Smooth Manifolds

The book explains the concepts of smooth manifolds and helps in doing further research in general relativity and Riemannian geometry. Topics on point set topology have been developed from the scratch at suitable places in the text for their proper applications. The book discusses topics on smooth structures, tangent vectors, tangent bundles, Lie groups, immersed and embedded submanifolds, Sard s theorem, Whitney embedding theorem, multivariable calculus, together with theorems based on these topics. Complete proofs of all results are real marvels of the book. The book discusses, for the first time, Sard s theorem in case n = 1. The theory of smooth manifolds is abstract one, so figures given are not true ones. At many occasions, they simply confuse and delimit the imagination of readers. To avoid confusion, figures are avoided at some places in the book. Several difficult theorems have been broken into many lemmas, and many notes (equivalent to sub-lemmas), so that readability of the book is enhanced. For pedagogical reasons, a concept, first introduced, occurs at many places in the book, so that it is reinforced. Rank theorem, a vital theorem of the smooth manifolds theory, occurs in many manifestations rank theorem for Euclidean space and global rank theorem. The book, however, aims at graduate students of mathematics, but is useful to researcher. The prerequisites for this text have been kept intentionally to a minimum so that undergraduate students can also be benefitted. It is a believed conviction that mathematical proofs are the core of all mathematical joy . The present book reflects this conviction so vividly.