will speak on
“Metric connections I: the Levi-Civita connection”
This is the first in a series of lectures on metric connections. On a smooth vector bundle, a connection provides a way of computing covariant derivatives of smooth sections of the bundle or of metrics on the bundle. If one fixes a metric and the covariant derivatives of the metric vanish with respect to a given connection, then this connection is said to be (compatible with the) metric. In this talk, I will begin by introducing connections on smooth vector bundles and present some of their properties. I will then specialize to affine connections on the tangent bundle, in particular giving a geometric interpretation of the curvature and the torsion of an affine connection. I will finally discuss the Levi-Civita connection on a Riemannian manifold, which is the unique torsion-free connection that is compatible with the Riemannian metric. Although I will briefly recall the notion of vector bundle, I will assume the knowledge of differential forms.