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.     

A RIGIDITY THEOREM FOR NON-NEGATIVELY IMMERSED SUBMANIFOLDS

The paper is to generalize the rigidity theorem that the special Weingarten surface isthe sphere to the case of submanifolds.It is proved that a non-negatively immersedcompact submaifnold in space form of constant curvature is a Riemannian product ofseveral totally umbilical submanifolds if the mean curvature and the scalar curvature ofthe submanifold satisfy a certain function relation.     

A Pinching Theorem for Minimal Submanifolds in Sphere

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关于局部对称空间中的伪脐子流形

本文讨论了局部对称完备黎曼流形中的紧致伪脐子流形,且具有平行平均山率向量场。得到了这类子流形成为全脐子流形及其余维数减小的几个拼挤定理。     

Anti-Invariant Submanifolds in a.Bochner-Kaehler Manifold

The Properties of submanifolds in a Bochner-Kaehler manifold have been studied mainly in the cases that the submanifolds are totally real by Yano, K., Houh, 0. S. and others. The main purpose of the present paper is to study whether the condition for the submanifold to be totolly real in their theorems is necessary, and to prove some theorems which are analogous to those mentioned above. A submanifold M^n of Kaehlerian manifold M^2m is called totally real or antiinvariant,if each tangent space of M^n is mapped into the normal space by the complex structure $\[{F_{\nu \mu }}\]$ of M^2m. Similarly, a submanifold M^n of Kaehlerian manifold M^2m is called anti-in variant with respect to L', if each tangent space of M^n is mapped into the normal space by the operator L' of M^2m. We obtain: (1) A necessary and sufficient condition for a totally umbilical submanifold M^n, n>3, in a Boohner-Kaehler manifold M^2m to be conformally flat is that the submanifold M^n is either a totally real submanifold or an anti-invariant submanifold with respect to L'. (2) Let M^n be the submanifold immersed in a Boohner-Kaehler manifold M^2m. If each tangent vector of M^n is Ricci principal direction and Ricci principal curvature $\[{\rho _h}\]$ does not equal $[\frac{{\tilde K}}{{4(m + 1)}}\]$ , then the anti-invariant submanifold with respect to L^' coincides with the totally real submanifold. (3) Let M^n be a totally umbilical submanifold immersed in a Boohner-Kaehler manifold M^2m If M^n is a totally real submanifold or an anti-invariant submanifold,then the sectional curvature of Mn is given by $[\rho (u,v) = \frac{1}{8}(\tilde K(u) + \tilde K(v)) + \sum\limits_{x = n + 1}^{2m} {{H^2}} ({e_x})\]$(A) where H(e_x) =H_x. Conversely, if the sectional curvature of M^n satisfying the condition mentioned in (2) is given by (A) for any two orthonormal tangent vectors u^\alpha and $v^\alpha$ then M^n is a totally real submanifold.     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

SUBMANIFOLDS OF A HIGHER DIMENSIONAL SPHERE

Let M be an m-dimensional manifold immersed in S~(m+k)(r).Then △X=μH-(m/r~2)X,where X is the position vector of M and H is a unit normal vector field which is orthogonalto X everywhere.If M is a compact connected manifold with parallel mean curvature vector field ξimmersed inS~(m+k)(r),and the sectional curvature of M is not less than (1/2)((1/r~2)+|ξ|~2),thenM is a small sphere.For a compact connected hypersurface M in S~(m+1)(r),if the sectional curvature is non-nesative and the scalar curvature is proportional to the mean curvature everywhere,then M isa totally umbilical hypersurface or the multiplication of two totally umbilical submanifolds.     

常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

de Sitter空间中法联络平坦的完备类空子流形

该文证明了de Sitter空间中具有平行平均曲率向量的常数量曲率完备类空子流形,如果其法联络是平坦的,且M的截面曲率小于0,或M的第二基本形式模长平方‖σ‖相似文献   

关于复射影空间中的全实伪脐子流形

设$M^n$是复射影空间${\bf C}P^{n+p}$中的全实子流形. 本文研究$M^n$的平行脐性法向量场在法丛中的位置. 在$p>0$的情形通过选取合适的标架场, 得到具有平行平均曲率向量的全实伪脐子流形关于第二基本形式模长平方的一个Pinching定理.     

