The principle aim of this unique text is to illuminate the beauty of the subject both with abstractions like proofs and mathematical text, and with visuals, such as abundant illustrations and diagrams. With few mathematical prerequisites, geometry is presented through the lens of linear fractional transformations. The exposition is motivational and the well-placed examples and exercises give students ample opportunity to pause and digest the material. The subject builds from the fundamentals of Euclidean geometry, to inversive geometry, and, finally, to hyperbolic geometry at the end. Throughout, the author aims to express the underlying philosophy behind the definitions and mathematical reasoning. This text may be used as primary for an undergraduate geometry course or a freshman seminar in geometry, or as supplemental to instructors in their undergraduate courses in complex analysis, algebra, and number theory. There are elective courses that bring together seemingly disparate topics and this text would be a welcome accompaniment.

A sequel to Lectures on Riemann Surfaces (Mathematical Notes, 1966), this volume continues the discussion of the dimensions of spaces of holomorphic cross-sections of complex line bundles over compact Riemann surfaces. Whereas the earlier treatment was limited to results obtainable chiefly by one-dimensional methods, the more detailed analysis presented here requires the use of various properties of Jacobi varieties and of symmetric products of Riemann surfaces, and so serves as a further introduction to these topics as well. The first chapter consists of a rather explicit description of a canonical basis for the Abelian differentials on a marked Riemann surface, and of the description of the canonical meromorphic differentials and the prime function of a marked Riemann surface. Chapter 2 treats Jacobi varieties of compact Riemann surfaces and various subvarieties that arise in determining the dimensions of spaces of holomorphic cross-sections of complex line bundles. In Chapter 3, the author discusses the relations between Jacobi varieties and symmetric products of Riemann surfaces relevant to the determination of dimensions of spaces of holomorphic cross-sections of complex line bundles. The final chapter derives Torelli's theorem following A. Weil, but in an analytical context. Originally published in 1973. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In this book, I explored differential equations for operation in Lie group and for representations of group Lie in a vector space.

I tell about different mathematical tool that is important in general relativity. The text of the book includes definition of geometric object, concept of reference frame, geometry of metric affinne manifold. Using this concept I learn dynamics in general relativity. We call a manifold with torsion and nonmetricity the metric affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection. The dynamics of a particle follows to the Frenet transport. The analysis of the Frenet transport leads to the concept of the Cartan connection which is compatible with the metric tensor. We need additional physical constraints to make a nonmetricity observable.

Robert Geroch's lecture notes on differential geometry reflect his original and successful style of teaching - explaining abstract concepts with the help of intuitive examples and many figures. The book introduces the most important concepts of differential geometry and can be used for self-study since each chapter contains examples and exercises, plus test and examination problems which are given in the Appendix. As these lecture notes are written by a theoretical physicist, who is an expert in general relativity, they can serve as a very helpful companion to Geroch's excellent "General Relativity: 1972 Lecture Notes."

Black holes present one of the most fascinating predictions of Einstein's general theory of relativity. There is strong evidence of their existence through observation of active galactic nuclei, including the centre of our galaxy, observations of gravitational waves, and others. There exists a large scientific literature on black holes, including many excellent textbooks at various levels. However, most of these steer clear from the mathematical niceties needed to make the theory of black holes a mathematical theory. Those which maintain a high mathematical standard are either focused on specific topics, or skip many details. The objective of this book is to fill this gap and present a detailed, mathematically oriented, extended introduction to the subject. The book provides a wide background to the current research on all mathematical aspects of the geometry of black hole spacetimes.

This book acquaints engineers with the basic concepts and terminology of modern global differential geometry. It introduces the Lie theory of differential equations and examines the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, and vector fields. 1990 edition.

This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone!

Suitable for advanced undergraduate and graduate students of mathematics, physics, and engineering, this text employs vector methods to explore the classical theory of curves and surfaces. Subsequent topics include the basic theory of tensor algebra, tensor calculus, the calculus of differential forms, and elements of Riemannian geometry. 1959 edition.

Quantum mechanics had been started with the theory of the hydrogen atom, so when considering the quantum mechanics in Riemannian spaces it is natural to turn first to just this simplest system. A common quantum-mechanical hydrogen atom description is based materially on the assumption of the Euclidean character of the physical 3-space geometry. In this context, natural questions arise: what in the description is determined by this special assumption, and which changes will be entailed by allowing for other spatial geometries. The questions are of fundamental significance, even beyond their possible experimental testing. In the present book, detailed analytical treatment and exact solutions are given to a number of problems of quantum mechanics and field theory in simplest non-Euclidean spacetime models. The main attention is focused on new themes created by non-vanishing curvature in classical physical topics and concepts.

This book presents the previously unpublished notes from a series of lectures given by the author at the Tata Institute of Fundamental Research in 1961. Basic material on affine connections and on locally or globally Riemannian and Hermitian symmetric spaces is covered. The final chapter proves the basic theorems on maximal compact subgroups of Lie groups. Readers should be familiar with differential manifolds and the elementary theory of Lie groups and Lie algebras.

