MA5314 Differential geometry of manifolds
Revision of multivariate calculus , inverse and implicit function theorems Smooth manifolds, manifolds with boundary : definitions ,examples, smooth maps, diffeomorphisms, tangent vector, tangent plane, derivative (differential) of a smooth map immersions, submersions, embeddings and submanifolds orienation on manifolds vector fields and Lie bracket, Lie derivative Riemmanian metric, Riemmanian manifold, definition, example, pullback of a metric, vector bundles, tangent and cotangent bundle, Lie groups definition and examples, left , right and bi-invariant Riemmanian metric, connections on manifolds, Riemmanian (Levi-Civita) connections, covariant derivative, connections on vector bundles geodesics, geodesic flow, exponential maps, map, minimizing properties of geodesics curvature, sectional curvature, scalar curvature, tensors on Riemmanian manifolds
Manifolds and Differential geometry by J.M. Lee, publisher AMS,Roode Island,2009
1) Riemmanin geometry by Manfredo Peridiago Do Carmo, Publisher Birkhauser, 1992
2) Foundations of Differentiable manifolds and Lie groups F.K.Warner, publisher Springer,1983.