Riemannian Manifolds

Author: John M. Lee
Publisher: Springer Science & Business Media
ISBN: 9780387227269
Release Date: 2006-04-06
Genre: Mathematics

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

The Laplacian on a Riemannian Manifold

Author: Steven Rosenberg
Publisher: Cambridge University Press
ISBN: 0521468310
Release Date: 1997-01-09
Genre: Mathematics

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Differential and Riemannian Manifolds

Author: Serge Lang
Publisher: Springer Science & Business Media
ISBN: 9781461241829
Release Date: 2012-12-06
Genre: Mathematics

This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.).

Convex Functions and Optimization Methods on Riemannian Manifolds

Author: C. Udriste
Publisher: Springer Science & Business Media
ISBN: 9789401583909
Release Date: 2013-11-11
Genre: Mathematics

The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately "nowhere" to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex.

Geometric Mechanics on Riemannian Manifolds

Author: Ovidiu Calin
Publisher: Springer Science & Business Media
ISBN: 9780817644215
Release Date: 2006-03-30
Genre: Mathematics

* A geometric approach to problems in physics, many of which cannot be solved by any other methods * Text is enriched with good examples and exercises at the end of every chapter * Fine for a course or seminar directed at grad and adv. undergrad students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics

Riemannian Manifolds of Conullity Two

Author: Eric Boeckx
Publisher: World Scientific
ISBN: 9789814498555
Release Date: 1996-11-09
Genre: Mathematics

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. Contents:IntroductionDefinition of Semi-Symmetric Spaces and Early DevelopmentLocal Structure of Semi-Symmetric SpacesExplicit Treatment of Foliated Semi-Symmetric SpacesCurvature Homogeneous Semi-Symmetric SpacesAsymptotic Foliations and Algebraic RankThree-Dimensional Riemannian Manifolds of Conullity TwoAsymptotically Foliated Semi-Symmetric SpacesElliptic Semi-Symmetric SpacesComplete Foliated Semi-Symmetric SpacesApplication: Local Rigidity Problems for Hypersurfaces with Type Number Two in IR4Three-Dimensional Riemannian Manifolds of c-Conullity TwoMore about Curvature Homogeneous SpacesBiolographyIndex Readership: Mathematicians and mathematical physicists. keywords:Riemannian Manifold;Curvature Homogeneous Space;Semi-Symmetric Space;Pseudo-Symmetric Space;Asymptotic Foliation;Hypersurface with Type Number Two;Gromov Conjecture;Lichnerowicz Formula;Nomizu Conjecture;Singer Number

Sobolev Spaces on Riemannian Manifolds

Author: Emmanuel Hebey
Publisher: Springer
ISBN: 9783540699934
Release Date: 2006-11-14
Genre: Mathematics

Several books deal with Sobolev spaces on open subsets of R (n), but none yet with Sobolev spaces on Riemannian manifolds, despite the fact that the theory of Sobolev spaces on Riemannian manifolds already goes back about 20 years. The book of Emmanuel Hebey will fill this gap, and become a necessary reading for all using Sobolev spaces on Riemannian manifolds. Hebey's presentation is very detailed, and includes the most recent developments due mainly to the author himself and to Hebey-Vaugon. He makes numerous things more precise, and discusses the hypotheses to test whether they can be weakened, and also presents new results.

Isometric Embedding of Riemannian Manifolds in Euclidean Spaces

Author: Qing Han
Publisher: American Mathematical Soc.
ISBN: 9780821840719
Release Date: 2006
Genre: Mathematics

The question of the existence of isometric embeddings of Riemannian manifolds in Euclidean space is already more than a century old. This book presents, in a systematic way, results both local and global and in arbitrary dimension but with a focus on the isometric embedding of surfaces in ${\mathbb R}^3$. The emphasis is on those PDE techniques which are essential to the most important results of the last century. The classic results in this book include the Janet-Cartan Theorem, Nirenberg's solution of the Weyl problem, and Nash's Embedding Theorem, with a simplified proof by Gunther.The book also includes the main results from the past twenty years, both local and global, on the isometric embedding of surfaces in Euclidean 3-space. The work will be indispensable to researchers in the area. Moreover, the authors integrate the results and techniques into a unified whole, providing a good entry point into the area for advanced graduate students or anyone interested in this subject. The authors avoid what is technically complicated. Background knowledge is kept to an essential minimum: a one-semester course in differential geometry and a one-year course in partial differential equations.

Foliations on Riemannian Manifolds and Submanifolds

Author: Vladimir Rovenski
Publisher: Springer Science & Business Media
ISBN: 9781461242703
Release Date: 2012-12-06
Genre: Mathematics

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.

Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable

Author: Kazuyoshi Kiyohara
Publisher: American Mathematical Soc.
ISBN: 9780821806401
Release Date: 1997
Genre: Mathematics

In this work, two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kahler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.

Introduction to Global Analysis Minimal Surfaces in Riemannian Manifolds

Author: John Douglas Moore
Publisher: American Mathematical Soc.
ISBN: 9781470429508
Release Date: 2017-12-15
Genre: Electronic books

During the last century, global analysis was one of the main sources of interaction between geometry and topology. One might argue that the core of this subject is Morse theory, according to which the critical points of a generic smooth proper function on a manifold determine the homology of the manifold. Morse envisioned applying this idea to the calculus of variations, including the theory of periodic motion in classical mechanics, by approximating the space of loops on by a finite-dimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinite-dimensional manifolds, and these infinite-dimensional manifolds were found useful for studying a wide variety of nonlinear PDEs. This book applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed parametrized minimal surfaces in a compact Riemannian manifold, establishing Morse inequalities for perturbed versions of the energy function on the mapping space. It studies the bubbling which occurs when the perturbation is turned off, together with applications to the existence of closed minimal surfaces. The Morse-Sard theorem is used to develop transversality theory for both closed geodesics and closed minimal surfaces. This book is based on lecture notes for graduate courses on “Topics in Differential Geometry”, taught by the author over several years. The reader is assumed to have taken basic graduate courses in differential geometry and algebraic topology.

Uncertainty Principles on Riemannian Manifolds

Author: Wolfgang Erb
Publisher: Logos Verlag Berlin GmbH
ISBN: 9783832527440
Release Date: 2011

In this thesis, the Heisenberg-Pauli-Weyl uncertainty principle on the real line and the Breitenberger uncertainty on the unit circle are generalized to Riemannian manifolds. The proof of these generalized uncertainty principles is based on an operator theoretic approach involving the commutator of two operators on a Hilbert space. As a momentum operator, a special differential-difference operator is constructed which plays the role of a generalized root of the radial part of the Laplace-Beltrami operator. Further, it is shown that the resulting uncertainty inequalities are sharp. In the final part of the thesis, these uncertainty principles are used to analyze the space-frequency behavior of polynomial kernels on compact symmetric spaces and to construct polynomials that are optimally localized in space with respect to the position variance of the uncertainty principle.

Homogeneous Structures on Riemannian Manifolds

Author: F. Tricerri
Publisher: Cambridge University Press
ISBN: 9780521274890
Release Date: 1983-06-23
Genre: Mathematics

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

Recent Developments in Pseudo Riemannian Geometry

Author: Dmitriĭ Vladimirovich Alekseevskiĭ
Publisher: European Mathematical Society
ISBN: 3037190515
Release Date: 2008-01-01
Genre: Mathematics

This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of the art in the field as well as open problems, which can stimulate further research.