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1.
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. 相似文献
2.
Spacelike hypersurfaces with constant scalar curvature 总被引:1,自引:0,他引:1
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 相似文献
3.
In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) 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 also obtain several rigidity theorems for such hypersurfaces. 相似文献
4.
In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds.
Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.)
for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation,
and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second
order mean curvature. 相似文献
5.
We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S
n
other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC. 相似文献
6.
ZhangJianfeng 《高校应用数学学报(英文版)》2005,20(2):183-196
Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n 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 M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n 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 M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated. 相似文献
7.
Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime 下载免费PDF全文
Henrique Fernandes de Lima Marco Antonio Lázaro Velásquez 《Mathematische Nachrichten》2014,287(11-12):1223-1240
We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so‐called generalized Robertson‐Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett 29 . 相似文献
8.
The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces
always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results
hold for the dual space .
(Received 27 August 1999; in revised form 18 November 1999) 相似文献
9.
In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ
n
which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal
vector fields in ℍ
n
.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean
curvature (CMC) hypersurface. Our definition coincides with previous ones.
Our main result describes which are the CMC hypersurfaces of revolution in ℍ
n
.The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential
equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart
in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean
space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ
n
. 相似文献
10.
We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP2, (C⊗O)P2, (H⊗O)P2 and (O⊗O)P2, which are symmetric spaces associated to the exceptional simple Lie groups F4, E6, E7 and E8 (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry. 相似文献
11.
Yun Tao Zhang 《Differential Geometry and its Applications》2011,29(6):730-736
Let Mn be a complete hypersurface in Sn+1(1) with constant mean curvature. Assume that Mn has n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C−(H), either S?S+(H) or RicM?0 or the fundamental group of Mn is infinite, then S is constant, S=S+(H) and Mn is isometric to a Clifford torus with . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature. 相似文献
12.
Bing-Le Wu 《Geometriae Dedicata》1994,50(3):247-250
In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS
n+1 with four distinct principal curvatures. 相似文献
13.
For a compact minimal hypersurface M in Sn+1 with the squared length of the second fundamental form S we confirm that there exists a positive constant δ(n) depending only on n, such that if n?S?n+δ(n), then S≡n, i.e., M is a Clifford minimal hypersurface, in particular, when n?6, the pinching constant . 相似文献
14.
The volume of a unit vector field V of the sphere (n odd) is the volume of its image V() in the unit tangent bundle. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fibre of a Hopf
fibration are well known to be critical for the volume functional. Moreover, Gluck and Ziller proved that these fields achieve the
minimum of the volume if n = 3 and they opened the question of whether this result would be true for all odd dimensional spheres. It was shown to be
inaccurate on spheres of radius one. Indeed, Pedersen exhibited smooth vector fields on the unit sphere with less volume than Hopf vector fields for a dimension greater than five. In this article, we consider the situation
for any odd dimensional spheres, but not necessarily of radius one. We show that the stability of the Hopf field actually depends on radius, instability occurs precisely if and only if In particular, the Hopf field cannot be minimum in this range. On the contrary, for r small, a computation shows that the volume of vector fields built by Pedersen is greater than the volume of the Hopf one
thus, in this case, the Hopf vector field remains a candidate to be a minimizer. We then study the asymptotic behaviour of
the volume; for small r it is ruled by the first term of the Taylor expansion of the volume. We call this term the twisting of the vector field. The lower this term is, the lower the volume of the vector field is for small r. It turns out that unit Hopf vector fields are absolute minima of the twisting. This fact, together with the stability result,
gives two positive arguments in favour of the Gluck and Ziller conjecture for small r. 相似文献
15.
In this work we prove the existence of totally geodesic two-dimensional foliation on the Lorentzian Heisenberg group H
3. We determine the Killing vector fields and the Lorentzian geodesics on H
3. 相似文献
16.
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
17.
《Mathematische Nachrichten》2017,290(16):2661-2672
Biconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant. 相似文献
18.
We characterize Hopf hypersurfaces inS
6 as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS
6. 相似文献
19.
本文研究了单位球中的数量曲率满足r=aH+b的完备超曲面的问题.利用极值原理的方法,获得了超曲面的一个刚性结果,推广了这一类具有常中曲率或者常数量曲率超曲面的结果. 相似文献
20.
Toshiaki Adachi 《Monatshefte für Mathematik》2008,153(4):283-293
In this paper, we study geodesics with null structure torsions on real hypersurfaces of type A
2 in a complex space form. These geodesics give a nice family of helices of order 3 generated by Killing vector fields on the
ambient complex space form.
Author’s address: Toshiaki Adachi, Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan 相似文献