Differential Geometry Lecture 1: Multilinear Algebra

Описание: First Lecture of Differential Geometry.

List of topics:
Multilinear Algebra
The dual space, tensor product space
Exterior k-forms, change of basis, and the wedge product. The determinant formula
Orientations on R^n, top forms. The interior product. The exterior algebra and the geometric algebra

Topology and Manifolds
Open and closed sets, and topologies. Continuity. Homeomorphisms. Open covers and compactness
Topological bases of open sets. Refined topologies. The separation axioms, Hausdorff spaces
First and second countable spaces. Locally finite, locally compact, and locally euclidean spaces
Manifolds. Charts and transition maps. Smooth maps between manifolds. Submanifolds
The tangent space to a manifold at a point. The tangent bundle and general vector bundle structures
The pushforward of a smooth map. Vector fields. Coordinate representations of vector fields.
The lie bracket of vector fields. Lie derivatives and the cartan formula.
Flows of vector fields. Integral curves.
The cotangent bundle. Wedge products of sections of the cotangent bundle. Smooth and differentiable k-forms (differential forms).
Pullback of forms and coordinate representations.
Volume forms and orientability of manifolds. Manifolds with boundary
Integrals of k forms. Stokes' theorem
Lie groups. The lie algebra of a lie group. Lie algebra axioms
Vector fields and flows on a lie group. Infinitesimal generators of flows. The structure coefficients

Differential Geometry: Classical to Modern
The hodge star operator. Self-dual and anti-self dual forms. The laplace-beltrami operator
Metrics. The musical isomorphism. The metric volume form. Riemannian manifolds
Divergence, gradient, and curl. Theorems of vector calculus
Frenet-Serret formulas. Curvature of a submanifold.
Shape tensor and principal curvature. Gaussian curvature of a manifold.
The geodesic equation. Arclength parametrization of geodesics
Parallel transport. Curvature and parallel transport. Connections on TM
The Levi-Civita connection. Riemannian curvature and torsion of a connection.
Killing forms of a connection. The Hopf-Rinow theorem.

Differential Geometry of Vector Bundles
Fibre bundles and fibre bundle charts.
Morphisms of fibre bundles. Sections of fibre bundles.
Vector bundles and the transition maps as matrices. Frames of vector bundles
Principal bundles and gauge transformations.
Connections on vector and principal bundles. The connection matrix. Curvature and torsion.