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 - Lec 1...
• Lecture 6: Fields (International Winter Sc...
(For Connection)
• Lecture 7: Connections (International Wint...
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