Prof. Alexander F. Ritter, Associate Professor, University of Oxford.

♦ Lecture notes, 1 page per side (version 66, Sep 2018)

♦ Lecture notes, 2 pages per side (version 66, Sep 2018)

♦ Handout: Dictionary of some terminology from topology and analysis

Question Sheets from 2017/2018:

♦ Question sheet 0

♦ Question sheet 1

♦ Question sheet 2

♦ Question sheet 3

♦ Question sheet 4

Question Sheets from 2015/2016:

♦ All question sheets 1-6

♦ Question sheet 1

♦ Question sheet 2

♦ Question sheet 3

♦ Question sheet 4

♦ Question sheet 5

♦ Question sheet 6

Examples of topological surfaces, smooth surfaces in R^3, abstract smooth surfaces, Riemann surfaces. Topological surfaces: cellular decompositions, the Euler characteristic, orientations. Classification theorem for compact surfaces (without proof).

Smooth surfaces: local parametrizations, the inverse function theorem, the implicit function theorem, local graphs.

Riemann surfaces: holomorphic maps between Riemann surfaces, the Riemann-Hurwitz formula, meromorphic functions, elliptic functions, the Weierstrass P-function. The uniformization theorem (classification of Riemann surfaces).

Smooth surfaces in Euclidean three-space, the first fundamental form, tangential derivatives, the Gauss map, the second fundamental form, the shape operator. Gaussian curvature. Riemannian metrics on abstract smooth surfaces. Riemann curvature. Gauss' Theorema Egregium.

Geodesics. Isometries. The Gauss-Bonnet Theorem (local version and global version). Critical points of real-valued functions, Morse functions, vector fields, Poincaré-Hopf theorem, hairy ball theorem.

The hyperbolic plane, its isometries and geodesics.

___________________________

Prof. Alexander F. Ritter. Contact me.

Associate Professor in Geometry, Mathematical Institute, Oxford.

The Roger Penrose Fellow and Tutor, Wadham College, Oxford.