|Riemannian Geometry on Four-Manifolds|
|Time：||Tue/Thu 18:30-20:00, 2013-09-26 ~ 2014-01-02 (except for public holidays)|
|Instructor：||Abul Masood-ul-Alam [Tsinghua University]|
|Place：||Conference Room 3, Floor 2, Jin Chun Yuan West Building|
The dimension four differs from higher dimensions in many respects. It is the first dimension Weyl curvature becomes non-trivial, quarternions come to play, rotation group SO(4) has non-trivial homomorphic images while SO(n), n>4 are simple or almost simple, Weyl tensor is decomposed into self-dual and anti-self-dual parts, Yang-Mills action is conformally invariant, and finally it is the dimension of our spacetime (we forget the metric signature for a moment). This course is mainly an effort to study Simon Salamon's "Topics in Four-Dimensional Riemannian Geometry (Lecture Notes in Mathematics vol. 1022, 1983)." Aim is to make the course suitable for both graduate and senior level undergraduate students in a self-contained way. We shall start with spheres, a review of group actions on manifolds, principal fibre bundles and some matrix groups.
Multivariable Calculus, Real Analysis, Linear Algebra and some basic knowledge of Riemannian Geometry (tensor, covariant derivative and curvature) or concurrent enrollment in a Riemannian Geometry or Relativity or Geometric Analysis course.