Scalar curvature and sectional curvature Theory of geodesics. Hopf-Rinow theorem and completeness theorem. First and second variation formulas. Constant curvature manifolds. Negative curvature means universal cover is Euclidean space.
Postitive curvature means compact and finite fundamental group.
Closed subgroup matrix group is Lie group. Exterior Algebra Tensors and differential forms. Generalized Stokes theorem. Brauer fixed point theorem. Combing hair on billiard ball.