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2019 | Buch

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Submanifold Theory

Beyond an Introduction

verfasst von: Marcos Dajczer, Ruy Tojeiro

Verlag: Springer US

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides a comprehensive introduction to Submanifold theory, focusing on general properties of isometric and conformal immersions of Riemannian manifolds into space forms. One main theme is the isometric and conformal deformation problem for submanifolds of arbitrary dimension and codimension. Several relevant classes of submanifolds are also discussed, including constant curvature submanifolds, submanifolds of nonpositive extrinsic curvature, conformally flat submanifolds and real Kaehler submanifolds. This is the first textbook to treat a substantial proportion of the material presented here. The first chapters are suitable for an introductory course on Submanifold theory for students with a basic background on Riemannian geometry. The remaining chapters could be used in a more advanced course by students aiming at initiating research on the subject, and are also intended to serve as a reference for specialists in the field.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Basic Equations of a Submanifold

Abstract

In this chapter we establish several basic facts of the theory of submanifolds that will be used throughout the book. We first introduce the second fundamental form and normal connection of an isometric immersion by means of the Gauss and Weingarten formulas. Then we derive their compatibility conditions, namely, the Gauss, Codazzi and Ricci equations. The main result of the chapter is the Fundamental theorem of submanifolds, which asserts that these data are sufficient to determine uniquely a submanifold of a Riemannian manifold with constant sectional curvature, up to isometries of the ambient space. As an application, we classify totally geodesic and umbilical submanifolds of space forms. We introduce the relative nullity distribution as well as the notion of principal normal vector fields of an isometric immersion, and derive some of their elementary properties. Submanifolds with flat normal bundle are briefly discussed.

Marcos Dajczer, Ruy Tojeiro

Chapter 2. Reduction of Codimension

Abstract

The study of isometric immersions becomes increasingly difficult for higher values of the codimension. Therefore, it is important to investigate whether the codimension of an isometric immersion into a space of constant sectional curvature can be reduced. That an isometric immersion \(f\colon M^n\to \mathbb {Q}_c^{n+p}\) admits a reduction of codimension to q < p means that there exists a totally geodesic submanifold \(\mathbb {Q}_c^{n+q}\) in \(\mathbb {Q}_c^{n+p}\) such that \(f(M)\subset \mathbb {Q}_c^{n+q}\). The possibility of reducing the codimension fits into the fundamental problem of determining the least possible codimension of an isometric immersion of a given Riemannian manifold into a space of constant sectional curvature.

Marcos Dajczer, Ruy Tojeiro

Chapter 3. Minimal Submanifolds

Abstract

The theory of minimal submanifolds is one of the most beautiful and developed subjects of differential geometry. The aim of this chapter is to introduce a few of its general aspects.

Marcos Dajczer, Ruy Tojeiro

Chapter 4. Local Rigidity of Submanifolds

Abstract

One of the basic problems in submanifold theory addressed in this book concerns the uniqueness of isometric immersions \(f\colon M^n\to \mathbb {Q}_c^m\) of Riemannian manifolds into space forms. Clearly, since g = τ ∘ f is also an isometric immersion for any isometry \(\tau \colon \mathbb {Q}_c^m\to \mathbb {Q}_c^m\), uniqueness should be understood to be up to congruences by isometries of the ambient space.

Marcos Dajczer, Ruy Tojeiro

Chapter 5. Constant Curvature Submanifolds

Abstract

The theory of flat bilinear forms has been developed in the previous chapter aiming at its applications to rigidity aspects of submanifolds in space forms. However, the initial motivation of Cartan’s theory of exteriorly orthogonal quadratic forms, which are equivalent to symmetric flat bilinear forms with respect to positive definite inner products, was to study isometric immersions \(f\colon M_c^n\to \mathbb {Q}_{c}^{n+p}\). Indeed, the second fundamental form of such an isometric immersion at any point \(x\in M_c^n\) provides the basic example of a symmetric bilinear form which is flat with respect to the positive definite inner product on N _f M(x). In fact, Cartan also used his theory to study isometric immersions \(f\colon M_c^n\to \mathbb {Q}_{\tilde c}^{n+p}\) with \(c<\tilde c\), just by looking at the composition \(\tilde f=i\circ f\) of f with an umbilical inclusion of \(\mathbb {Q}_{\tilde c}^{n+p}\) into \(\mathbb {Q}_{c}^{n+p+1}\).

