Curvature in Mathematics and Physics

Overview:

The idea of curvature plays an important role in many areas of mathematics and physics. In this course we shall look at some of its definitions and manifestations in geometry, topology, and mathematical physics. Notice: this is a mathematics course and no familiarity with physics will be needed. Similarly, no knowledge of di erential geometry will be assumed. I shall provide all the necessary background.

Topics to be covered:

The following topics will be covered.

• Gauss's theorema egregium and what it means for mankind. Riemann cur- vature tensor and its shadows (Ricci and scalar curvature), sectional curvature. An analysis of Riemann's 1854 paper. Connections on the tangent bundle, Levi- Civita's theorem. Connections on principal G-bundles and on vector bundles. Chern-Weil and Chern-Simons theories. Gauss-Bonnet theorem.
• Bi-invariant metrics on Lie groups and Maurer-Cartan equations.
• Einstein-Hilbert action and Einstein's eld equations in general relativity, special solutions, introduction to black holes.
• Gauge theory and Yang-Mills equations, Higgs mechanism, applications to elementary particle physics.

Marking Policy:  %60 take home assignments (2 sets of assignments), %40 presentation.

Textbook:

Curvature in Mathematics and Physics, by Shlomo Sternberg (Dover Publication, 2012).