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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."
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.
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.
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."
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