SE420 : Differential Geometry
Differential geometry is a branch of mathematics which has wide applications in Physics, Engineering, Architecture, and Economics. This course introduces important concepts of differential geometry by understandings of curves and surfaces.
- Curves : regular curves, Frenet formula, isoperimetric inequality, Cauchy-Crofton Formula
- Surfaces : regular surfaces, differentiable functions, regular values, tangent planes, differentials, area, orientation
- The first/second fundamental form : isometry, Gauss map, (principal, mean, Gaussian) curvatures,
- Vector fields : Christofffel symbols, Mainardi-Codazzi equations, covariant derivatives
- Parallel transport : geodesics, Gauss-Bonnet theorem, length-minimizing curves
- Exponential map : normal neighborhoods, complete surface
- Riemannian Geometry : Riemannian manifold, Levi-Civita connection, tangent bundle
- Tensor : curvature, sectional curvature
- Basic Lie group theory : left-invariant vector fields, Lie derivatives