Canonical Connections in Riemannian and Hermitian geometry
After giving an outline of the talk and the background material, we will focus our attention on various naturally occurring geometrical objects – namely, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, the Chern connection associated with a Hermitian holomorphic line bundle over a complex manifold, and the ambient connection on a vector bundle where the fibers are naturally embedded in a fixed Euclidean space. We will perform explicit computations and show that the Chern connection and the ambient connection are equal on the tautological line bundle over \(CP^1\). We will also demonstrate that the Levi-Civita connection and the ambient connection are equal on the tangent bundle of \(S^2\). Finally, we will show that the Chern connection on \(S^2\) (where we regard the tangent bundle of \(S^2\) as a Hermitian holomorphic line bundle) is equal to the Levi-Civita connection on \(S^2\).