Semi-Riemannian Maps and Their Applications

by Garcia-Rio, Eduardo; Kupeli, Demir N.

• ISBN13: 9780792356615
• ISBN10: 0792356616
• Format: Hardcover
• Copyright: 1999-06-01
• Publisher: Kluwer Academic Pub

Summary

A major flaw in semi-Riemannian geometry is a shortage of suitable types of maps between semi-Riemannian manifolds that will compare their geometric properties. Here, a class of such maps called semi-Riemannian maps is introduced. The main purpose of this book is to present results in semi-Riemannian geometry obtained by the existence of such a map between semi-Riemannian manifolds, as well as to encourage the reader to explore these maps. The first three chapters are devoted to the development of fundamental concepts and formulas in semi-Riemannian geometry which are used throughout the work. In Chapters 4 and 5 semi-Riemannian maps and such maps with respect to a semi-Riemannian foliation are studied. Chapter 6 studies the maps from a semi-Riemannian manifold to 1-dimensional semi- Euclidean space. In Chapter 7 some splitting theorems are obtained by using the existence of a semi-Riemannian map. Audience: This volume will be of interest to mathematicians and physicists whose work involves differential geometry, global analysis, or relativity and gravitation.

Table of Contents

Preface	ix
Linear Algebra of Indefinite Inner Product Spaces	1	(18)
Inner Product Spaces	1	(6)
Linear Transformations Between Inner Product Spaces	7	(12)
Semi-Riemannian Manifolds	19	(40)
Metric Tensors	19	(6)
Connections on Semi-Riemannian Manifolds	25	(11)
Curvatures of Semi-Riemannian Manifolds	36	(7)
Some Differential Operators	43	(4)
The Semi-Riemannian Divergence Theorem	47	(12)
Second Fundamental Form of a Map	59	(24)
The Second Fundamental Form	59	(5)
Affine and Harmonic Maps	64	(5)
The Generalized Semi-Riemannian Divergence Theorem	69	(3)
The Bochner Identity	72	(11)
Semi-Riemannian Maps	83	(18)
The Generalized Eikonal Equation and Semi-Riemannian Maps	84	(2)
Semi-Riemannian Submersions	86	(4)
Riemannian Maps	90	(5)
Affine Riemannian Maps	95	(6)
Semi-Riemannian Transversal Maps	101	(16)
The Transversal Generalized Eikonal Equation and Semi-Riemannian Maps With Respect To a Semi-Riemannian Foliation	102	(3)
Semi-Riemannian Transversal Maps	105	(12)
Semi-Riemannian Eikonal Equations and The Semi-Riemannian Regular Interval Theorem	117	(26)
Semi-Riemannian Eikonal Equations	118	(8)
Affine Solutions of Semi-Riemannian Eikonal Equations	126	(6)
The Semi-Riemannian Regular Interval Theorem	132	(11)
Applications To Splitting Theorems	143	(20)
The Semi-Riemannian Splitting Theorem	143	(2)
Riemannian Splitting Theorems	145	(2)
Lorentzian Splitting Theorems	147	(16)
A Submanifolds of Semi-Riemannian Manifolds	163	(18)
A.1 Semi-Riemannian Submanifolds	163	(6)
A.2 Degenerate Submanifolds	169	(12)
B Riemannian and Lorentzian Geometry	181	(12)
B.1 Riemannian Geometry	181	(2)
B.2 Lorentzian Geometry	183	(10)
Bibliography	193	(3)
Index	196