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1.
We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian
metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of classC
2 where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them
there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233,
(1972)) saying that such a hypersurface of classC
4 has to be a round sphere. In particular, the sphere is notII-rigid in the class of all convexC
2-hypersurfaces. 相似文献
2.
In this paper we study the topological and metric rigidity of hypersurfaces in ℍ
n+1, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface
in ℍ
n+1 to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface
with constant norm of the second fundamental form to be totally umbilic. 相似文献
3.
A. A. Borisenko 《Mathematical Notes》2000,67(4):425-432
Locally convex compact hypersurfaces immersed in a hollow simply connected Riemannian space of nonpositive sectional curvature
are considered. They are proved to be convex hypersurfaces homeomorphic to the sphere. A similar result for immersed hypersurfaces
with nonpositive definite second quadratic form of rank no smaller than one is obtained.
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 498–507, April, 2000. 相似文献
4.
BING YE WU 《Proceedings Mathematical Sciences》2011,121(4):435-446
We investigate the immersed hypersurfaces in space forms ℕ
n + 1(c), n ≥ 4 with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant.
We prove that any immersed hypersurface in space forms with two distinct non-simple principal curvatures is locally conformal
to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for the Clifford hypersurfaces
in terms of the trace free part of the second fundamental form. 相似文献
5.
Julien Roth 《Differential Geometry and its Applications》2007,25(5):485-499
We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L∞-norm. 相似文献
6.
The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.Mathematics Subject Classifications (2000): 53C20. 相似文献
7.
We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental
form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed,
two-sided index one hypersurfaces with scalar curvature one in the real projective space.
Received: October 12, 2001 相似文献
8.
Ivko Dimitri 《Differential Geometry and its Applications》2009,27(4):507-526
We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n5. For example, the complete minimal hypersurfaces of the unit sphere Sn+1 which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface and the Clifford minimal hypersurfaces Mk,n−k for n≠2k. An interesting characterization of horospheres in is also obtained. 相似文献
9.
The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized. 相似文献
10.
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
11.
We establish a C2 a priori estimate for convex hypersurfaces whose principal curvatures κ=(κ1,…, κn) satisfy σk(κ(X))=f(X,ν(X)), the Weingarten curvature equation. We also obtain such an estimate for admissible 2‐convex hypersurfaces in the case k=2. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.© 2015 Wiley Periodicals, Inc. 相似文献
12.
1IntroductionTherehavebeensomeinterestingresultsinstudyingtheflowofconvexhypersurfacesintheEuclideanspacebyfunctionsOftheirprincipalcurvatures.BeingviewedasanextensionOfthetheoremOfGageandHedton[3],Huiskellprovedin16]thatdeformingconvexhypersurforesbytheirmeancurysturefunctiollsconvergetoaroundsphereinasense.FollowingthemethodsOfHuiskell[6]andTso[IOI,Chowshowedin[1]thatthesame.statementasin.[6]reconstrueifthemeancurvatureisreplacedbythen-throotoftheGauss-Kroneckercurvature.FOrgenerality… 相似文献
13.
Hanno Rund 《Annali di Matematica Pura ed Applicata》1972,93(1):99-110
Summary A special class of hypersurfaces of a Riemannian space is examined, this class being defined by the stipulation that the coefficients
of the third fundamental form be expressible as linear combinations of the coefficients of the first and second fundamental
forms. It is jound that these so-called C-hypersurfaces are umbilical provided that certain conditions (which may depend on
dimension) are satisfied. An (n-1)-dimensional Einstein space imbedded in an n-dimensional space of constant curvature is
such a C-hypersurface; accordingly the theory may be applied to the problem of the local imbedding of such spaces.
Entrata in Redazione il 23 giugno 1971. 相似文献
14.
We consider embedded hypersurfacesM in hyperbolic space with compact boundaryC and somer
th mean curvature functionH
r a positive constant. We investigate when symmetries ofC are symmetries ofM. We prove that if 0H
r1 andC is a sphere thenM is a part of an equidistant sphere. Forr=1 (H
1 is the mean curvature) we obtain results whenC is convex. 相似文献
15.
The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in Rn. They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces. In particular, they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere. 相似文献
16.
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
17.
We classify hypersurfaces of the hyperbolic space ?n+1(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mn is a complete hypersurfaces 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 ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces. 相似文献
18.
19.
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
20.
In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating
interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper
bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation
problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC
1, cubicC
2, and quarticC
3 splines on refined grids. 相似文献