Marcos Dajczer, Ruy Tojeiro

Chapter 6. Submanifolds with Nonpositive Extrinsic Curvature

Abstract

The results of this chapter show that isometric immersions \(f\colon M^n\to \tilde {M}^m\) with low codimension and nonpositive extrinsic curvature at any point must satisfy strong geometric conditions. That f has nonpositive extrinsic curvature at any point means that the sectional curvature K _M(σ) of M ⁿ along any plane σ does not exceed the corresponding sectional curvature \(K_{\tilde M}(f_*\sigma )\) of \(\tilde {M}^m\). The simplest result along this line is that a two-dimensional surface with nonpositive curvature in \(\mathbb {R}^3\) cannot be compact. This is a consequence of the fact that at a point of maximum of a distance function on a compact surface in \(\mathbb {R}^3\) the Gaussian curvature must be positive. It turns out that the simple idea in the proof of this elementary fact has far-reaching generalizations for non-necessarily compact submanifolds in fairly general ambient Riemannian manifolds.

Marcos Dajczer, Ruy Tojeiro

Chapter 7. Submanifolds with Relative Nullity

Abstract

Several of the results of Chaps. 4 and 6 have provided relevant geometric conditions under which a submanifold of a space form must have positive index of relative nullity at any point. The aim of this chapter is to study submanifolds that have this property.

Marcos Dajczer, Ruy Tojeiro

Chapter 8. Isometric Immersions of Riemannian Products

Abstract

The simplest way of constructing an immersion of a product manifold into a space form is to take an extrinsic product of immersions of the factors, a concept discussed in this chapter. The metric induced on a product manifold by an extrinsic product of immersions is the Riemannian product of the metrics induced by the immersions of the factors, and its second fundamental form is adapted to the product net of the manifold in the sense that the tangent spaces to each factor are preserved by all shape operators.

Marcos Dajczer, Ruy Tojeiro

Chapter 9. Conformal Immersions

Abstract

In this chapter we initiate the study of conformal immersions. Our approach is based on the fact that, to any conformal immersion \(f\colon M^n\to \mathbb {R}^m\) of a Riemannian manifold M ⁿ into Euclidean space, one can naturally associate an isometric immersion \(F\colon M^n\to \mathbb {V}^{m+1}\subset \mathbb {L}^{m+2}\) into the light-cone \(\mathbb {V}^{m+1}\) of Lorentzian space \(\mathbb {L}^{m+2}\), called its isometric light-cone representative.

Marcos Dajczer, Ruy Tojeiro

Chapter 10. Isometric Immersions of Warped Products

Abstract

In this chapter we discuss two other useful ways of constructing immersions of product manifolds from immersions of the factors, with an increasing degree of generality. Namely, we introduce the notions of (extrinsic) warped products of immersions and, more generally, of partial tubes over extrinsic products of immersions.

Marcos Dajczer, Ruy Tojeiro

Chapter 11. The Sbrana–Cartan Hypersurfaces

Abstract

By the classical Beez-Killing theorem, a hypersurface \(f\colon M^n\to \mathbb {Q}_c^{n+1}\) is rigid if it has type number τ ≥ 3 at any point. Therefore, if \(f\colon M^n\to \mathbb {Q}_c^{n+1}\) is an isometric immersion such that M ⁿ admits another isometric immersion \(\tilde f\colon M^n\to \mathbb {Q}_c^{n+1}\) that is not congruent to f on any open subset of M ⁿ, then f must have type number τ ≤ 2 at any point. Notice that f has type number τ ≤ 1 at a point of M ⁿ if and only if all sectional curvatures of M ⁿ at that point are equal to c, as follows from the Gauss equation. Totally geodesic hypersurfaces have already been classified in Chap. 1, whereas hypersurfaces of constant type number τ = 1 can locally be explicitly parametrized by means of the Gauss parametrization; see Corollaries 7.20 and 7.23.

