Details of MA3202 (Spring 2017)

Level: 3 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
MA3202 Geometry of Curves and Surfaces Somnath Basu

Part I : Curves :

  • Curves : Parametrized and regular curves, arc length, parametrization by arc length.

  • Local Theory : Tangent-Normal-Binormal frame, curvature, torsion, fundamental theorems of local theory of plane and space curves.

  • Global Theory : Simple curves, the Jordan curve theorem (without proof), isoperimetric inequality, the four-vertex theorem.

Part II : Surfaces :

  • Surfaces : Parametrization, change of parameters, smooth functions, tangent plane, differential, diffeomorphism, inverse and implicit function theorems.

  • Second Fundamental Form and Curvature : Gauss map, oriented surfaces, second fundamental form, Gauss, mean and principal curvatures, normal sections.

  • Integration on Surface : Definition of integral, partitions of unity, change of variables formula, divergence theorem.

  • Global Extrinsic Geometry : Positively curved surfaces, Minkowski formulas, Aleksandrov theorem, isoperimetric inequality.

  • Intrinsic Geometry : Rigid motions and isometries, Gausss theorema egregium, geodesics, existence and uniqueness of geodesics, Hopf-Rinows theorem.

  • Gauss-Bonnet theorem : Degree of maps between surfaces, index of a vector field at an isolated zero, Gauss-Bonnet formula, Euler characteristic.

Analysis I, Analysis II, Linear algebra.


  1. M. Berger, and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, Springer-Verlag, 1987.

  2. M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

  3. S. Montiel, and A. Ros, Curves and Surfaces, Graduate Studies in Mathematics, Vol 69., American Mathematical Society, 2005.

  4. B. ONeill, Elementary Differential Geometry, (2nd Edition), Academic Press, 1997.

  5. A. Pressley, Elementary Differential Geometry, Springer International Edition, Springer-Verlag, 2006.

Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 2 Not Allowed
2 IP 4 Not Allowed
3 IP 6 Not Allowed
4 MR 2 Not Allowed
5 MR 4 Not Allowed
6 MS 6 Core
7 RS 1 Elective
8 RS 2 Not Allowed