Differential Geometry III

Michaelmas 2016/Epiphany 2017

Textbooks:

The lectures are based on the following books. Although we will not follow any of these strictly, the material can be found in them.

• J. Bolton and L.M. Woodward, Differential Geometry Lecture Notes. Copies are available from the Maths office, the electronic version can be found on duo
• 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; isometries and conformal maps, the Weingarten map, the second fundamental form, Gauss curvature and mean curvature, minimal surfaces, Theorema Egregium, Christoffel symbols, normal and geodesic curvatures, Meusnier's theorem, asymptotic curves, lines of curvature, geodesics, Clairaut's relations, global and local Gauss--Bonnet theorems.

Schedule (preliminary):

• 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, arc length, existence of arc length parametrization, examples. B&W Sections 1.1, 1.2
• Week 2: Tangent and normal vectors, curvature of a plane curve; vertices and inflection points of plane curves, four vertex theorem, fundamental theorem of local theory of plane curves. B&W 1.3
• Week 3: Radius and center of curvature, evolute and involute of a plane curve; space curves, principal normal vector, binormal vector, curvature, torsion, Serret-Frenet equations, curvature and torsion for unit speed space curves. B&W 1.4, 1.5
• Week 4: 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. B&W 1.5
• Week 5: Open sets in R^n, smooth maps R^n -> R^m, Jacobi matrix, differential; implicit function theorem.
• Week 6: Parametrized surfaces in Euclidean space, regular surfaces as level sets, examples; change of parameters. B&W 2.1, 2.2
• Week 7: Special surfaces: surfaces of revolution, canal surfaces, ruled surfaces; tangent vectors and tangent plane. B&W 2.2, 3.1
• Week 8: First fundamental form, coefficients of the first fundamental form; arc length of a curve on a surface, metric, coordinate curves and angles. B&W 3.2, 3.3
• Week 9: Area of subsets of surfaces, calculation in terms of the coefficients of the first fundamental form, examples; families of curves on surfaces. B&W 3.4, 3.5
• Week 10: Smooth maps between surfaces; the Gauss map. B&W 4.1, 4.2
• Week 11: Revision of Michaelmas term; isometries and conformal maps. B&W 4.1, 4.2, 4.3
• Week 12: The Weingarten map, the second fundamental form; Gauss curvature and mean curvature. B&W 5.1, 5.2
• Week 13: More about curvature.
• Week 14: Global theorems about curvature, Theorema Egregium; Christoffel symbols.
• Week 15: Proof of Theorema Egregium, curves on surfaces; normal and geodesic curvatures.
• Week 16: Meusnier's theorem, asymptotic curves; lines of curvature.
• Week 17: Geodesics.
• Week 18: Clairaut's relations; global Gauss--Bonnet theorem.
• Week 19: Local Gauss--Bonnet theorem; minimal surfaces.

Handouts:

• Regular curves in R^n
• Plane curves
• Space curves
• A bit of Analysis
• Surfaces
• Tangent plane, first fundamental form and area
• Maps between surfaces
• Geometry of the Gauss map
• Theorema Egregium
• Curves on surfaces
• Geodesics
• Gauss -- Bonnet Theorem
• Everything in one file

Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 3, 5, 7, 9, 13, 15, 17 and 19

• Homework 1 (due Thursday, October 27)               
• Homework 2 (due Thursday, October 27)               
• Homework 3 (due Thursday, November 10)           
• Homework 4 (due Thursday, November 10)           
• Homework 5 (due Thursday, November 24)           
• Homework 6 (due Thursday, November 24)           
• Homework 7 (due Thursday, December 8)             
• Homework 8 (due Thursday, December 8)             
• Homework 9                                                          
• Homework 10                                                        
• Homework 11 (due Thursday, February 2)             
• Homework 12 (due Thursday, February 2)                (Corrected!)
• Homework 13 (due Friday, February 17)                
• Homework 14 (due Friday, February 17)                
• Homework 15 (due Thursday, March 2)                  
• Homework 16 (due Thursday, March 2)                  
• Homework 17 (due Thursday, March 16)                
• Homework 18 (due Thursday, March 16)                
• Homework 19