- Note di Matematica
- 1 Introduction
- Riemannian Submanifolds - PDF Free Download
- Online A Basic Inequality For Submanifolds In Locally Conformal Almost Cosymplectic Manifolds 2002
- Table of Contents

Schouten , D. Calabi in the early 's. These classes of submanifolds have many interesting properties and many important results have been discovered in the last quarter of this century from this standpoint. Submanifold theory is a very active vast research field which in turn plays an important role in the development of modem differential geometry in this century.

This important branch of differential geometry is still so far from being exhausted; only a small portion of an exceedingly fruitful field has been cultivated, much more remains to be discovered in the coming centuries. Riemannian submanifolds H. In this case, M is called a Riemannian submanifold or simply a submanifold of M. We shall identify the image f M with M when there is no danger of confusion. One of fundamental problems in submanifold theory is the problem of isometric immersibility. The earliest publication on isometric embedding appeared in by L.

The problem of isometric immersion or embedding admits an obvious analytic interpretation; namely, if gij x , x -- Xl.

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Xn , are the components of the metric tensor g in local coordinates x l. Apparently it is appropriate to assume that he had in mind of analytic metrics and local analytic embeddings. Cartan-Janet's theorem In M.

## Note di Matematica

Cartan revised Janet's paper with the same title; while Janet wrote the problem in the form of a system of partial differential equations which he investigated using rather complicated methods, Cartan applied his own theory of Pfaffian systems in involution.

Both Janet's and Cartan's proofs contained obscurities. In C. Burstin got rid of them. Not every Riemannian n-manifold can be isometrically immersed in E m with m B. Then there exists a neighborhood of x which can be smoothly isometrically embedded into E 6.

### 1 Introduction

Nakamura and Y. Then there exists a local smooth isometric embedding of a neighborhood of x into E 6. Jacobowitz and J. Nash's embedding theorem A global isometric embedding theorem was proved by J. E Nash [ ] which states as follows. Furthermore, Greene [] and M. Gromov and V. Gromov showed that every Riemannian n-manifold can be C 1 isometrically immersed into E 2n cf.

This result has been extended to almost all symmetric spaces of compact type [Nagano, ; Kobayashi, ]. In J.

## Riemannian Submanifolds - PDF Free Download

Moore proved that every compact Riemannian homogeneous manifold admits an equivariant isometric embedding in some Euclidean space. Riemannian submanifolds 2. Isometric immersions with the smallest possible codimension According to Nash's embedding theorem, every Riemannian manifold can be isometrically embedded in a Euclidean space of sufficiently large dimension, it is thus natural to look for a Euclidean space of smallest possible dimension in which a Riemannian manifold can be isometrically embedded.

Hilbert proved in that a complete surface of constant negative curvature cannot be C4-isometrically immersed in Euclidean 3-space. A result of S. Chern and N.

Kuiper [] states that a compact Riemannian n-manifold with nonpositive sectional curvature cannot be isometrically immersed in E 2n A generalization by B. Weyl is that whether a Riemannian 2-manifold of positive Gaussian curvature that is diffeomorphic to a sphere can be realized as a smooth ovaloid in E3?

Weyl himself suggested an incomplete solution of this problem for analytic surfaces. In fact, he solved the problem in the case of analytic metrics sufficiently close to the metric of a sphere. A complete solution of Weyl's problem for analytic case was given by H. Lewy in Hong and C. Hong [ ] established isometric embedding in E 3 of complete noncompact nonnegatively curved surfaces. Lin [] considered the problem of isometric embedding of two-dimensional metrics of curvature that changes sign and proved the following: Let the curvature K of a Riemannian 2-manifold be equal to zero at the point P, but the gradient of the curvature VK be nonzero.

Chen neighborhood of P admits an isometric embedding of class C s - 3 in E 3. Greub and D.

For instance, sufficient conditions were found under which this problem can be solved, either by topological methods [Oliker, ; Caffarelli et al. Isometric immersions with prescribed m e a n curvature Given a closed n - 1 -dimensional submanifold F in a Riemannian manifold N, the problem of finding an oriented n-dimensional submanifold M with a prescribed mean curvature vector and with F as its boundary has been investigated by many mathematicians.

The first necessary conditions for parametric surfaces were given by E. Heinz in for surfaces of constant mean curvature in Euclidean 3-space with a prescribed rectifiable boundary. Gulliver gave in a necessary condition on the magnitude of the mean curvature vector field for there to exist an oriented submanifold of a Riemannian manifold M having prescribed mean curvature vector and a given closed submanifold as boundary. Bakelman and B. Kantor [], A. Treibergs and W. Wei [ ], A. Treibergs [], and K. Tso [] established the existence of closed convex hypersurfaces in a Euclidean space with prescribed mean curvature.

Fundamental theorems, basic notions and results 3.

Denote by V and V the metric connections of M and M, respectively. Formula 3. The metric connection on the normal bundle T-t-M induced from the metric connection of M is called the normal connection of M or of f. Let D denote covariant differentiation with respect to the normal connection. Weingarten Then the integrability condition for 3.

## Online A Basic Inequality For Submanifolds In Locally Conformal Almost Cosymplectic Manifolds 2002

The tangential and normal components of 3. Chen As a special case, suppose the ambient space M is a Riemannian manifold of constant sectional curvature c.

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The formulas can be found in a paper by D. These formulas at that time already published by G. Mainardi [Mainardi, ]. The fundamental importance of these formulas was fully recognized by O. Bonnet [Bonnet, ]. The equations of Gauss and Codazzi for general submanifolds were first given by A.

Voss in Ricci in Fundamental theorems The fundamental theorems of submanifolds are the following. Let M, g be a simply-connected Riemannian n-manifold and suppose there is a given m-dimensional Riemannian vector bundle v M over M with curvature tensor R o and a v M -valued symmetric 0, 2 tensor h on M. If they satisfy 3. The first to give a proof of the fundamental theorems was O. Bonnet cf. Riemannian submanifolds 3.

A point x e M is called a geodesic point if the second fundamental form h vanishes at x. The submanifold is said to be totally geodesic if every point of M is a geodesic point. A Riemannian submanifold M is a totally geodesic submanifold of M if and only if every geodesic of M is a geodesic of M.

Let M be a submanifold of M and let el. M is called a minimal submanifold of M if the mean curvature vector field vanishes identically. The submanifold is said to be totally umbilical if every point of the submanifold is an umbilical point. A submanifold M is called an isotropic submanifold if every point of M is an isotropic point.

### Table of Contents

It is clear that umbilical points are isotropic points. A submanifold M is said to have parallel mean curvature vector if the mean curvature vector field of M is a parallel normal vector field. A Riemannian submanifold M is said to have flat normal connection if the curvature tensor R O of the normal connection D vanishes at each point x e M. A general inequality Let M be a Riemannian n-manifold.

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The cardinal number S n of S n is equal to p n - 1, where p n denotes the number of partition of n which increases quite rapidly with n.