Course Outline
Geometry
GEOMETRY
Parts of Analysis II will be found useful for this course.
- Euclidean and spherical geometry; length, lines and groups of isometries; M¨obius maps and stereographic projection. [3]
- Triangulations of the sphere and the torus. *Informal discussion of abstract smooth surfaces, orientability and statement of the classification of compact smooth surfaces.* [2]
- Riemannian metrics on open subsets of the plane. The hyperbolic plane. Poincar´e models and their metrics. The isometry group. Hyperbolic triangles and the Gauss–Bonnet theorem. The hyperboloid model. [4]
- Embedded surfaces in R3. The first fundamental form. Length and area. Examples. [1]
- Length and energy. Geodesics for general Riemannian metrics as stationary points of the energy. First variation of the energy and geodesics as solutions of the corresponding Euler–Lagrange equations. Geodesic polar coordinates (informal proof of existence). Surfaces of revolution. [3]
- The second fundamental form and Gaussian curvature. For metrics of the form du2 + G(u,v)dv2, expression of the curvature as −(√G)uu/√G. Abstract smooth surfaces and isometries, with examples. Euler numbers and statement of Gauss–Bonnet theorem, examples and applications. [3]
Appropriate books
- † P.M.H. Wilson Curved Spaces. CUP, January 2008
- M. Do Carmo Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc.,
- Englewood Cliffs, N.J., 1976 A. Pressley Elementary Differential Geometry. Springer Undergraduate Mathematics Series, SpringerVerlag London Ltd., 2001
- E. Rees Notes on Geometry. Springer, 1983
- M. Reid and B. Szendroi Geometry and Topology. CUP, 2005
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