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1.
In this paper, we study closed k-maximal spacelike hypersurfaces M
n
in anti-de Sitter space H
1
n+1 (−1) with two distinct principal curvatures and give some integral formulas about these hypersurfaces.
The first author was supported by Japan Society for Promotion of Science. The third author was supported by grant Proj. No.
R17-2008-001-01000-0 from Korea Science & Engineering Foundation. 相似文献
2.
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
3.
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
4.
Oscar Palmas 《Bulletin of the Brazilian Mathematical Society》1999,30(2):139-161
In this paper we classify all complete rotation hypersurfaces withH
k
constant in
n+1 andH
n+1, is the normalizedk-th symmetric function of the principal curvatures. Partial results are also given forH
n+1.Partially supported by DGAPA-UNAM, México, CONACYT, México, under Project 1068P, and CNPp, Brazil. 相似文献
5.
Guoxin Wei 《Monatshefte für Mathematik》2006,149(4):343-350
By investigating hypersurfaces M
n
in the unit sphere S
n+1(1) with H
k
= 0 and with two distinct principal curvatures, we give a characterization of torus the
. We extend recent results of Perdomo [9], Wang [10] and Otsuki [8]. 相似文献
6.
7.
The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in S n+1 under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in S 4. 相似文献
8.
A hypersurface x : M → S n+1 without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ?2∑ i (e i (H) + ∑ j (h ij ?Hδ ij )e j (log ρ))θ i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S
n
+1 without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ−2∑
i
(e
i
(H) + ∑
j
(h
ij
−Hδ
ij
)e
j
(log ρ))θ
i
vanishes and its M?bius shape operator ? = ρ−1(S−Hid) has constant eigenvalues. Here {e
i
} is a local orthonormal basis for I = dx·dx with dual basis {θ
i
}, II = ∑
ij
h
ij
θ
i
⊗θ
i
is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S
n
+1 is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric
hypersurfaces in S
n
+1 with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show
that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S
n
+1 can take only the values 2, 3, 4, 6.
Received September 7, 2001, Accepted January 30, 2002 相似文献
9.
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
10.
The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a
compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex
hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a
classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique.
Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics
which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(= 1, 2, 3) distinct principal curvatures.
Dedicated to Professor Hajime Urakawa on his sixtieth birthday.
H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially
supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006. 相似文献
11.
Guoxin Wei 《Monatshefte für Mathematik》2006,59(2):343-350
By investigating hypersurfaces M
n
in the unit sphere S
n+1(1) with H
k
= 0 and with two distinct principal curvatures, we give a characterization of torus the
S1(?{k/n})×Sn-1(?{(n-k)/n})S^1(\sqrt{k/n})\times S^{n-1}(\sqrt{(n-k)/n})
. We extend recent results of Perdomo [9], Wang [10] and Otsuki [8]. 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.
Rolf Böning 《manuscripta mathematica》1995,87(1):449-458
We study the principal curvatures of a Hopf hypersurfaceM in ℂP
n
or ℂH
n
. The respective eigenspaces of the shape operator often turn out to induce totally real foliations ofM, whose leaves are spherical in the ambient space. Finally we classify the Hopf hypersurfaces with three distinct principal
curvatures satisfying a certain non-degeneracy condition. 相似文献
14.
Stefan Immervoll 《Archiv der Mathematik》2006,87(5):478-480
We generalize a result of Kramer, see [7, 10.7 and 10.10], on generalized quadrangles associated with isoparametric hypersurfaces
of Clifford type to Tits buildings of type C2 derived from arbitrary isoparametric hypersurfaces with four distinct principal curvatures in spheres: two distinct points
p and q of a generalized quadrangle associated with an isoparametric hypersurface in the unit sphere of a Euclidean vector space
can be joined by a line K if and only if (p − q)/||p − q|| is a line. This line is orthogonal to K. Dually, two distinct lines L and K intersect if and only if (L − K)/||L − K|| is point.
Received: 14 October 2005 相似文献
15.
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
16.
Li Zhenqi 《数学年刊B辑(英文版)》1988,9(4):485-493
This paper proves that the number of distinct principal curvatures of a realisoparametric hypersurface in CP~n with constant principal curvatures can be only 2, 3 or 5.The prehnage of such hypersurface under the Hopf fibration is an isoparametrichypersarface in S~(2n+l) with 2 or 4 distinct principal curvatures. For real isoparametrichypersurfaces in CP~n with 5 distinct constant principal curvatures a local structuretheorem is given. 相似文献
17.
We study compact minimal hypersurfaces Mn in Sn+1S^{n+1} with two distinct principal curvatures and prove that if the squared norm S of the second fundamental form of Mn satisfies S \geqq nS \geqq n, then S o nS \equiv n and Mn is a minimal Clifford torus. 相似文献
18.
Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k~v(k ≥ 1) of a submanifold M~n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k~v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold. 相似文献
19.
In this paper, by Nomizu’s method and some technical treatment of the asymmetry of the F-Weingarten operator, we obtain a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces
with constant anisotropic principal curvatures, in Euclidean spaces, which is a generalization of the classical case for isoparametric
hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed
by B. Palmer, we find that in general anisotropic isoparametric hypersurfaces have both local and global aspects as in the
theory of proper Dupin hypersurfaces, which differs from classical isoparametric hypersurfaces. 相似文献
20.
Guoxin Wei 《Monatshefte für Mathematik》2006,23(6):251-258
By investigating hypersurfaces M
n
in the unit sphere S
n+1(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S
1(a) ×
Sn-1(?{1-a2})S^{n-1}(\sqrt{1-a^2})
, where
a2=\frac2+nH2±?{n2H4+4(n-1)H2}2n(1+H2)a^2=\frac{2+nH^2\pm\sqrt{n^2H^4+4(n-1)H^2}}{2n(1+H^2)}
. We extend recent results of Hasanis et al. [5] and Otsuki [10]. 相似文献