|Applications of Topology to Differential Geometry|
|Time：||Mon/Wed 15:10-17:00, 2016-09-12~ 2016-12-07 (No classes on public holidays)|
|Instructor：||Thomas Farrell [Tsinghua University]|
|Place：||Conference Room 3, Floor 2, Jin Chun Yuan West Building|
We will develop a theory of smooth fiber bundles p: E→B with extra geometric structure; i.e. whose concrete fibers — the fiber of p over x in B — are equipped with Riemannian metrics which vary continuously with x; but whose sectional curvatures are constrained to lie in some fixed interval S of real numbers. For example, when S is or these are called negatively curved or positively curved bundles, respectively. As background for this study, we will discuss the failure of smooth rigidity for negatively curved manifolds and the Nielsen realization question for aspherical manifolds.
A course in algebraic topology plus the basics of differential topology as contained ( for example ) in Milnor’s short book “Topology from the differentiable viewpoint”.
N. Steenrod: The Topology of Fiber Bundles
M. Hirsch: Differential Topology