Differentiable manifolds

Abstract

Definition of a topological manifold, structures on a manifold, morphisms of manifolds, examples of differentiable manifolds. Submanifolds, embeddings, immersions. The tangent space of a differentiable manifold, the tangent map. Differential forms on a manifold, transformations of differential forms, complex-valued differential forms. Partitions of unity on a manifold, construction of a Riemannian metric, construction of a tubular neighbourhood of a closed submanifold. Orientation of manifolds. Every complex analytic manifold has a canonical orientation. The integral of a differential form on an oriented manifold. Integration on a manifold with boundary and Stokes’ formula. Appendix on complex analytic sets.