Analysis on Manifolds

Abstract

The document is aimed at students planning to pursue their studies at the graduate level. The first part of the document is a rigorous introduction to integration, in particular to integration on manifolds. It includes a chapter on manifolds and differential forms. A chapter is devoted to applications and the link between modern differential geometry and classical vector calculus. The second part of the document is an introduction to modern differential geometry. There is a chapter on De Rham cohomology and a chapter on homology and cohomology, both simplicial and singular, with a proof of the relation between the simplicial and singular cohomology and de Rham cohomology. The last chapter is on Riemannian geometry and covers Cartan structural equations and geodesics, including a proof of Gauss-Bonnet theorem. The document ends with an introduction to non-euclidean geometries.