# 45. The Fundamental Theorem of Riemannian Geometry

We begin this lecture by proving the (somewhat grandiosely named) *Fundamental Theorem of Riemannian Geometry*, which states that if $(M,m)$ is a Riemannian manifold then there exists a *unique *connection $ \nabla$ on $M$ which is simultaneously Riemannian with respect to $m$ and torsion-free. This connection is usually called the *Levi-Civita connection*** **of $m$.

We then discuss the celebrated *Berger Classification Theorem*** **which enumerates the seven possible options for the holonomy group of a Riemannian metric.

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