Differential Geometry III

Michaelmas 2019 - Epiphany 2020

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.

• M. Do Carmo, Differential Geometry of Curves and Surfaces, in the library
• S. Gudmundsson, An Introduction to Gaussian Geometry, Lecture Notes, Lund University (2017).
• L.M.Woodward and J. Bolton, A First Course in Differential Geometry, in the library

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. [DC: 1-2,1-3. G: p.7. B&W: Section 1.1]
• Week 2: Length of a curve, arc length, existence of arc length parametrization, examples. Tangent and normal vectors, curvature of a plane curve. [DC: 1-3. G: p.8-12. B&W: 1.2, 1.3.]
• 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. [DC: 1-7B. B&W: 1.4, 1.5.]
• Week 4: Space curves, principal normal vector, binormal vector, curvature, torsion, Serret-Frenet equations, curvature and torsion for unit speed space curves. Curvature and torsion for non unit speed space curves; geometric meaning of curvature and torsion. Fundamental theorem of local theory of space curves. [DC: 1-5. G: p.17-23. B&W: 1.5]
• Week 5: Local canonical form of space curves [DC: 1-6]. 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. [DC: 2-2, 2-3. G: p.25-36. B&W: 2.1, 2.2]
• Week 7: Special surfaces: surfaces of revolution, canal surfaces, ruled surfaces; tangent vectors and tangent plane. [DC: 2.4. G: p.32-35.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. [DC: 2-5. G: p.36-38. 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. [DC: 2-5. G: p.38. B&W: 3.4, 3.5]
• Week 10: Smooth maps between surfaces; the Gauss map. [DC: 3-2. G: p.41. B&W: 4.1, 4.2]
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• Week 11: Revision of Michaelmas term; isometries and conformal maps. [DC: 4-2. G: p.37-38. B&W: 4.1, 4.2, 4.3]
• Week 12: The Weingarten map, the second fundamental form; Gauss curvature and mean curvature. [DC: 3-2. G: p41-45. B&W: 5.1, 5.2]
• Week 13: More about curvature. [DC: 3-2, 3-3. G: p46-49. B&W: 5.2]
• Week 14: Global theorems about curvature, Theorema Egregium; Christoffel symbols. [DC: 4-3. G:p53-56. B&W: 6.1, 6.2]
• Week 15: Theorema Egregium: examples. [B&W: 6.1, 6.2.]
• Week 16: Curves on surfaces; normal and geodesic curvatures. Meusnier's theorem, asymptotic curves; lines of curvature. [DC: 3-2, B&W: 5.3, 5.5]
• Week 17: More on lines of curvature. Geodesics. [DC: 4-4. G: p59-63. B&W: 7.2, 7.3, 7.4]
• Week 18: More on geodesics. Clairaut's relations. [DC: 4-4. G: p64-70. B&W: 7.3, 7.5]
• Week 19: Local and global Gauss--Bonnet theorem. [DC: 4-5. G: p79-85, B&W: 8]
• Week 20: Hilbert's Theorem. [DC: 5-11.]

If you have any questions you are very welcome to ask (during the lectures, after a lecture, during office hours, in any other convenient time or via e-mail)!!!

Homeworks:

• Week 1-2: --- - Exercises ---- ----
• Week 3-4: --- - Exercises ---- ----
• Week 5-6: --- - Exercises ---- ----
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• Week 9-10: --- ----
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• Week 11-12: - Exercises ---- ----
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