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.
- Animov Y. Differential Geometry and Topology of Curves
- Csikós B. Differential Geometry (FREE!)
- do Carmo M.P. Differential Geometry of Curves and Surfaces - Quite popular for introductory level. Beware that ^ means cross product, and means a dot b or inner product in this text. Check out the errata list by Bjorn Poonen.
- Hicks N.J. Notes on Differential Geometry [.pdf] (FREE!)
- Kreyszig E. Differential Geometry - Neither do Carmo nor O'Neill introduce the matrix notation when they first discuss the Frenet formulae, Kreyszig does that, which is nice.
- Millman R.S. and Parker G.D. Elements of Differential Geometry
- O'Neill B. Elementary Differential Geometry
- Pressley A. Elementary Differential Geometry - Solution at the back.
- Sharipov R. Course of Differential Geometry (FREE!)
- Struik D.J. Lectures on Classical Differential Geometry
- Zaitsev D. Differential Geometry: Lecture Notes [.pdf] (FREE!)
Tensor Analysis and Manifolds:
- Abraham R., Marsden J.E. and Ratiu T. Manifolds, Tensors, Analysis and Applications (FREE!)
- Bishop R.L. and Goldberg S.I. Tensor Analysis on Manifolds
- Lebedev L.P. and Cloud M.J. Tensor Analysis
- Munkres J.R. Analysis on Manifolds
- Sharipov R. Quick Introduction to Tensor Analysis (FREE!)
- Spivak M. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus - Not as excellent as his books on single variable calculus and the five volumes differential geometry, still better than many authors.
- Arapura D. Introduction to Differential Forms [.pdf] (FREE!)
- Bachman D. A Geometric Approach to Differential Forms
- Cartan H. Differential Forms - Member of Bourbaki, get a taste of the so called French style.
- Darling R.W.R. Differential Forms and Connections
- do Carmo M.P. Differential Forms and Applications
- Flanders H. Differential Forms with Applications to the Physical Sciences
- Morita S. Geometry of Differential Forms