SECTIONAL CURVATURE OF KAEHLER SUBMANIFOLDS OF A COMPLEX PROJCETIVE SPACE

Let M~n (n≥2) be a complex Kaehler submanifold immersed in the complex projective space CP~m(1). Let K be the sectional curvature of M~n. Then K≥1/8 if and only if M~n is an imbedding submanifold congruent to the standard imbedding CP~n (1) or CP~n(1/2).     

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.     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds

Let $M^{n}(n\geq4)$ be an oriented compact submanifold with parallel mean curvature in an $(n+p)$-dimensional complete simply connected Riemannian manifold $N^{n+p}$. Then there exists a constant $\delta(n,p)\in(0,1)$ such that if the sectional curvature of $N$ satisfies $\ov{K}_{N}\in[\delta(n,p), 1]$, and if $M$ has a lower bound for Ricci curvature and an upper bound for scalar curvature, then $N$ is isometric to $S^{n+p}$. Moreover, $M$ is either a totally umbilic sphere $S^n\big(\frac{1}{\sqrt{1+H^2}}\big)$, a Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)$ in the totally umbilic sphere $S^{n+1}\big(\frac{1}{\sqrt{1+H^2}}\big)$ with $n=2m$, or $\mathbb{C}P^{2}\big(\frac{4}{3}(1+H^2)\big)$ in $S^7\big(\frac{1}{\sqrt{1+H^2}}\big)$. This is a generalization of Ejiri''s rigidity theorem.     

MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE

A Riemannian manifold V~m which admits isometric imbedding into two spaces V~(m+p)ofdifferent constant curvatures is called a manifold of quasi constant curvature.TheRiemannian curvature of V~m is expressible in the formand conversely.In this paper it is proved that if M~n is any compact minimal submanifoldwithout boundary in a Riemannian manifold V~(n+p)of quasi constant curvature,then∫_(M~u)(2-1/p)σ~2-[na+1/2(b-丨b丨)(n+1)]σ+n(n-1)b~2*丨≥0,where σ is the square of the norm of the second fundamental form of M~n When V~(n+p)is amanifold of constant curvature,b=0,the above inequality reduces to that of Simons.     

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.     

ON TFE THIRD CONJECTURE OF K.OGIUE

In this paper, the authors prove following result:Let M~n be a complete Bechner-Kaehler submanifold of complex dimension (n≥4) in a complex projective space CP~(n p)(1) of complex dimension n p, endowed with the FubiniStudy metric of constant holomorphic sectional curvature 1. If the sectional curvature K of M~n satisfies K<1, then codimension p of M~n is not less then n(n 1)/2.     

New examples of Willmore submanifolds in the unit sphere via isoparametric functions

An isometric immersion ${x:M^n\rightarrow S^{n+p}}$ is called Willmore if it is an extremal submanifold of the Willmore functional: ${W(x)=\int\nolimits_{M^n} (S-nH^2)^{\frac{n}{2}}dv}$ , where S is the norm square of the second fundamental form and H is the mean curvature. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. This article gives a series of new examples of Willmore submanifolds in the unit sphere via isoparametric functions of FKM-type.     

Special Slant Surfaces and a Basic Inequality

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H²(p) ≥ 2K(p) ? 2(1 + 3 cos²θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.     

Differential geometry of real submanifolds in a Kähler manifold

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.     

Submanifolds of generalized $$(k,\mu )$$-space-forms

In this paper, we have studied submanifolds especially, totally umbilical submanifolds of generalized \((k,\mu )\)-space-forms. We have found a necessary and sufficient condition for such submanifolds to be either invariant or anti-invariant. It is also shown that every totally umbilical submanifold of a generalized \((k,\mu )\)-space-form is a pseudo quasi-Einstein manifold.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

On vertical skew symmetric almost contact 3-structures

We consider a (2m + 3)-dimensional Riemannian manifold M(ξ _r, η^r, g ) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector field is an isoparametric function. If, in addition, M(ξ _r, η^r, g ) is endowed with an f -structure φ, M, turns out to be a framed f−CR-manifold. The fundamental 2-form Ω associated with φ is a presymplectic form. Locally, M is the Riemannian product of two totally geodesic submanifolds, where is a 2m-dimensional Kaehlerian submanifold and is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of Ω is obtained.     

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.     

多面体截面和小覆盖的闭子流形

设π:M~n→P~n是P~n上的小覆盖,S是P~n的任意一个n-1维截面.给出了π~(-1)(S)是n-1维闭子流形(或者两个相互同胚n-1维闭子流形的不交并),以及π~(-1)(S)是n-1维伪流形的充要条件.     

3—Sasakian流形的3—C—全实子流形

引入并研究了３－Ｓａｓａｋｉａｎ流形的３－Ｃ－全实子流形，借助了第二基本形式的长度，给出了单位球面Ｓ＾４ｍ＋３－Ｃ－全实子流形的一个Ｐｉｎｃｈｉｎｇ定理。     

On the Tangent Bundle of a Hypersurface in a Riemannian Manifold

Let(Mn, g) and(Nn+1, G) be Riemannian manifolds. Let TMn and TNn+1 be the associated tangent bundles. Let f :(Mn,g) →(Nn+1,G) be an isometrical immersion with g = f*G, F =(f, df) :(TMn, ■) →(TNn+1, Gs) be the isometrical immersion with ■= F*Gs where (df)x: TxM → Tf(x)N for any x ∈M is the differential map, and Gs be the Sasaki metric on TN induced from G. This paper deals with the geometry of TMn as a submanifold of TNn+1 by the moving frame method. The authors firstly study the extrinsic geometry of TMn in TNn+1. Then the integrability of the induced almost complex structure of TM is discussed.     

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.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold (M) of non-negative (n-1)-th Ricci curvature.The Liouville type theorem about the p-harmonic map with finite Lq-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

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.