Let $(M,g)$ be a smooth manifold with connection $\nabla$. Then the definition for Christoffel symbols I am familiar with is
$\nabla_{\partial_i} \partial_j = \Gamma^h_{ij} \partial_h$
where $\{\partial_1,...,\partial_n\}$ is some smooth chart. Now I have seen an expression in a paper of the form
$\Gamma_f(h,k)$
where $h,k$ are tangent vectors and $f$ a point on the manifold (as far as I understood). What is the definition for this expression here? Also they call this the Christoffel forms, not symbols. Is there a difference? Are the definitions analogous?
Link to paper: https://arxiv.org/pdf/math/0312384.pdf