Riemannian Manifolds Of Conullity Two
Riemannian Manifolds Of Conullity Two

Author: Eric Boeckx

Publisher: World Scientific

Published: 1996-11-09

Total Pages: 319

ISBN-13: 9814498556

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This book deals with Riemannian manifolds for which the nullity space of the curvature tensor has codimension two. These manifolds are “semi-symmetric spaces foliated by Euclidean leaves of codimension two” in the sense of Z I Szabó. The authors concentrate on the rich geometrical structure and explicit descriptions of these remarkable spaces. Also parallel theories are developed for manifolds of “relative conullity two”. This makes a bridge to a survey on curvature homogeneous spaces introduced by I M Singer. As an application of the main topic, interesting hypersurfaces with type number two in Euclidean space are discovered, namely those which are locally rigid or “almost rigid”. The unifying method is solving explicitly particular systems of nonlinear PDE.

Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics
Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics

Author: Stancho Dimiev

Publisher: World Scientific

Published: 2001

Total Pages: 220

ISBN-13: 9812810145

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The Deligne-Simpson problem for zero index of rigidity / V.P. Rostov -- Theorems for extension on manifolds with almost complex structures / L.N. Apostolova, M.S. Marinov and K.P. Petrov -- The theorem on analytic representation on hypersurface with singularities / A.M. Kytmanov and S.G. Myslivets -- Pseudogroup structures on Spencer manifolds / S. Dimiev -- Type-changing transformations of Hurwitz pairs, quasiregular functions, and hyper Kahlerian holomorphic chains I / J. Lawrynowicz and L.M. Tovar -- Embedding of the moduli space of Riemann surfaces with Igeta structures into the Sato Grassmann manifold / Y. Hashimoto and K. Ohba -- On the quotient spaces of S2 x S2 under the natural action of subgroups of D4 / K. Kikuchi -- Existence of spin structures on cyclic branched covering spaces over four-manifolds / S. Nagami -- Length spectrum of geodesic spheres in rank one symmetric spaces / T. Adachi -- Grassmann geometry of 6-dimensional sphere, II / H. Hashimoto and K. Mashimo -- Hypersurfaces in Euclidean space which are one-parameter families of spheres / G. Ganchev and V. Mihova -- Hypersurfaces of conullity two in Euclidean space which are one-parameter systems of torses / G. Ganchev and V. Milousheva -- Real hypersurfaces of a Kaehler manifold (the sixteen classes) / G. Ganchev and M. Hristov -- Almost contact B-metric hypersurfaces of Kaehlerian manifolds with B-metric / M. Manev -- Projective formalism and some methods from algebraic geometry in the theory of gravitation / B.G. Dimitrov -- Geometry of manifolds and dark matter / I.B. Pestov -- Lagrangian fluid mechanics / S. Manoff -- Transformation of connectednesses / G. Zlatanov

Semiparallel Submanifolds in Space Forms
Semiparallel Submanifolds in Space Forms

Author: Ülo Lumiste

Publisher: Springer Science & Business Media

Published: 2009-03-02

Total Pages: 306

ISBN-13: 038749913X

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Quite simply, this book offers the most comprehensive survey to date of the theory of semiparallel submanifolds. It begins with the necessary background material, detailing symmetric and semisymmetric Riemannian manifolds, smooth manifolds in space forms, and parallel submanifolds. The book then introduces semiparallel submanifolds and gives some characterizations for their class as well as several subclasses. The coverage moves on to discuss the concept of main symmetric orbit and presents all known results concerning umbilic-like main symmetric orbits. With more than 40 published papers under his belt on the subject, Lumiste provides readers with the most authoritative treatment.

The Geometry Of Curvature Homogeneous Pseudo-riemannian Manifolds
The Geometry Of Curvature Homogeneous Pseudo-riemannian Manifolds

Author: Peter B Gilkey

Publisher: World Scientific

Published: 2007-04-26

Total Pages: 389

ISBN-13: 1908979275

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Pseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and Stanilov-Tsankov-Videv theory./a

Foliations on Riemannian Manifolds and Submanifolds
Foliations on Riemannian Manifolds and Submanifolds

Author: Vladimir Rovenski

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 296

ISBN-13: 1461242703

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This monograph is based on the author's results on the Riemannian ge ometry of foliations with nonnegative mixed curvature and on the geometry of sub manifolds with generators (rulings) in a Riemannian space of nonnegative curvature. The main idea is that such foliated (sub) manifolds can be decom posed when the dimension of the leaves (generators) is large. The methods of investigation are mostly synthetic. The work is divided into two parts, consisting of seven chapters and three appendices. Appendix A was written jointly with V. Toponogov. Part 1 is devoted to the Riemannian geometry of foliations. In the first few sections of Chapter I we give a survey of the basic results on foliated smooth manifolds (Sections 1.1-1.3), and finish in Section 1.4 with a discussion of the key problem of this work: the role of Riemannian curvature in the study of foliations on manifolds and submanifolds.

Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor
Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor

Author: Peter B. Gilkey

Publisher: World Scientific

Published: 2001

Total Pages: 316

ISBN-13: 9812799699

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A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition. The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed. Contents: Algebraic Curvature Tensors; The Skew-Symmetric Curvature Operator; The Jacobi Operator; Controlling the Eigenvalue Structure. Readership: Researchers and graduate students in geometry and topology.