Riemannian Geometry-5 (Vector Fields and Covariant Derivative (Connection))

In this lecture, we shall discuss the concepts of vector field and covariant derivative. The topic of the covariant derivative is presented in a way that is suitable for beginners.

Timestamps:

0:00 - Introduction 
0:57 - Vector Field 
16:00 - Derivative of f in the direction of vector field X
21:50 - Is there a notion of derivative of vector field Y in the direction of tangent v?
28:30 - Case of M = R^n 
48:00 - The operator D (Covariant Derivative) and its properties in R^n
55:57 - Definition of Covariant derivative (affine connection) on any smooth manifold M
1:01:56 - Local expression for covariant derivative 
1:11:24 - Physicist's notation for covariant derivative

A few References:

Videos:

(For Bundles and fields)
   • Construction of the tangent bundle - ...  

   • Lecture 6: Fields (International Wint...  

(For Connection)
   • Lecture 7: Connections (International...  

Books:

Smooth Manifolds , John M. Lee

Riemannian Manifolds: An introduction to curvature, John M. Lee

Differential Geometry and Lie Groups, S. Kumaresan

Riemannian Geometry, Manfredo P. do Carmo