Course Details
Subject {L-T-P / C} : MA5119 : Differential Manifolds { 3-0-0 / 3}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
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, 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, 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.
Course Objectives
- A continuation of the course MA 5111, the course will enable the students to think in a more abstract fashion and apply it to several problems in PDEs
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Course Outcomes
Excellent opportunity for students to venture into the research field in a very strong topic in Mathematics.
Essential Reading
- W.M. Boothby, An Introduction to Differential Manifolds and Riemannian Geometry, 2nd Edition, Elsevier.
- S. Kumaresan, Intriduction to Differentiable Mnifolds and Lie groups, Hindustan Book Agency
Supplementary Reading
- L. Conlon, Differentiable Manifolds, Springer-Verlag, 2001.
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