Parametrized curves in R3, 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 R3, gradient vector field of a smooth function, vector field along a smooth curve, integral curve of a vector field. Existence theorem of an integral curve of a vector field through a point, maximal integral curve through a point. Level sets, examples of surfaces in R3. 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. |
Reference Books:
- Manfredo P. Do Carmo. Differential Geometry of Curves and Surfaces, Prentice Hall
- McClearyJ. Geometry from a Differentiable Viewpoint, Cambridge University Press
- SpivakM. A Comprehensive Introduction to Differential Geometry, Publish or Perish
- Pesic P., Gauss C. F. General Investigations of Curved Surfaces, Edited with an Introduction and Notes, Dover
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