A TREATISE ON THE DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES BY IJTTHEU PFAIILKR EISENHAIIT PKOPKHHOIt OK MATUKMATKH IN 1 ltINCMTON UNIVJCIWITY GOT AND COMPANY BOSTON NEW YOKE CHICAGO LONDON PREFACE This book is a development from courses which I have given in Princeton for a number of years. During this time I have come to feel that more would be accomplished by my students if they had an introductory treatise written in English and otherwise adapted to the use of men beginning their graduate work. Chapter I is devoted to the theory of twisted curves, the method in general being that which is usually followed in discussions of this subject. But in addition I have introduced the idea of moving axes, and have derived the formulas pertaining thereto from the previously obtained Freiiet-Serret fornmlas. In this way the student is made familiar with a method which is similar to that used by Darboux in the tirst volume of his Lepons, and to that of Cesaro in his Gcomctria Ittiriiiseca. This method is not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu able iu developing geometrical thinking. The remainder of the book may be divided into threo parts. The iirst, consisting of Chapters II-VI, deals with the geometry of a sur face in the neighborhood of a point and the developments therefrom, such as curves and systems of curves defined by differential equa tions. To a large extent the method is that of Gauss, by which the properties of a surface are derived from the discussion of two qxiad ratie differential forms. However, little or no space is given to the algebraic treatment of differential forms and their invariants. In addition, the method of moving axes, as defined in the first chapter, has been extended so as to be applicable to an investigation of the properties of surf ac. es and groups of surfaces. The extent of the theory concerning ordinary points is so great that no attempt has been made to consider the exceptional problems. Por a discussion of uch questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. lu Chapters VII and VIII the theory previously developed is applied to several groups of surfaces, such as the quadrics, ruled surfaces, minimal surfaces, surfaces of constant total curvature, and surfaces with plane and spherical lines of curvature iii iv PREFACE The idea of applicability of surfaces is introduced in Chapter IIT as a particular case of conformal representation, and throughout the book attention is called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely with the recent method of Weingarten and its developments. The remaining four chapters are devoted to a discussion of infinitesimal deformation of surfaces, congruences of straight Hues and of circles, and triply orthogonal systems of surfaces. It will be noticed that the book contains many examples, and the student will find that whereas certain of them are merely direct applications of the formulas, others constitute extensions of the theory which might properly be included as portions of a more ex tensive treatise. At first I felt constrained to give such references as would enable thereader to consult the journals and treatises from which some of these problems were taken, but finally it seemed best to furnish, no such key, only to remark that the flncyklopadie der mathematisc7ien Wissensckaften may be of assistance. And the same may be said about references to the sources of the subject-matter of the book. Many important citations have been made, but there has not been an attempt to give every reference. However, I desire to acknowledge niy indebtedness to the treatises of Uarboux, Biancln, and Scheffers...

This review presents the differential-geometric theory of homogeneous structures (mainly Poisson and symplectic structures)on loop spaces of smooth manifolds, their natural generalizations and applications in mathematical physics and field theory.

A TREATISE ON THE DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES BY IJTTHEU PFAIILKR EISENHAIIT PKOPKHHOIt OK MATUKMATKH IN 1 ltINCMTON UNIVJCIWITY GOT AND COMPANY BOSTON NEW YOKE CHICAGO LONDON PREFACE This book is a development from courses which I have given in Princeton for a number of years. During this time I have come to feel that more would be accomplished by my students if they had an introductory treatise written in English and otherwise adapted to the use of men beginning their graduate work. Chapter I is devoted to the theory of twisted curves, the method in general being that which is usually followed in discussions of this subject. But in addition I have introduced the idea of moving axes, and have derived the formulas pertaining thereto from the previously obtained Freiiet-Serret fornmlas. In this way the student is made familiar with a method which is similar to that used by Darboux in the tirst volume of his Lepons, and to that of Cesaro in his Gcomctria Ittiriiiseca. This method is not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu able iu developing geometrical thinking. The remainder of the book may be divided into threo parts. The iirst, consisting of Chapters II-VI, deals with the geometry of a sur face in the neighborhood of a point and the developments therefrom, such as curves and systems of curves defined by differential equa tions. To a large extent the method is that of Gauss, by which the properties of a surface are derived from the discussion of two qxiad ratie differential forms. However, little or no space is given to the algebraic treatment of differential forms and their invariants. In addition, the method of moving axes, as defined in the first chapter, has been extended so as to be applicable to an investigation of the properties of surf ac. es and groups of surfaces. The extent of the theory concerning ordinary points is so great that no attempt has been made to consider the exceptional problems. Por a discussion of uch questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. lu Chapters VII and VIII the theory previously developed is applied to several groups of surfaces, such as the quadrics, ruled surfaces, minimal surfaces, surfaces of constant total curvature, and surfaces with plane and spherical lines of curvature iii iv PREFACE The idea of applicability of surfaces is introduced in Chapter IIT as a particular case of conformal representation, and throughout the book attention is called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely with the recent method of Weingarten and its developments. The remaining four chapters are devoted to a discussion of infinitesimal deformation of surfaces, congruences of straight Hues and of circles, and triply orthogonal systems of surfaces. It will be noticed that the book contains many examples, and the student will find that whereas certain of them are merely direct applications of the formulas, others constitute extensions of the theory which might properly be included as portions of a more ex tensive treatise. At first I felt constrained to give such references as would enable thereader to consult the journals and treatises from which some of these problems were taken, but finally it seemed best to furnish, no such key, only to remark that the flncyklopadie der mathematisc7ien Wissensckaften may be of assistance. And the same may be said about references to the sources of the subject-matter of the book. Many important citations have been made, but there has not been an attempt to give every reference. However, I desire to acknowledge niy indebtedness to the treatises of Uarboux, Biancln, and Scheffers...