TITLE: Conceptual understanding of Christoffel symbols
QUESTION [7 upvotes]: The basis vectors on a manifold are defined as partial derivative operators of any function that can be locally mapped around a point to $\mathbb R^n.$ The Christoffel symbols come about when a vector field on a such a manifold is differentiated, and the product rule calls for differentiating not just the components, but also these basis vectors, since they change from point to point.
Is there any sense, therefore, in which the Christoffel symbols can be understood as a second derivative?

REPLY [3 votes]: The Christoffel symbols arise naturally when you want to differentiate a scalar function $f$ twice and want the resulting Hessian to be a $2$-tensor. When you work out the details, you discover that with respect to local coordinates, the Hessian of $f$ is given by
$$ \nabla^2_{ij}f = \partial^2_{ij}f - \Gamma^k_{ij}\partial_kf. $$
In particular, if you set $f(x) = x^k$, you get
$$
-\nabla^2_{ij}x^k = \Gamma^k_{ij}.
$$
Also, notice that the Hessian of $f$ will always be a symmetric $2$-tensor if $\Gamma^k_{ij} = \Gamma^k_{ji}$, which is equivalent to saying that the connection is torsion-free. That's one of the reasons you want the Levi-Civita connection to be torsion-free.