Fundamentals of Differential Geometry

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas. Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds", the focus has now changed from the general theory of manifolds to general differential geometry.

New edition of a successful Serge Lang title Written in the authors unique and engaging style, with clear and elegant proofs Covers the fundamentals of differential geometry, differential topology, and differential equations Includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, and an example of semi-negative curvature New chapters, sections, examples, and exercises have been added

Autorentext

Zusammenfassung

"There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books. ...
It can be warmly recommended to a wide audience."
EMS Newsletter, Issue 41, September 2001

"The text provides a valuable introduction to basic concepts and fundamental results in differential geometry. A special feature of the book is that it deals with infinite-dimensional manifolds, modeled on a Banach space in general, and a Hilbert space for Riemannian geometry. The set-up works well on basic theorems such as the existence, uniqueness and smoothness theorem for differential equations and the flow of a vector field, existence of tubular neighborhoods for a submanifold, and the Cartan-Hadamard theorem. A major exception is the Hopf-Rinow theorem. Curvature and basic comparison theorems are discussed. In the finite-dimensional case, volume forms, the Hodge star operator, and integration of differentialforms are expounded. The book ends with the Stokes theorem and some of its applications."-- MATHEMATICAL REVIEWS

Inhalt
I General Differential Theory.- I Differential Calculus.- II Manifolds.- III Vector Bundles.- IV Vector Fields and Differential Equations.- V Operations on Vector Fields and Differential Forms.- VI The Theorem of Frobenius.- II Metrics, Covariant Derivatives, and Riemannian Geometry.- VII Metrics.- VIII Covariant Derivatives and Geodesics.- IX Curvature.- X Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle.- XI Curvature and the Variation Formula.- XII An Example of Seminegative Curvature.- XIII Automorphisms and Symmetries.- XIV Immersions and Submersions.- III Volume Forms and Integration.- XV Volume Forms.- XVI Integration of Differential Forms.- XVII Stokes' Theorem.- XVIII Applications of Stokes' Theorem.