Course Contents

Theory of surfaces, Riemannian space, Fundamental forms, Principal curvature, Euler’s theorem, Transformation of coordinates, Contravariant and covariant vectors, Metric as a tensor, Contravariant, covariant and mixed tensors, Multiplication of tensors, Manifolds, Derivations and dual derivations, The affine connection and their transformation, Covariant differentiation, Differential forms, Generalized Stokes’ and divergence theorem, Curves on manifolds, Intrinsic derivative, Lie derivative, Parallel and Lie transport, Geodesics, Curvature tensor, The Bianchi identities, The Ricci tensor and Ricci scalar, The Einstein tensor, Geodesic deviation, Conformal transformations, The Weyl tensor, Isometries and Killing vectors, The uniform vector field, The condition for flat spacetime, Parallel displacement and affine connections, Affine connection for covariant vector, Affine connection for metric tensor, Parallel displacement and covariant differentiation, Energy momentum tensor for a perfect fluid.

Course Learning Outcomes

This course is designed to enable the students to understand and apply the techniques of Lie derivative and basic concept of geometry related to applied mathematics.