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This book presents a survey of Topology and Differential Geometry and also, Lie Groups and Algebras, and their Representations. The first topic is indispensable to students of gravitation and related areas of modern physics (including string theory), while the second has applications in gauge theory and particle physics, integrable systems and nuclear physics.Part I provides a simple introduction to basic topology, followed by a survey of homotopy. Calculus of differentiable manifolds is then de...

This volume presents eleven papers dealing with geometric PDEs, geometric flows, and related subject areas. Among the authors and topics are: Richard S. Hamilton on three-manifolds with positive Ricci curvature; Gerhard Huisken on flow by mean curvature of convex surfaces into spheres; L. C. Evans and J. Spruck on motion of level sets by mean curvature; Yun Gang Chen, Yoshikazu Giga and Shun'ichi Goto on uniqueness and existence of viscosity solutions of generalized mean curvature flow equations...

Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L...~[ji - [ji~ ,;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in- gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). Bu...

Beyond Einstein: Perspectives on Geometry, Gravitation, and Cosmology explores the rich interplay between mathematical and physical ideas by studying the interactions of major actors and the roles of important research communities over the course of the last century.

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity.The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects.The m...

This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these conce...

This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

This volume is an updated edition of "Essays on Mirror Manifolds", the first book of papers published after the phenomenon of mirror symmetry was discovered. The two major groups who made the discovery reported their papers here. Greene, Plesser and Candelas gave details on their findings; Witten gave his interpretation which was vital for future development; Vafa introduced the concept of quantum cohomology and several mathematicians, including Katz, Morrison, Wilson, Roan, Tian, Hubsch, Yau an...

Luis Antonio Santalo (Spain 1911 - Argentina 2001) contributed to several branches of Geometry, his laying of the mathematical foundations of Stereology and its applications perhaps being his most outstanding achievement. A considerable power of abstraction, a brilliant geometric intuition and an outstanding gift as a disseminator of science were among his virtues. The present volume contains a selection of his best papers. Part I consists of a short biography and some photographs together with...

This volume of proceedings contains selected and refereed articles - both surveys and original research articles - on geometric structures, global analysis, differential operators on manifolds, cohomology theories and other topics in differential geometry.

Writing this book, I had in my mind areader trying to get some knowledge of a part of the modern differential geometry. I concentrate myself on the study of sur­ faces in the Euclidean 3-space, this being the most natural object for investigation. The global differential geometry of surfaces in E3 is based on two classical results: (i) the ovaloids (i.e., closed surfaces with positive Gauss curvature) with constant Gauss or mean curvature are the spheres, (ü) two isometrie ovaloids are congruent...

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry - except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, i...

Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham...

This book gathers contributions by respected experts on the theory of isometric immersions between Riemannian manifolds, and focuses on the geometry of CR structures on submanifolds in Hermitian manifolds. CR structures are a bundle theoretic recast of the tangential Cauchy-Riemann equations in complex analysis involving several complex variables. The book covers a wide range of topics such as Sasakian geometry, Kaehler and locally conformal Kaehler geometry, the tangential CR equations, Lorentz...

This book is an introduction to the theory of quasiconformal and quasiregular mappings in the euclidean n-dimensional space, (where n is greater than 2). There are many ways to develop this theory as the literature shows. The authors' approach is based on the use of metrics, in particular conformally invariant metrics, which will have a key role throughout the whole book. The intended readership consists of mathematicians from beginning graduate students to researchers. The prerequisite requirem...

This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of exist ence of r linearly independent vector fields. For r = 1 the classical result of H. Hopf asserts that the vanishing of the Euler characteristic is the necessary and sufficient condition, and our results will give partial extens ions of Hopf's theorem to the case r > 1. Arecent article by E. Thomas [10] gives a good survey of work in this general are...

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson...

Diffeology is an extension of differential geometry. With a minimal set of axioms, diffeology allows us to deal simply but rigorously with objects which do not fall within the usual field of differential geometry: quotients of manifolds (even non-Hausdorff), spaces of functions, groups of diffeomorphisms, etc. The category of diffeology objects is stable under standard set-theoretic operations, such as quotients, products, co-products, subsets, limits, and co-limits. With its right balance betwe...