### MA5314 Differential geometry of manifolds

#### Course Details

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

#### Course References:

TextBooks:

Manifolds and Differential geometry by J.M. Lee, publisher AMS,Roode Island,2009

ReferenceBooks:

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.

Prerequisite: MA5310,MA5370,MA5380