常曲率空间中具平行平均曲率向量的子流形

本文利用第二基本形式的长度平方和平均曲率的关系研究常曲率空间中具平行平均曲率向量的子流形为全脐的ｐｉｎｃｈｉｎｇ问题，获得了一定条件下的最佳ｐｉｎｃｈｉｎｇ区间，并确定了ｐｈｉｎｃｎｉｎｇ区间端点处对应非全脐子流形的分类．     

Complete space-like submanifolds in locally symmetric semi-definite spaces

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized     

Contact totally umbilical submanifolds of a sasakian space form

Summary We define a notion of contact totally umbilical submanifolds of Sasakian space forms corresponds to those of totally umbilical submanifolds of complex space forms. We study a contact totally umbilical submanifold M of a Sasakian space form (c ≠ −3) and prove that M is an invariant submanifold or an anti-invariant submanifold. Furthermore we study a submanifold M with parallel second fundamental form of a Sasakian space form (c ≠ 1) and prove that M is invariant or anti-invariant. Entrata in Redazione il 7 settembre 1976.     

复射影空间中法丛平坦的全实伪脐子流形

该文证明了复射影空间中两种类型的法丛平坦全实伪脐子流形必是极小的,并在紧致的情形确定了它们的具体形状．此外,还说明了复射影空间中的全实全脐子流形一定不是法丛平坦的.     

局部对称共形平坦黎曼流形中具有平行平均曲率向量的子流形

本文把[1]的结论推广到了环绕空间是局部对称共形平坦的情形,即获得了:设M~是局部对称共形平坦黎曼流形N~+p(p>1)中具有平行平均曲率向量的紧致子流形,如果则M~位于N~+p的全测地子流形N~+1中。其中S,H分别是M~的第二基本形式长度的平方和M~的平均曲率,T_C、t_c分别是N~+p的Ricci曲率的上、下确界,K是N~+p的数量曲率。     

On compact submanifolds of nonpositive external curvature in Riemannian spaces

In this paper we consider compact multidimensional surfaces of nonpositive external curvature in a Riemannian space. If the curvature of the underlying space is ≥ 1 and the curvature of the surface is ≤ 1, then in small codimension the surface is a totally geodesic submanifold that is locally isometric to the sphere. Under stricter restrictions on the curvature of the underlying space, the submanifold is globally isometric to the unit sphere. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 3–10, July, 1996.     

Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator D^s is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.     

Totally umbilical manifolds inG(2,n). I

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds F^l of dimension l 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H 0, H 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S¹ × S¹ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic S^l+1 × S^l+1 G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold F^l E^l+2 (l 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 83–98, 1991.     

KAEHLER SUBMANIFOLDS IN A LOCALLY SYMMETRIC BOCHNER-KAEHLER MANIFOLD

This paper gives some sufficient conditions for a compact Kaehler submanifold M~n in a locally symmetric Bochner-Kaehler manifold ~(n p) to be totally geodesic. The conditions are given by inequalities which are established between. the sectional curvature(resp, holomorphic sectional curvature) of M~n and the Ricci curvature of ~(n p). In particular, similar results in the case where ~(n p) is a complex projective spathe are contained.     

The first eigenvalue of Laplace operator under powers of mean curvature flow

In this paper, we derive the evolution equation for the first eigenvalue of Laplace operator along powers of mean curvature flow. Considering a compact, strictly convex n-dimensional surface M without boundary, which is smoothly immersed in R n+1 , we prove that if the initial 2-dimensional surface M is totally umbilical, then the first eigenvalue is nondecreasing along the unnormalized H k -flow. Moreover, as applications of the evolution equation, we construct some monotonic quantities along this kind of flow.