Fundamentals Of Differential Geometry

The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. 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. This is the new edition of Serge Lang's "Differential and Riemannian Manifolds. " 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: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations.