We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained results to slant submanifolds.
We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds of bounded sectional curvature endowed with a weighted measure.
本文从外在的观点研究了常截面曲率实空间型$κ\leq 0$中等距浸入的开子流形在无穷远处的结构。我们将使用衰减的第二基本型的所谓驯服浸入，以获得一个描述在无穷远的子流形线的结构结果的文件Internat。Math.Res.Notices1994，no.9，作者R. E. Greene，P. Petersen和S. Zhou and Math.Ann.2001，321（4），authenticated by A. Petrunin和W.图希曼我们也将从下面得到一个估计的数量，其端部的体积增长方面的一类特殊的外部域，外部球。
From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the $18$seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the $19$nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships betwe…
We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive…
In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
Volume Growth, Number of Ends and the Topology of a Complete Submanifold
给定一个完全等距浸入$ϕ: P^m \longrightarrow N^n$在周围黎曼流形$N^n$中，该流形具有一个极点和径向截面曲率，该径向截面曲率由径向对称空间$M^n_w$的相应径向截面曲率从上到下界定，我们确定了关于$P$的外曲率的一组条件，该条件保证浸入是正确的，并且$P$具有有限拓扑，在论文“On Submanifold With Tamed第二基本型“（Glasgow Mathematical Journal，51，2009）中，G. Pacelli Bessa和M. Silencia Costa.当周围流形是一个径向对称空间时，证明了一个完备的极小子流形的（外在的）体积增长与它的端点数之间的不等式，这个不等式推广了Anderson的论文“The compactification of a minimal submanifold by the Gauss Map”（Preprint IEHS，1984）中关于$\erre^n$中完备的极小子流形的经典不等式.作为推论，我们得到了双曲空间中完备和极小子流形的（外在）体积增长与端点数之间的相应不等式，以及欧氏空间和双曲空间中这类子流形的Bernstein型结果，这是A. Kasue和K.徐文，“欧氏空间和双曲空间形式中某些子流形的间隙定理”，（大坂数学杂志，24，1987）。
Local and global conformal invariants of submanifolds
Penn State University（宾夕法尼亚州立大学）；Rutgers University（罗格斯大学）；Wichita State University（威奇托州立大学）；University of Notre Dame（圣母大学）；Sun Yat-sen University（中山大学）
For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least …
In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and $k$-Ricci curvature are also obtained.
In this work, we prove a compactness theorem on the space of all Hamiltonian stationay Lagrangian submanifolds in a compact symplectic manifold with uniform bounds on area and total extrinsic curvature.
In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of discrete Dirac operators, the latter on discrete surfaces with polygonal faces and normals defined on each face, and show that many key properties of the smooth theory are preserved. In particular, the corresponding spin transformations, conformal invariants for them, and the relatio…
The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar confor…
In this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants ($δ$-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. For further research directions, we address a…
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper we provide a comprehensive survey on this important class of submanifolds.
