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

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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