1
Smooth Manifolds
Definition of smooth manifolds
2
Examples of Differentiable Manifolds
Various examples of differentiable manifolds
3
Tangent Space
Tangent Vectors and Tangent Space
4
Cotangent Space
Tangent Vectors and Tangent Space
5
Differentials
The differential between two tangent spaces
6
Examples of Differentials
Examples of smooth functions and differentials
7
Submanifolds and the Inverse Function Theorem
Substructures of differentiable manifolds
8
Uniqueness of Submanifolds
The topological structure of immersed submanifolds and the factorization of smooth functions
9
Implicit Function Theorem
The implicit function theorem on differentiable manifolds and its consequences
10
Tangent and Cotangent Bundles
Definition of vector bundles and tangent, cotangent bundles
11
Vector Fields
Vector fields
12
Differential Forms
Differential form
13
Distribution
The definition of a distribution and the Frobenius theorem
14
Lie Derivative
Lie derivative and Lie bracket
15
Differential Ideal
Differential ideals and Frobenius’s theorem
16
Orientation
Orientation on a manifold