|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Differential Geometry I
Time: Currently not offered
| Syllabus |
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).
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