The torsion tensor of a connection $\nabla$ on $M$ is the alternating tensor of type $(1,2)$ defined by $$T^{\nabla}(X,Y) := \nabla_X(Y) - \nabla_Y(X) - [X,Y].$$

A connection is said to be torsion-free if $T^{\nabla} = 0$. We prove that a connection is uniquely determined by its geodesics and its torsion tensor. We then strengthen the Ambrose-Palais-Singer Spray Theorem from last lecture and prove that if $\mathbb{S}$ is any spray on $M$ and $T$ is any alternating tensor of type $(1,2)$ then there exists a unique connection $\nabla$ on $M$ with geodesic spray $\mathbb{S}$ and torsion tensor equal to $T$.

Finally, we briefly survey the question as to which Lie groups can occur as holonomy groups of torsion-free connections, and introduce the notion of a Berger group.

Enjoy your spring break! 😀