|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Differential Geometry I
Time: Currently not offered
| Syllabus |
i) Smooth manifolds: Manifolds in Rn, submanifolds, manifolds with boundary. Smooth maps between manifolds. Regular values. Examples of manifolds: A) Curves and surfaces in R2 and R3. B) Level surfaces in Rn+1, C) Inverse image of regular values. Tangent spaces, derivatives of smooth maps, smooth vector fields, Existence of integral curves of a vector field near a point.
ii) Geometry of curves and surfaces: Parametrized curves in R3, length, integral formula for smooth curves, regular curves, parametrization by arc length. Os- culating plane of a space curve, Frenet frame, Frenet formula, curvature, invari- ance under isometry and reparametrization. Discussion of the cases for plane curves, rotation number of a closed curve, osculating circle, Umlaufsatz.
iii) Surfaces in R3: Existence of a normal vector of a connected surface. Gauss map. The notion of a geodesic on a surface. The existence and uniqueness of a geodesic on a surface through a given point with a given velocity vec- tor 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 surface at a point in a given direction. Principal curvatures, first and second fundamental forms, Gauss cur- vature and mean curvature. Gauss-Bonnet theorem (statement only).
iv) Differential forms and orientation: Differential Forms, Orientation of mani- folds, Integration of forms, Stokes Theorem. (proof to be given if time per- mits). Proof of Gauss-Bonnet theorem (if time permits).
Suggested Texts :
(a) B O Neill, Elementary Differential Geometry, Academic Press (1997).
(b) A. Pressley, Elementary Differential Geometry, Springer (Indian Reprint 2004).
(c) J. A. Thorpe, Elementary topics in Differential Geometry, Springer (Indian reprint, 2004).
(d) V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall.
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