Submanifolds with Flat Normal Bundle

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.     

复射影空间中法丛平坦的全实迷向子流形

证明了复射影空间中两种类型法丛平坦的全实迷向予流形必是极小的,并在紧致的情形确定了它们的具体形状．     

欧氏空间中完备极小子流形

在一个类似于稳定不等式的条件下,得到了欧氏空间中完备极小子流形的Bernstein型定理.我们的结果部分推广了Li H.Z.和Wei G.X.的定理.     

Uniqueness of Minimal Submanifolds in Euclidean Space

Let M be a properly immersed n-dimensional complete minimal submanifold in Euclidean space R^n+p of dimension n+p. Let A be the second fundamental form of the immersion, and r the extrinsic distance from the origin. Suppose M has one end and inf_t sup_r(x)>t r²(x) |A|²(x) < C(n,p), then M is an affine n-plane, where C(n,p) are constants given by C(n,1) = n – 1 and C(n,p) = (2/3)(n – 1) when p > 1.     

Almost Extrinsically Homogeneous Submanifolds of Euclidean Space

Consider a closed manifold M immersed in R^m. Suppose that the trivial bundle M × R^m = T M ⊗ ν M is equipped with an almost metric connection ~ ∇ which almost preserves the decomposition of M × R^m into the tangent and the normal bundle. Assume moreover that the difference Γ = ∂~∇ with the usual derivative ∂ in R^m is almost ~∇-parallel. Then M admits an extrinsically homogeneous immersion into R^m. Mathematics Subject Classifications (2000): 53C20, 53C24, 53C30, 53C42, 53C40.     

欧氏空间子流形的第一特征值的估计

本文利用浸入在欧氏空间中的子流形的第二基本形式的长度平方估计其Laplace算子的第一特征值的上界,从而建立紧致子流形等距同构于球面的一个特征.     

Lagrangian Submanifolds with Zero Scalar Curvature in Complex Euclidean Space

In this paper we show how to embed a time slice of the Schwarzchild spacetime that models the outer space around a massive star, as a Lagrangian submanifold invariant under the standard action of the special orthogonal group on complex Euclidean space. This result is generalized for rotationally invariant metrics that can be considered as higher-dimensional versions of Schwarzchild's and these submanifolds are locally characterized as the only ones with zero scalar curvature inside the above family.     

Minimal Submanifolds in a Locally Symmetric and Conformally Flat Space

We study minimal submanifolds in the locally symmetric and conformally flat Riemannian manifold and generalize Yau's result obtained in J. Amer. Math. 97 (1975), 76–100.     

Double Structures and Normal Bundle of Space Curves

It is shown that the general plane section of a double structureon an integral curve CP³ has a connected numerical character(that is, its Hilbert function is of decreasing type). The papergives applications, particularly to the stability of the normalbundle of space curves.     

Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

On the Structure of Submanifolds with Degenerate Gauss Maps

An n-dimensional submanifold X of a projective space P ^N (C) is called tangentially degenerate if the rank of its Gauss mapping gamma;; X G(n, N) satisfies 0 < rank < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P ^N (C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible.     

Submanifolds of Weyl Flat Manifolds

 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.     

Submanifolds of Weyl Flat Manifolds

 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     

关于局部对称共形平坦空间中具有常数量曲率的子流形

该文研究了局部对称共形平坦空间中具有常数量曲率的紧致子流形,证明了这类子流形的某些内蕴刚性定理.     

On Pairs of Hypersurfaces in Euclidean Space

Mathematical Notes -     

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.     

Genuine Rigidity of Euclidean Submanifolds in Codimension Two

An isometric deformation of an Euclidean submanifold is called genuine if the submanifold cannot be included into a submanifold of larger dimension in such a way that the deformation of the former is given by an isometric deformation of the latter. The submanifold is said to be genuinely rigid if it has no genuine deformations. In this paper we study the deformation problem in codimension two for the classes of elliptic and parabolic submanifolds. In spite of having a second fundamental form as degenerate as possible without being flat, i.e., the Gauss map has rank two everywhere, our main result says that generically these submanifolds are genuinely rigid. An additional unexpected deformation phenomenon for elliptic submanifolds carrying a Kaehler structure is described.