Module I (12 hours)
Differentiable Manifolds: Topological manifolds, Chart, Atlas, Maximal atlas, Differentiable structures, Differentiable functions, Diffeomorphisms, Germs of local smooth functions, Algebra of smooth germs, Derivation, Tangent and cotangent spaces, Differential of smooth map, Immersion, Vector bundles, Examples of smooth vector bundles.

Module II (10 hours)
Differential forms: Alternate k-linear functions, Grassmann algebras, Universal property of exterior algebra, Differential forms, Differential k-forms, Exterior multiplication, Exterior differentiation, De Rham cohomolgy groups, Induced transformations, Poincare’s lemma

Module III (8 hours)
Riemannian manifolds: Inner products, Riemannian structures, Riemannian metric, Riemannian connection, Geodesics, Convex neighbourhoods, De Rahm’s theorem: Singular homology groups, Real singular cohomology groups, De Rham’s theorem.

To extend the study of multivariable calculus.

CO1. They will learn about manifolds, differential forms, homology and cohomology groups.

CO2. They will get familiar with the concept of differentiation on manifolds.

CO3. They will be able to apply the learned concepts to several problems in PDEs.

CO4. It will work as a preparatory course for Differential Topology, Lie Groups, Ergodic Theory and other more advanced geometry courses.

W. M. Boothby, An Introduction to Differential Manifolds and Riemannian Geometry, Elsevier

S. Kumaresan, Introduction to Differentiable Manifolds and Lie groups, Hindustan Book Agency

L. Conlon, Differentiable Manifolds, Springer-Verlag

J. R. Munkres, Analysis on Manifolds, Addison Wesley