This book surveys Topology and Differential Geometry and also, Lie Groups and Algebras, and their Representations. The former topic is indispensable to students of gravitation and related areas of modern physics, including string theory. The latter has applications in gauge theory and particle physics, integrable systems and nuclear physics, among many others. The style of presentation is such that the mathematical statements are succinct and precise, but skip involved proofs that are not of primary importance to the physics reader.
Part I provides a simple introduction to basic topology, followed by a survey of homotopy. Calculus of differentiable manifolds is developed, after which a Riemannian metric is introduced along with the key concepts of connections and curvature. The final chapters lay out the basic notions of homology and De Rham cohomology as well as fibre bundles, particularly tangent and cotangent bundles.
Part II starts with a review of group theory followed by the basics of representation theory. A thorough description of Lie groups and algebras is presented with their structure constants and linear representations. Next, root systems and their classifications are detailed, concluding with the description of representations of simple Lie algebras emphasizing on spinor representations of orthogonal and pseudo-orthogonal groups.