Differential geometry

cosmos 1st October 2018 at 1:33am

See Curves, Surfaces, for the the theory in 1 and 2 D respectively.

Differentiable manifold generalize these spaces in such a way that one can still do calculus on them. They are Topological manifolds with extra structure (a Differentiable structure)

Riemannian geometry – Riemannian manifolds extend differentiable manifolds to do the rest of geometry: distances, angles, curvature, etc.

Applications to General relativity and many other fields.

Learn from waifu

Analysis

Coordinate transformation

We could take coordinates as fundamental as David Wallace proposed..

Diffeomorphism

Differentiable function –> Smooth function

Fiber bundles

Vector bundles

Tangent bundle

Cotangent bundle

Derivatives

Differential (Jacobian, for maps between manifolds)

Vector fields. Derivative of a function (to $\mathbb{R}$ ) on a manifold, along a flow on the manifold, resulting on a new function on the manifold (to $\mathbb{R}$ ). This is basically a field of rank-1 differentials (Jacobians which are just vectors).