Manifolds and Differential Geometry By Jeffrey M Lee ISBN 9780821887134
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume and developments in topology. The basic object is a smooth manifold to which some extra structure has been attached such as Riemannian metric a symplectic form, a distinguished group of symmetries or a connection on the tangent bundle.
This book is a graduate- level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds such as vector bundles, tensors differentials forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory.