Differential Geometry III
Michaelmas 2013
| Textbooks: | The following is a list of books on which the lectures are based. Although we will not follow any of these books strictly, the material can be found in them. |
- L.M. Woodward and J. Bolton, Differential Geometry Lecture
Notes. Copies are available from the Maths office
- M. Do Carmo, Differential Geometry of Curves and Surfaces
Preliminary course content (subject to change):
Plane and space curves, arc length, tangent and normal vectors, curvature, local and global properties; embedded surfaces, tangent planes, curves on surfaces; intrinsic geometry of a surface, metric, length, area, first fundamental form; maps between surfaces, Gauss map.
Schedule:
Week 1: Introduction and overview of the course, idea of curvature of a curve and surface; definition of a curve, trace, tangent vector, regular curve, length of a curve, examples
Week 2: Arc length, existence of arc length parametrization, examples; tangent and normal vectors, curvature of a plane curve
Week 3: Vertices and inflection points of plane curves, four vertex theorem, fundamental theorem of local theory of plane curves; radius and center of curvature, evolute and involute of a plane curve
Week 4: Problems class; space curves, principal normal vector, binormal vector, curvature, torsion, Serret-Frenet equations, curvature and torsion for unit speed space curves
Week 5: Curvature and torsion for non unit speed space curves, geometric meaning of curvature and torsion, fundamental theorem of local theory of space curves; local canonical form of space curves, open sets in R^n, smooth maps R^n -> R^m, Jacobi matrix, differential
Week 6: Implicit function theorem; parametrized surfaces in Euclidean space, regular surfaces as level sets, examples
Week 7: Regular values, regular surfaces as level sets; change of parameters, special surfaces: surfaces of revolution, canal surfaces, ruled surfaces; tangent vectors and tangent plane
Week 8: Tangent vector and tangent plane. Problems class
Week 9: First fundamental form, coefficients of the first fundamental form; arc length of a curve on a surface, metric, coordinate curves and angles
Week 10: Area of subsets of surfaces, calculation in terms of the coefficients of the first fundamental form, examples; smooth maps between surfaces, the Gauss map
Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 3, 5, 7, and 9