复射影空间中具有平坦法丛的一般极小子流形的注记

设Mn是复射影空间CPn+p/2中具有平坦法丛的一般极小子流形.该文研究了这种子流形的曲率性质与几何性质之间的关系.运用活动标架法,得到关于Ricci曲率和第二基本形式模长的刚性定理,完善了已有文献的相关结果.此外,该文还得到具有平坦法丛的一般子流形一个重要性质.     

ON CURVATURE PINCHING FOR MINIMAL AND KAEHLER SUBMANIFOLDSiWITH ISOTROPIC SECOND FUNDAMENTAL FORM

An isometrically immersed submanifold is said to have isotropic second fundamental form if the length of the second fundamental form related into any normal vector is the same one. In this note, some curvature pinching theorems for compact minimal (resp. Kaehler) submanifolds in $S^{n+p}(c)$(resp. $CP^{n+p}(c)$) with isotropic second fundamental form are given.     

A Pinching Theorem for Minimal Submanifolds in Sphere

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＭｂｅａｎｎ－ｄｉｍｅｎｓｉｏｎａｌｃｌｏｓｅｄｍｉｎｉｍａｌｙｉｍｍｅｒｓｅｄｓｕｂｍａｎｉｆｏｌｄｉｎｔｈｅｕｎｉｔｓｐｈｅｒｅＳｎ＋ｐ，Ｓｔｈｅｓｅｑｕｒｅｏｆｔｈｅｌｅｎｇｔｈｏｆｔｈｅｓｅｃｏｎｄｆｕｎｄ...     

2—Harmonic Totally Real Submanifolds in a Complex Projective Space

２－ＨａｒｍｏｎｉｃＴｏｔａｌｌｙＲｅａｌＳｕｂｍａｎｉｆｏｌｄｓｉｎａＣｏｍｐｌｅｘ　Ｐｒｏｊｅｃｔｉｖｅ　ＳｐａｃｅＳｕｎＨｏｎｇａｎ（孙弘安）（ＳｏｕｔｈｅｒｎＩｎｓｔｉｔｕｔｅｏｆＭｅｔａｌｌｕｒｇｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ...     

单位球面中极小子流形的C∞紧性

本文研究了单位球面中极小子流形的C∞紧性,并得到两个紧性定理.作为应用,我们证明了存在正数δ(n),如果单位球面中极小子流形的第2基本形式的长度平方小于;2/3+δ(n),则它必须是全测地的或微分同胚于Veronese曲面.     

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Möbius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the Möbius geometry of submanifolds. In this paper, we give a classification of the Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues one of which is simple.     

Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map

We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p. In particular, it is shown that any variety in p>0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p=2.     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.     

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.     

ON COMPLETE SPACE-LIKE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＮｎ＋ｐｐｂｅａｎ（ｎ＋ｐ）－ｄｉｍｅｎｓｉｏｎａｌｃｏｎｎｅｃｔｅｄｐｓｅｕｄｏ－Ｒｉｅｍａｎｎｉａｎｍａｎｉｆｏｌｄｏｆｉｎｄｅｘｐ．ＩｆＮｎ＋ｐｐｉｓｃｏｍｐｌｅｔｅａｎｄｈａｓｃｏｎｓｔａｎｔｓｅｃｔｉｏｎ...     

de Sitter空间中具平行平均曲率向量的完备类空子流形

本文证明了ｄｅＳｉｔｔｅｒ空间中具平行平均曲率向量的完备类空子流形在Ｈ２〉Ｃ时其第二基本形式模长平方是上有界的，从而推广了Ｕ－ＨａｎｇＫＩ＾（４）及Ｑ．Ｍ．Ｃｈｅｎｇ＾（３）中的结果。     

双曲空间中完备子流形的特征值估计

研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献   

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

In this paper we prove that a submanifold with parallel mean curvature of a space of constant curvature, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, using properties of Clifford systems, we give a short and alternative proof of a result of Cartan asserting that a compact isoparametric hypersurface of the sphere with three distinct principal curvatures is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes. Received: 22 April 1998     

复射影空间的2-调和全实子流形

研究了复射影空间中2-调和全实子流形,得到了这类子流形的一个积分公式,讨论了伪脐条件下的情形,通过计算第二基本形式模长平方的Laplacian得到一个刚性定理.     

CR immersions and Lorentzian geometry

We study the geometry of the second fundamental form of pseudohermitian immersions among nondegenerate CR manifolds. In particular we study existence and uniqueness of pseudohermitian immersions $\phi : M \rightarrow S^{2n+3}$ of a strictly pseudoconvex CR manifold $M$ into an odd dimensional sphere, as determined by the pseudohermitian Gauss and Weingarten equations.     

The Topoligical Sphere Theorem for Complete Submanifolds

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold Mn in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of Mn has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable k-currents, we eliminate Hk (Mn, Z) for all 1 ` k ` n – 1.We then observe that the fundamental group of Mn is trivial. It should be emphasized that our result is optimal.     

Some properties of the distance function and a conjecture of De Giorgi

In the article [2] Ennio De Giorgi conjectured that any compact n-dimensional regular submanifold M of ℝ ^n+m ,moving by the gradient of the functional

DDVV-type inequality for skew-symmetric matrices and Simons-type inequality for Riemannian submersions

In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang–Mills horizontal distributions. In this way, we show phenomenons of duality between submanifold geometry and Riemannian submersion, particularly between second fundamental form of a submanifold and integrability tensor of a Riemannian submersion.     

On the Geometry of the 7-Sphere

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.