Modern Geometry ― Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics (93))

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised. manifolds, transformation groups, and Lie algebras, as well as the basic concepts of visual topology. It was also agreed that the course should be given in as simple and concrete a language as possible, and that wherever practic­ able the terminology should be that used by physicists. Thus it was along these lines that the archetypal course was taught. It was given more permanent form as duplicated lecture notes published under the auspices of Moscow State University as: Differential Geometry, Parts I and II, by S. P. Novikov, Division of Mechanics, Moscow State University, 1972. Subsequently various parts of the course were altered, and new topics added. This supplementary material was published (also in duplicated form) as Differential Geometry, Part III, by S. P. Novikov and A. T. Fomenko, Division of Mechanics, Moscow State University, 1974. The present book is the outcome of a reworking, re-ordering, and ex­ tensive elaboration of the above-mentioned lecture notes. It is the authors'view that it will serve as a basic text from which the essentials for a course in modern geometry may be easily extracted. To S. P. Novikov are due the original conception and the overall plan of the book. The work of organizing the material contained in the duplicated lecture notes in accordance with this plan was carried out by B. A. Dubrovin. Manifolds, Transformation Groups, And Lie Algebras, As Well As The Basic Concepts Of Visual Topology. It Was Also Agreed That The Course Should Be Given In As Simple And Concrete A Language As Possible, And That Wherever Practic Able The Terminology Should Be That Used By Physicists. Thus It Was Along These Lines That The Archetypal Course Was Taught. It Was Given More Permanent Form As Duplicated Lecture Notes Published Under The Auspices Of Moscow State University As: Differential Geometry, Parts I And Ii, By S. P. Novikov, Division Of Mechanics, Moscow State University, 1972. Subsequently Various Parts Of The Course Were Altered, And New Topics Added. This Supplementary Material Was Published (also In Duplicated Form) As Differential Geometry, Part Iii, By S. P. Novikov And A. T. Fomenko, Division Of Mechanics, Moscow State University, 1974. The Present Book Is The Outcome Of A Reworking, Re-ordering, And Ex Tensive Elaboration Of The Above-mentioned Lecture Notes. It Is The Authors' View That It Will Serve As A Basic Text From Which The Essentials For A Course In Modern Geometry May Be Easily Extracted. To S. P. Novikov Are Due The Original Conception And The Overall Plan Of The Book. The Work Of Organizing The Material Contained In The Duplicated Lecture Notes In Accordance With This Plan Was Carried Out By B. A. Dubrovin. Manifolds, transformation groups, and Lie algebras, as well as the basic concepts of visual topology. It was also agreed that the course should be given in as simple and concrete a language as possible, and that wherever practicƯ able the terminology should be that used by physicists. Thus it was along these lines that the archetypal course was taught. It was given more permanent form as duplicated lecture notes published under the auspices of Moscow State University as: Differential Geometry, Parts I and II, by S.P. Novikov, Division of Mechanics, Moscow State University, 1972. Subsequently various parts of the course were altered, and new topics added. This supplementary material was published (also in duplicated form) as Differential Geometry, Part III, by S.P. Novikov and A.T. Fomenko, Division of Mechanics, Moscow State University, 1974. The present book is the outcome of a reworking, re-ordering, and exƯ tensive elaboration of the above-mentioned lecture notes. It is the authors' view that it will serve as a basic text from which the essentials for a course in modern geometry may be easily extracted. To S.P. Novikov are due the original conception and the overall plan of the book. The work of organizing the material contained in the duplicated lecture notes in accordance with this plan was carried out by B.A. Dubrovin This is the first volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This new edition offers substantial revisions, and the material is written in concrete language with terminology acceptable to physicists. Front Matter....Pages i-xv Geometry in Regions of a Space. Basic Concepts....Pages 1-60 The Theory of Surfaces....Pages 61-144 Tensors: The Algebraic Theory....Pages 145-233 The Differential Calculus of Tensors....Pages 234-312 The Elements of the Calculus of Variations....Pages 313-374 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants....Pages 375-454 Back Matter....Pages 455-464 Pt. 1. The Geometry Of Surfaces, Transformation Groups, And Fields. B.a. Dubrovin, A.t. Fomenko, S.p. Novikov ; Translated By Robert G. Burns. Original Russian Edition: Sovremennaja Geometria: Metody I Priloz̆enia. Moskva: Nauka, 1979--t.p. Verso. Includes Bibliographies And Index. The concept of a manifold is in essence a generalization of the idea, first formulated in mathematical terms by Gauss, underlying the usual procedure used in cartography (i.e. the drawing of maps of the earth's surface, or portions of it). Over the past fifteen years, the geometrical and topological methods of the theory of manifolds have as- sumed a central role in the most advanced areas of pure and applied mathematics as well as theoretical physics.