Differential Geometry Working Seminar

Nicholas Kayban, Department of Pure Mathematics, University of Waterloo

"Riemannian Submersions and the O'Neill Tensors"

In an introductory Riemannian geometry course, one typically encounters the Euler, Gauss, and Codazzi equations, which relate the curvature of a submanifold to the curvature of the ambient manifold via the second fundamental form. The O'Neill equations are analogous equations for the case of a Riemannian submersion.

In this talk we define Riemannian submersions and discuss the Fubini Study metric on $CP^n$ as an example. We also consider a vector bundle $E$ over a Riemannian manifold $(M,g)$ where the $E$ is endowed with a Riemannian metric induced from a fibre metric on $E$, a connection on $E$, and the Riemannian metric $g$ on $M$, such that the projection is a Riemannian submersion. The O'Neill tensors are defined, and we state the fundamental equations. We determine the O'Neill tensors of the Fubini-Study metric and the Riemannian metric on $E$. The O'Neill tensors are then applied to show that the sectional curvature of the Fubini-Study metric is bounded between 1 and 4.

Zoom meeting:

• Meeting ID: 958 7361 8652
• Passcode: 577854