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1.
Let x:M→ be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold and let ρ
i
(i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].?If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced
by E and P are stated.?The principal curvatures ρ
i
are isoparametric functions and the set (ρ1,...,ρ
n
) defines an isoparametric system [10].?In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P
♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b
2≥1.
Received: April 7, 1999; in final form: January 7, 2000?Published online: May 10, 2001 相似文献
2.
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
3.
Let M be a compact oriented minimal
hypersurface of the unit n-dimensional sphere
Sn.
It is known that if the norm squared of the second fundamental form,
, satisfies that
for all
, then M is isometric to a Clifford
minimal hypersurface ([2], [5]). In this paper we will generalize this result
for minimal hypersurfaces with two principal curvatures and dimension greater
than 2. For these hypersurfaces we will show that if the average of the function
is n - 1, then M
must be a Clifford hypersurface.
Received: 24 December 2002 相似文献
4.
A submanifold M
n
r
of Minkowski space
is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of
to the tangent space of M
n
r
at every point of M
n
r
. In this paper we completely classify hypersurfaces of restricted type in
. More precisely, we prove that a hypersurface of
is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S
k
×
, S
k
1
×
, H
k
×
, S
n
1
, H
n
, with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium. 相似文献
5.
Gabjin Yun 《Geometriae Dedicata》2002,89(1):133-139
Let M
n
, n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R
n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C
s
, where C
s
is the Sobolev constant for M, then there are no L
2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane. 相似文献
6.
Let M
n
be a compact (two-sided) minimal hypersurface in a Riemannian manifold . It is a simple fact that if has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.?We prove that if is the real projective space , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface obtained as the product of two spheres of dimensions n
1, n
2 and radius R
1, R
2, with and .
Received: June 6, 1998 相似文献
7.
Majid Ali Choudhary 《Russian Mathematics (Iz VUZ)》2014,58(8):56-64
S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ 1 of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S n+1. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S n+1 with λ 1 = n is either isometric to the sphere S n or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate. 相似文献
8.
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
9.
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
10.
We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S
n+1. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius
curvature. 相似文献
11.
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
12.
If M is an isoparametric hypersurface in a sphere S
n
with four distinct principal curvatures, then the principal curvatures κ1, . . . , κ4 can be ordered so that their multiplicities satisfy m
1 = m
2 and m
3 = m
4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R
n
(or S
n
) with four distinct principal curvatures with multiplicities m
1 = m
2 ≥ 1 and m
3 = m
4 = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the
assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.
相似文献
13.
Let M
n
be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere
, then M
n
is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M
n
under the M?bius transformation group of
. A classical theorem of M?bius geometry states that M
n
(n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All
the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter
are Dupin hypersurfaces.
In this paper, we determine all M?bius isoparametric hypersurfaces in
by proving the following classification theorem: If
is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in
; or (ii) the pre-image of the stereographic projection of the cone in
over the Cartan isoparametric hypersurface in
with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures
in
. The classification of hypersurfaces in
with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart
of the classification for Dupin hypersurfaces in
up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen.
Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM.
Partially supported by the Zhongdian grant No. 10531090 of NSFC.
Partially supported by RFDP, 973 Project and Jiechu grant of NSFC. 相似文献
14.
Let M be a closed Willmore hypersurface in the sphere S^n+1(1) (n ≥ 2) with the same mean curvature of the Willmore torus Wm,n-m, if SpecP(M) = Spec^P(Wm,n-m ) (p = 0, 1,2), then M is Wm,n-m. 相似文献
15.
Zhen Guo 《数学学报(英文版)》2009,25(1):77-84
Let x : Mn^n→ R^n+1 be an n(≥2)-dimensional hypersurface immersed in Euclidean space Rn+1. Let σi(0≤ i≤ n) be the ith mean curvature and Qn = ∑i=0^n(-1)^i+1 (n^i)σ1^n-iσi. Recently, the author showed that Wn(x) = ∫M QndM is a conformal invariant under conformal group of R^n+1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper. 相似文献
16.
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
17.
In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following
situation. Letc(t) be a curve in a space formM
λ
n
of sectional curvature λ. LetP
0 be a totally geodesic hypersurface ofM
λ
n
throughc(0) and orthogonal toc(t). LetC
0 be a hypersurface ofP
0. LetC be the hypersurface ofM
λ
n
obtained by a motion ofC
0 alongc(t). We shall denote it byC
PorC
Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC
0, then volume(C) ≥ volume(C),P),and the equality holds whenC
0 is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion).
Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052. 相似文献
18.
Qintao Deng 《Archiv der Mathematik》2008,90(4):360-373
In this paper, we consider complete hypersurfaces in R
n+1 with constant mean curvature H and prove that M
n
is a hyperplane if the L
2 norm curvature of M
n
satisfies some growth condition and M
n
is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M
n
is a hyperplane (or a round sphere) under the condition that M
n
is strongly stable (or weakly stable) and has some finite L
p
norm curvature.
Received: 14 July 2007 相似文献
19.
Pak Tung Ho 《Differential Geometry and its Applications》2008,26(3):273-276
Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2λ1>k−(n−1)maxM|H| where λ1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M. 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