Torsion of a connection as an obstruction to integrability

Solution 1:

The torsion of an affine connection is pointwise, in the sense that if two affine connections, $\nabla$ and $\nabla',$ agree at a point $p$, then so do their respective torsions. This property distinguishes torsion from curvature, for example, as the curvature of a connection at a point depends on the connection on a neighborhood. Therefore, it makes sense that torsion is the obstruction for some pointwise property, as opposed to integrability of some sort.

Indeed, we have the following simple observation. The torsion of an affine connection $\nabla$ vanishes at a point $p$ if and only if there exist local coordinates around $p$ whose induced affine connection coincides with $\nabla$ at $p$. The "if" part is trivial, as any connection induced by coordinates is torsion-free. Conversely, suppose the torsion of $\nabla$ vanishes at $p$. Let $\exp_p$ denote the exponential map with respect to $\nabla$, and use it as a local parametrization around $p$. Let $\Gamma_{ij}^k$ denote the Christoffel symbols of $\nabla$ with respect to this parametrization. By construction, any curve $\gamma$ whose coordinate expression with respect to $\exp_p$ is a straight line through the origin is a geodesic passing through $p$. As the geodesic equation in coordinates reads $$\ddot{\gamma}^k+\dot{\gamma}^i\dot{\gamma}^j\Gamma_{ij}^k=0,$$this means that for any tangent vector $v=v^i\partial/\partial x^i\in T_pM$ and $k=1,\ldots,n$ we have $$v^iv^j\Gamma_{ij}^k=0.$$In other words, the tensor $\Gamma$ is anti-symmetric at $p$. But, by assumption, $\nabla$ is torsion-free, which means that $\Gamma$ is symmetric, and the proof is complete.

The above observation can also be thought of as the following, more geometric, interpretation. Let $M$ be a smooth manifold, let $p\in M$, and let $\nabla$ be an affine connection on $M$. Let $L\subset T_pM$ be a linear subspace, and let $U\subset L$ be a neighborhood of $0$, such that $\exp_p|_U$ is an immersion, where $\exp_p$ is taken with respect to $\nabla$. Write $\Lambda:=\exp_p(U)$. If the torsion of $\nabla$ vanishes at $p$, then $\Lambda$ is "totally geodesic" at $p$, in the sense that we have $$\nabla_XY(p)\in L=T_p\Lambda$$ for any two vector fields $X$ and $Y$ which restrict to vector fields on $\Lambda$. (This follows from the above observation by taking exponential coordinates around $p$).