This book, which focuses on the study of curvature, is an introduction to various aspects of pseudo-Riemannian geometry. We shall use Walker manifolds (pseudo-Riemannian manifolds which admit a non-trivial parallel null plane field) to exemplify some of the main differences between the geometry of Riemannian manifolds and the geometry of pseudo-Riemannian manifolds and thereby illustrate phenomena in pseudo-Riemannian geometry that are quite different from those which occur in Riemannian geometry, i.e. for indefinite as opposed to positive definite metrics. Indefinite metrics are important in many diverse physical contexts: classical cosmological models (general relativity) and string theory to name but two. Walker manifolds appear naturally in numerous physical settings and provide examples of extremal mathematical situations as will be discussed presently. To describe the geometry of a pseudo-Riemannian manifold, one must first understand the curvature of the manifold. We shall analyze a wide variety of curvature properties and we shall derive both geometrical and topological results. Special attention will be paid to manifolds of dimension 3 as these are quite tractable. We then pass to the 4 dimensional setting as a gateway to higher dimensions. Since the book is aimed at a very general audience (and in particular to an advanced undergraduate or to a beginning graduate student), no more than a basic course in differential geometry is required in the way of background. To keep our treatment as self-contained as possible, we shall begin with two elementary chapters that provide an introduction to basic aspects of pseudo-Riemannian geometry before beginning on our study of Walker geometry. An extensive bibliography is provided for further reading.

Math subject classifications : Primary: 53B20 -- (PACS: 02.40.Hw) Secondary: 32Q15, 51F25, 51P05, 53B30, 53C50, 53C80, 58A30, 83F05, 85A04

### Table of Contents

Basic Algebraic Notions

Basic Geometrical Notions

Walker Structures

Three-Dimensional Lorentzian Walker Manifolds

Four-Dimensional Walker Manifolds

The Spectral Geometry of the Curvature Tensor

Hermitian Geometry

Special Walker Manifolds

### About the Author(s)

**Peter Gilkey**, Mathematics Department, University of Oregon

Peter Gilkey is a Professor of Mathematics and a member of the Institute of Theoretical Science at the University of Oregon (USA). He is a fellow of the American Mathematical Society and is a member of the editorial board of Results in Mathematics, Differential Geometry and Applications, and the International Journal of Geometric Methods to Mathematical Physics. He received his Ph.D. in 1972 from Harvard University under the direction of L. Nirenberg. His research specialties are Differential Geometry, Elliptic Partial Differential Equations, and Algebraic topology. He has published more than 230 research articles and books.

**Miguel Brozos-Vazquez**, Department of Mathematics, University of Coruna

Miguel Brozos-Vazquez got his Ph.D. in 2007 from the University of Santiago de Compostela under the direction of Eduardo Garcia-Rio and Ramon Vazquez-Lorenzo. In 2005 and 2006, respectively, he visited the University of Oregon and the Max-Planck-Institut fur Mathematik in den Naturwissenschaften. After a year working as a teacher in Secondary School, he became Assistant Professor at the Universidade da Coruna in 2008. His research focuses mainly on Pseudo-Riemannian Geometry.

**Eduardo Garcia-Rio**, Faculty of Mathematics, University of Santiago de Compostela, Spain

Eduardo Garcia-Rio is a Professor of Mathematics and a member of the Institute of Mathematics of the University of Santiago de Compostela (Spain). He received his Ph.D. degree in 1992 from the University of Santiago de Compostela and is a member of the editorial board of the Journal of Geometric Analysis. His research specialities are Differential Geometry and Mathematical Physics.

**Stana Nikcevic**, Mathematical Institute, Belgrade, Serbia

Stana Nikcevic is a Professor of Mathematics at the University of Belgrade (Serbia). She received her Ph.D. from the University of Belgrade at the Mathematical Faculty and has been working at the University since 1974. During this period, she also sporadically worked at the University of Banja Luka (Bosnia and Hercegovina) and at the Mathematical Faculty in Kragujevac (Serbia). Her research mainly focuses on Differential Geometry. She has maintained international cooperation and has gone on short visits to the TU Berlin, Charles University Prague, Universitate Pierre et Marie Curie (Paris VI), University of Oregon (USA), and University of Santiago de Compostela (Spain).