Singular Semi-Riemannian Geometry

1. Front Matter

   Pages i-x

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2. Singular Semi-Riemannian Manifolds

   1. Front Matter

      Pages 1-1

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   2. Preliminaries I: The Linear Algebra of Real Inner Product Spaces

      • Demir N. Kupeli

      Pages 3-21

   3. A Review of Covariant Derivative Operators in Real Vector Bundles

      • Demir N. Kupeli

      Pages 23-37

   4. Singular Semi-Riemannian Manifolds

      • Demir N. Kupeli

      Pages 39-60

   5. Semi-Riemannian Submanifolds in Nondegenerate Semi-Riemannian Manifolds

      • Demir N. Kupeli

      Pages 61-88

3. Singular Kähler Manifolds

   1. Front Matter

      Pages 89-89

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   2. Preliminaries II: Linear Algebra of Hermitian Inner Product Spaces

      • Demir N. Kupeli

      Pages 91-109

   3. A Review of Covariant Derivative Operators in Complex Vector Bundles

      • Demir N. Kupeli

      Pages 111-118

   4. Singular Kähler Manifolds

      • Demir N. Kupeli

      Pages 119-131

   5. Hermitian Submanifolds of Nondegenerate Kähler Manifolds

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      Pages 133-139

4. Singular Quaternionic Kähler Manifolds

   1. Front Matter

      Pages 141-141

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   2. Preliminaries III: Linear Algebra of Quaternionic Inner Product Spaces

      • Demir N. Kupeli

      Pages 143-161

   3. Singular Quaternionic Kähler Manifolds

      • Demir N. Kupeli

      Pages 163-170

   4. Quaternionic Semi-Riemannian Submanifolds of Nondegenerate Quaternionic Kähler Manifolds

      • Demir N. Kupeli

      Pages 171-173

5. Back Matter

   Pages 174-181

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This book is an exposition of "Singular Semi-Riemannian Geometry"- the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. The main topic of interest is those cases where the metric tensor is assumed to be nondegenerate. In the literature, manifolds with degenerate metric tensors have been studied extrinsically as degenerate submanifolds of semi­ Riemannian manifolds. One major aspect of this book is first to study the intrinsic structure of a manifold with a degenerate metric tensor and then to study it extrinsically by considering it as a degenerate submanifold of a semi-Riemannian manifold. This book is divided into three parts. Part I deals with singular semi­ Riemannian manifolds in four chapters. In Chapter I, the linear algebra of indefinite real inner product spaces is reviewed. In general, properties of certain geometric tensor fields are obtained purely from the algebraic point of view without referring to their geometric origin. Chapter II is devoted to a review of covariant derivative operators in real vector bundles. Chapter III is the main part of this book where, intrinsically, the Koszul connection is introduced and its curvature identities are obtained. In Chapter IV, an application of Chapter III is made to degenerate submanifolds of semi-Riemannian manifolds and Gauss, Codazzi and Ricci equations are obtained. Part II deals with singular Kahler manifolds in four chapters parallel to Part I.

• Riemannian geometry
• Signatur
• Tensor
• covariant derivative
• differential geometry
• manifold

• Department of Mathematics, Middle East Technical University, Ankara, Turkey

  Demir N. Kupeli

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