Content uploaded by Bang-Yen Chen
Author content
All content in this area was uploaded by Bang-Yen Chen on Aug 20, 2014
Content may be subject to copyright.
Totally umbilical submanifolds : Soochow Journal of Mathematics
This chapter presents a collection of problems on Riemannian geometry, covering various topics such as: Riemannian connections; geodesics; the Exponential map; curvature and Ricci tensors; characteristic classes; isometries; homogeneous Riemannian manifolds and Riemannian symmetric spaces; and left-invariant metrics on Lie groups.
To put some examples of the topics studied, some problems are dedicated to operators on Riemannian manifolds: gradient, divergence, codifferential, curl, Laplacian, and Hodge star operator. In other problems, affine, Killing, conformal, projective, harmonic or Jacobi fields are considered. Furthermore, we study some cases of submanifolds, surfaces in ℝ3, and pseudo-Riemannian manifolds.
Cartan’s method of moving frames is used in a number of problems. This is intended to familiarise the reader with this powerful method.
As an instance of aim of this chapter, some problems are related to the condition of constant curvature, showing whether the Riemannian manifold under study has or not this property, and thus permitting to the reader to familiarise with the different techniques used.
On 1995, T. Ikawa and J. B. Jun proved the following: If a doubly warped product manifold is conformally flat, then the manifolds and are conformally flat, too. As the corollary of this theorem, we can get the same properties in the case of product and warped product manifolds. But, about the case of a twisted product manifold, we can not see the similar theorem, yet. So, in this paper, we prove the same theorem in a twisted product manifold, that is, Let be a conformally flat twisted product manifold with the associated function f. Then the manifolds and are conformally flat, too.
In this paper, we consider a special type of twisted product man-ifold and we give the main result.
The conharmonic transformation is a conformal trans-formation which satisfies a specified differential equation. Such a transformation was defined by Y. Ishii and we have generalized his results. Twisted product space is a generalized warped product space with a warping function defined on a whole space. In this paper, we partially classified the twisted product space and ob-tain a sufficient condition for a twisted product space to be locally Riemannian products.
The conharmonic transformation is a conformal trans-formation which satisfies a specified differential equation. Such a transformation was defined by Y. Ishii and we have generalized his results. Twisted product space is a generalized warped product space with a warping function defined on a whole space. In this paper, we partially classified the twisted product space and ob-tain a sufficient condition for a twisted product space to be locally Riemannian products.
Hypercylinders in conformally symmetric manifolds are considered. The main result is the following theorem: Let (M,g) be a hypercylinder in a parabolic essentially conformally symmetric manifold (N,g ˜), dimN≥5 and let U ˜ be the subset of N consisting of all points of N at which the Ricci tensor S ˜ of (N,g ˜) is not recurrent. If U ˜∩M is a dense subset of M, then (M,g) is a conformally recurrent manifold.
The main purpose of this paper is to study totally umbilical submanifolds in a Kahler manifold.
By using antipodal sets and fixed point sets, we introduce a new method to study compact symmetry spaces. In this paper, we also provide several applications of our method.
The following main results are proved: An n-dimensional submanifold N in a Riemannian manifold M ̃ is an extrinsic sphere in M ̃ if and only if M ̃ admits an (n + 1)-dimensional totally geodesic submanifold M such that N lies in M as an extrinsic hypersphere. If M ̃ is a locally symmetric space, then N is an extrinsic sphere in M ̃ if and only if N is an extrinsic hypersphere in an (n + 1)-dimensional totally geodesic submanifold of constant curvature. If M ̃ is a Kaehler manifold and N lies in a totally geodesic submanifold M of constant curvature as an extrinsic hypersphere, then either N is a purely real manifold in M ̃ (i.e. TN ∩ J(TN) = {0}) or M is flat.
Totally umbilical submanifolds of dimension greater than four in quaternion-space-forms are completely classified.
From MATHSCINET.
This book is a welcome addition to the literature in differential geometry. It is concerned with many of the recent developments in differential geometry of submanifolds, to which the author has made a significant contribution. It presupposes a general knowledge of Riemannian geometry but is otherwise self-contained. Most of the background material is summarized in Chapter 1 (Riemannian manifolds) and Chapter 2 (Submanifolds).
The content of the main portion is as follows. Chapter 3 (Minimal submanifolds) includes classical results on the first variation of the volume integral and the Bernstein theorem for minimal surfaces in E^3 as well as recent results of Calabi, Cherndo Carmo-Kobayashi, Lawson, Simons, Takahashi and others. Chapter 4 (Submanifolds with parallel mean survature vector) describes the recent development in the theory of such submanifolds (particularly, surfaces that can be studied by the use of analytic functions) in spaces of constant curvature. Chapter 5 (Conformally flat submanifolds) is based mostly on the author's joint work with K. Yano. Chapter 6 (Umbilical submanifolds) contains results on pseudo-umbilical submanifolds and further related generalizations. Chapter 7 (Geometric inequalities) starts with a review of Morse theory and the results of Chern-Lashof on total absolute curvature. It then proceeds to the study of interesting inequalities concerning the total mean curvature.
Reviewer: Nomizu, K.