Differential Geometry of Curves and Surfaces

Code: MA611 | L-T-P-C: 3-0-0-6

 Prerequisites: MA541 or equivalent

Local theory of plane and space curves, Curvature and torsion formulas, Serret-Frenet formulas, Fundamental Theorem of space curves. Regular surfaces, Change of parameters, Differentiable functions, Tangent plane, Differential of a map. First and second fundamental form. Orientation, Gauss map and its properties, Euler's Theorem on principal curvatures. Isometries, and Gauss' Theorema Egregium. Parallel transport, Geodesics, Gauss-Bonnet theorem and its applications to surfaces of constant curvature. Hopf-Rinow's theorem, Bonnet's theorem, Jacobi fields, Theorems of Hadamard. Riemann's Habilitationsvortrag.

Texts/References:

1. Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.
2. John McCleary, Geometry from a Differentiable Viewpoint, Cambridge University Press, 1994.
3. Michael Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, 1994.
4. Barret O'Neill, Elementary Differential Geometry, Academic Press, Second Edition, 1997
5. Carl Friedrich Gauss, General Investigations of Curved Surfaces, Edited with an Introduction and Notes by Peter Pesic, Dover 2005.
6. Andrew Pressley, Elementary Differential Geometry, Springer 2002.