct principal curvatures, we show that such a hypersurface is either Möbius isoparametric or isotropic if the Blaschke tensor
has constant eigenvalues. 相似文献
7.
Mehmet Erdoğan 《Geometriae Dedicata》1996,61(3):221-225
We give an estimate for the Ricci curvature of a complete hypersurface M in a hyperbolic space H and in a sphere S under the same condition. As its application, we give the condition for unboundedness of a complete hypersurface M. 相似文献
8.
Luis J. Alías Takashi Kurose Gil Solanes 《Differential Geometry and its Applications》2006,24(5):492-502
We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface. 相似文献
9.
Changyu Xia 《manuscripta mathematica》1994,85(1):79-87
LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity
theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to
the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound.
Supported by the JSPS postdoctoral fellowship and NSF of China 相似文献
10.
E. Aguirre-Dabán 《Differential Geometry and its Applications》2006,24(2):91-100
Consider a smooth manifold with a smooth metric which changes bilinear type on a hypersurface Σ and whose radical line field is everywhere tangent to Σ. We describe two natural tensors on Σ and use them to describe “integrability conditions” which are similar to the Gauss-Codazzi conditions. We show that these forms control the smooth extendibility to Σ of ambient curvatures. 相似文献
11.
Naoyuki Koike 《Geometriae Dedicata》1995,54(1):1-11
We define the concept of a curvature netted hypersurface and investigate in what case the hypersurface is covered by a twisted product of spheres (or topological product of spheres). All hypersurfaces in a space form such that the number of mutually distinct principal curvatures is constant (i.e. each principal curvature has constant multiplicity) are curvature netted hypersurfaces. Also, we state some inductive constructions of the hypersurfaces, where we use the discussion related to the tube. 相似文献
12.
Quo-Shin Chi 《Differential Geometry and its Applications》2004,21(3):253-281
Up to now all the known torsion-free G3-connections have been analytic. We provide an explicit PDE-solving approach to construct a family of smooth torsion-free G3-connections that are not equivalent to any analytic ones, although each of these connections is locally homogeneous away from a hypersurface of the base manifold. 相似文献
13.
M.Carmen Domingo-Juan 《Differential Geometry and its Applications》2004,21(2):229-251
We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form Mnλ. We show: (a) volume(D) depends only on the second fundamental form of P, whereas volume(C) depends on all the ith fundamental forms of P, (b) when the domain that we move D0 has its q-centre of mass on P, volume(D) does not depend on the mean curvature of P, (c) when D0 is q-symmetric, volume(D) depends only on the intrinsic curvature tensor of P; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(n−q−d), and C is closed, then volume(C) does not depend on the ith fundamental forms of P for i>2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (n−q)-plane orthogonal to P) around a d-dimensional geodesic plane. 相似文献
14.
Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?″?max{H?′,0} along M, then Mn is either maximal or a spacelike slice Mt0={t0}×F, for some t0∈I. 相似文献
15.
Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p∈M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff convergence theory, that, if model volume growth is sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional space. Hence, our main theorem has various advantages of the Cheeger-Colding diffeomorphism theorem via the Euclidean volume growth. Our main theorem also contains a result of do Carmo and Changyu as a special case. 相似文献
16.
Luis J. Alías Henrique F. de Lima Josué Meléndez Fábio R. dos Santos 《Mathematische Nachrichten》2016,289(11-12):1309-1324
Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator. 相似文献
17.
Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape. 相似文献
18.
F.E.C. Camargo R.M.B. Chaves L.A.M. Sousa Jr. 《Differential Geometry and its Applications》2008,26(6):592-599
In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space , n?3, with constant normalized scalar curvature R satisfying totally umbilical? 相似文献
19.
Motivated by a conjecture of Steinhaus, we consider the mapping F, associating to each point x of a convex hypersurface the set of all points at maximal intrinsic distance from x. We first provide two large classes of hypersurfaces with the mapping F single-valued and involutive. Afterwards we show that a convex body is smooth and has constant width if its double has the above properties of F, and we prove a partial converse to this result. Additional conditions are given, to characterize the (doubly covered) balls. 相似文献
20.
Pak Tung Ho 《Differential Geometry and its Applications》2009,27(1):104-108
Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H3, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n2(n−2)/(7n−6). 相似文献