D. Martin 
Manifold Theory [PDF ebook] 
An Introduction for Mathematical Physicists

This account of basic manifold theory and global analysis, based on senior undergraduate and post-graduate courses at Glasgow University for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology. Topology is included in two appendices because many courses on mathematics for physics students do not include this subject. – Provides a comprehensive account of basic manifold theory for post-graduate students- Introduces the basic theory of differential geometry to students in theoretical physics and mathematics- Contains more than 130 exercises, with helpful hints and solutions