### Stage 3 Differential Geometry

Differential Geometry:
You may wonder, geometry takes up a large portion in high school mathematics, why isn't there any geometry course in the first two stages? In fact, geometry is kind of imbedded in stage two calculus (several variables) and linear algebra courses, they are usually assumed and will be used for this course. Here differential calculus is used to study geometry. Key things to study: multilinear algebra, curvature and torsion, Serret-Frenet equation, fundamental theorem of curves, Poincaré Index theorem (plane and surface), exterior calculus, Gauss' theorema egregium, geodesics, Gauss-Bonnet theorem.

Moving frame, due to Élie Cartan, is an approach to geometry of surface. According to my lecturer John Steele, it is "computationally the easiest, notationally the neatest, aesthetically the best, makes the definitions more natural and the proofs of the two major theorems easier. The moving frame method also points the way towards several important ideas in modern differential geometry and theoretical physics. The downside (if there is one) is the reliance on exterior calculus of differential forms." O'Neill, for example, uses this approach and he manages to prove Gauss' theorema egregium in half page, see p.281.

In general:

Tensor Analysis and Manifolds:

Differential Forms: