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Classification of locally symmetric spaces which admit a totally umbilical hypersurface
... The existence of proper totally umbilical hypersurface is a strong restriction for Riemannian symmetric spaces. Chen [2] proved that if a Riemannian symmetric space of dimension ≥ 4 admits non-totally geodesic totally umbilical hypersurfaces then it is conformally flat. In this section, we prove the nonexistence of totally umbilical hypersurfaces in Sol 4 1 . ...
In this paper, we prove the non-existence of Codazzi and totally umbilical hypersurfaces, especially totally geodesic hypersurfaces, in the 4 -dimensional model space .
In this article the author reports on recent results concerning reflections and involutions on Riemannian manifolds. This report is divided into two parts. One is on reflections of submanifolds and the other is on the interaction between two involutions.
This vita includes a curriculum vita and a list of publication published in the special issue dedicated to Bang-Yen Chen’s 60th birthday.
A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be
decomposes into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds
plays very important roles not only in geometry but also in mathematical physics, especially in general
relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution
and the Robertson–Walker models, are warped product manifolds.
The first part of this volume provides a self-contained and accessible introduction to the important subject of
pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account
on important results of warped product manifolds, including several important spacetimes such as Robertson–Walker’s and Schwarzschild’s.
The famous John Nash’s imbedding theorem published in 1956 implies that every warped product manifold
can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product
submanifolds in various important ambient spaces from extrinsic point of view was initiated by the author
around the beginning of this century.
The last part of this volume contains an extensive and comprehensive survey of numerous important results
on the geometry of warped product submanifolds done during this century by many geometers.
We establish the weak continuity of the Gauss-Coddazi-Ricci sys-tem for isometric embedding with respect to the uniform L p -bounded solution sequence for p > 2, which implies that the weak limit of the isometric embed-dings of the manifold in a fixed coordinate chart is an isometric immersion. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any se-quence of approximate solutions to this system which is uniformly bounded in L 2 and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of H −1 loc), the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.
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 first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on δ-invariants, which were introduced in the early 1990s by the author.
The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M. Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants or Chen invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.
The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on δ-invariants, which were introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M. Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants or Chen invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.
The main purpose of this paper is to study totally umbilical submanifolds in a Kahler manifold.
By applying Lie triple system we classify totally geodesic submanifolds of complex quadrics Q^n=SO(n+2)/SO(2) x SO(n) for n ≥ 2.
A submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result.
A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.
The main purpose of this paper is to study totally umbilical submanifolds in a Kahler manifold.