Course Archives    Theoretical Statistics and Mathematics Unit
Course: Differential Geometry I
Level: Postgraduate
Time: Currently not offered
Syllabus
Past Exams


Syllabus: Parametrized curves in R^3, length of curves, integral formula for smooth curves, regular curves, parametrization by arc length. Osculating plane of a space curve, Frenet frame, Frnet formula, curvatures, invariance under isometry and reparametrization. Discussion of the cases for plane curves, rotation number of a closed curve, osculating circle, Umlaufsatz.

Smooth vector fields on an open subset of R^3, gradient vector field of a smooth function, vector field along a smooth curve, integral curve of a vectorfield. Existence theorem of an integral curve of a vector field through a point, maximal integral curve through apoint.

Level sets, examples of surfaces in R^3. Tangent spaces at a point. Vector fields on surfaces. Existence theorem of integral curve of a smooth vector field on a surface through a point. Existence of a normal vector of a connected surface. Orientation, Gauss map. The notion of geodesic on a surface. The existence and uniqueness of geodesic on a surface through a given point and with a given velocity vector thereof. Covariant derivative of a smooth vector field. Parallel vector field along a curve. Existence and uniqueness theorem of a parallel vector field along a curve with a given initial vector. The Weingarten map of a surface at a point, its self-adjointness property.

Normal curvature of a surafce at a point in a given direction. Principal curvatures, first and second fundamental forms, Gauss curvature and mean curvature. Surface area and volume. Surfaces with boundary, local and global stokes theorem. Gauss-Bonnet theorem.

Suggested Texts :

1. B. ONeill, Elementary Differential Geometry, Academic Press (1997).

2. A. Pressley, Elementary Differential Geometry, Springer (Indian reprint 2004).

3. J.A. Thorpe, Elementary topics in Differential Geometry, Springer (Indian reprint 2004).

Top of the page

Past Exams
Midterm
04.pdf 07.pdf 10.pdf 12.pdf 14.pdf 16.pdf
Semestral
03.pdf 04.pdf 05.pdf 10.pdf 12.pdf 16.pdf
Supplementary and Back Paper
04.pdf 09.pdf 10.pdf

Top of the page

[ Semester Schedule ][ SMU ] [Indian Statistical Institute]