Marcos Dajczer, Ruy Tojeiro

Chapter 12. Genuine Deformations

Abstract

In order to find necessary conditions for a submanifold in a space form with codimension greater than one to admit isometric deformations, one has to take into account that any submanifold of a deformable submanifold already possesses the isometric deformations induced by the latter. Therefore, when studying the isometric deformations of a submanifold, one should look for the “genuine” ones, that is, those which are not induced by isometric deformations of an “extended” submanifold of higher dimension. Besides, it is also of interest to consider isometric deformations of a submanifold that take place in a possibly different codimension.

Marcos Dajczer, Ruy Tojeiro

Chapter 13. Deformations of Complete Submanifolds

Abstract

The main theorems of this chapter are of global nature and show that complete Euclidean submanifolds with low codimension that allow isometric deformations are rather special. A first basic result in this direction is a theorem due to Sacksteder, which asserts that any compact Euclidean hypersurface \(f\colon M^n\to \mathbb {R}^{n+1}\), n ≥ 3, is isometrically rigid, provided that the subset of totally geodesic points of f does not disconnect M ⁿ. Even if that subset disconnects the manifold, only discrete isometric deformations are possible. In fact, any such deformation is a reflection with respect to an affine hyperplane. The corresponding versions of that result for hypersurfaces of the sphere and hyperbolic space are also discussed.

Marcos Dajczer, Ruy Tojeiro

Chapter 14. Infinitesimal Bendings

Abstract

Around the time that Sbrana obtained the local description of the isometrically deformable hypersurfaces discussed in Chap. 11, he also considered the problem of locally describing, in terms of the Gauss parametrization, the Euclidean hypersurfaces that are infinitesimally bendable, that is, the ones that admit nontrivial infinitesimal deformations. Roughly speaking, this means that the hypersurface admits a nontrivial, smooth, one-parameter variation by hypersurfaces that are isometric only “up to the first order.”

Marcos Dajczer, Ruy Tojeiro

Chapter 15. Real Kaehler Submanifolds

Abstract

The purpose of this chapter is to present several results on isometric immersions of Kaehler manifolds into real space forms. In fact, most of the results are about real Kaehler submanifolds. By a real Kaehler submanifold \(f\colon M^{2n}\to \mathbb {R}^m\) we mean an isometric immersion of a Kaehler manifold M ²ⁿ of complex dimension n ≥ 2 into Euclidean space.

Marcos Dajczer, Ruy Tojeiro

Chapter 16. Conformally Flat Submanifolds

Abstract

This chapter brings us back to the conformal realm. Here our main interest is on geometric and topological properties of conformally flat submanifolds of Euclidean space, that is, isometric immersions into Euclidean space of Riemannian manifolds that are locally conformally diffeomorphic to an open subset of Euclidean space.

Marcos Dajczer, Ruy Tojeiro

Chapter 17. Conformally Deformable Hypersurfaces

Abstract

This chapter is devoted to provide a modern presentation of Cartan’s classification of Euclidean hypersurfaces M ⁿ of dimension n ≥ 5 that admit nontrivial conformal deformations. Besides conformally flat hypersurfaces, the simplest examples are those that are conformally congruent to cylinders and rotation hypersurfaces over surfaces in \(\mathbb {R}^3\), and to cylinders over three-dimensional hypersurfaces of \(\mathbb {R}^4\) that are cones over surfaces in \(\mathbb {S}^3\). These examples are called conformally surface-like hypersurfaces.

Marcos Dajczer, Ruy Tojeiro

Backmatter

Titel: Submanifold Theory
verfasst von: Marcos Dajczer
Ruy Tojeiro
Verlag: Springer US
Electronic ISBN: 978-1-4939-9644-5
Print ISBN: 978-1-4939-9642-1
DOI: https://doi.org/10.1007/978-1-4939-9644-5