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1.
Finite Type Non—Minimal Submanifolds 总被引:1,自引:1,他引:0
The notion of finite type xubmanifolds was introduced by B.Y.Chen.In this paper we consider the characteristics and the classifications of finite type non-minimal submanifolds.The characteristic theorems of 2-type Chen submanifolds、mass-symmetric hypersurfaces and Dupin hypersurfaces in Es^m are obtained.The classification theorems of 3-type hypersurfaces and null 2-type curves in Es^m are also proved. 相似文献
2.
Rolf Böning 《Results in Mathematics》1995,27(1-2):27-34
We assume that on an open subset of a submanifold M of an arbitrary Riemannian ambient space N the eigenspaces of the shape operator of M induce a foliation L whose leaves are spherical submanifolds of N. In this situation we derive a condition which characterizes when the leaves of L are complete Riemannian submanifolds of M (see Theorem 2.4). We apply this result to real hypersurfaces of complex space forms, in particular Hopf hypersurfaces (see Theorem 3.2 and Proposition 3.3). 相似文献
3.
We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski
space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local
sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties
of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike
hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space. 相似文献
4.
It is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too. 相似文献
5.
Paola Matzeu 《Monatshefte für Mathematik》2002,13(5):297-311
We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces.
In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds. 相似文献
6.
We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification. 相似文献
7.
Zhangjie Liu 《Frontiers of Mathematics in China》2007,2(3):417-438
In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in ℂPn, compact minimal quaternionic coisotropic submanifolds in ℚPn and compact minimal hypersurfaces in P2 (Cay).
相似文献
8.
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
9.
金天寿 《数学的实践与认识》2012,42(11):227-229
球面中等参超曲面理论近期获得蓬勃发展,它们在等距意义下的分类至今未完全解决,将证明球面中等参超曲面及其焦流形的微分同胚型有限. 相似文献
10.
Jianquan Ge 《Journal of Functional Analysis》2010,258(5):1682-1691
In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counterexample of the Brézis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina. 相似文献
11.
Baouendi M. S. Mir Nordine Rothschild Linda Preiss 《Journal of Geometric Analysis》2002,12(4):543-580
Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ
N
are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map
is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic
submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs
of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that
if two real-algebraic hypersurfaces in ℂ
N
are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general
context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds. 相似文献
12.
Paola Matzeu 《Monatshefte für Mathematik》2002,136(4):297-311
We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces.
In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds.
Received July 16, 2001; in revised form February 6, 2002 Published online August 9, 2002 相似文献
13.
In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities. 相似文献
14.
15.
We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension
2 inℂP
n
and are topologically “glued” out of algebraic hypersurfaces in (ℂ*)
n
. Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the
combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem
a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex
subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same
topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.
A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF
Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161.
The work was completed during the stay of both authors at Max-Planck-Institu für Mathematik. The authors thank these funds
and institutions for hospitality and financial support. 相似文献
16.
《Expositiones Mathematicae》2020,38(3):337-364
In this paper, we present two applications of the theory of singular connections developed by Harvey and Lawson (1993). The first one is a version of the Lelong–Poincaré formula with estimates for sections of vector bundles over an almost complex manifold. The second one is a convergence theorem for divisors associated to a general family of symplectic submanifolds constructed by Donaldson (1996) (the case of hypersurfaces) and by Auroux in (1997) (for arbitrary dimensional submanifolds). 相似文献
17.
<正> §1.如所周知,黎曼空间中关于平面公理的嘉当(E.Cartan)定理可以拓广到更一般的空间中去,满足平面公理的 m 维黎曼空间在它的每点容有∞~(m-1)张全测地超曲面.柏尔特拉米(Beltrami)给出常曲率空间的另一特征,只有常曲率空间才能与欧氏空 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.
Bang-yen Chen 《Monatshefte für Mathematik》1981,91(4):257-274
LetN be a real submanifold in a complex manifoldM. If the maximal complex subspaces of the tangent spaces ofM contained in the tangent spaces ofN are of constant dimension and they define a differentiable distribution, thenN is called a generic submanifold. The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds andCR-submanifolds. In this paper we initiate a study of generic submanifolds in a Kähler manifold from differential geometric point of view. Some fundamental results in this respect will be obtained. 相似文献
20.
A criterion for reduction of variables in the Willmore-Chen variational problem and its applications
Manuel Barros Angel Ferrá ndez Pascual Lucas Miguel A. Meroñ o 《Transactions of the American Mathematical Society》2000,352(7):3015-3027
We exhibit a criterion for a reduction of variables for Willmore-Chen submanifolds in conformal classes associated with generalized Kaluza-Klein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.