Geodesic christoffelian terms name Is there some canonical notation for the Christoffelian term $\Gamma(\cdot, \cdot) \in \mbox{Hom}(\mathcal{S}, T_p \mathcal{S})$ of a geodesic equation as denoted below rather than derived adjective term as I did?
$\ddot{x} + \Gamma(x, \dot{x}) = 0$
 A: Writing the geodesic equation like this is something only doable locally.
The situation is explained by the following general fact: the space of affine connections $\nabla$ on a manifold $M$ is an affine space, whose associated translation space is the space of type $(1,2)$-tensor fields $A$ (i.e., $A\colon \mathfrak{X}(M)\times\mathfrak{X}(M)\to \mathfrak{X}(M)$ is $C^\infty(M)$-bilinear). In layman terms, this means that if $\nabla$ is an affine connection, so is $\nabla+A$, and if $\nabla,\nabla'$ are two affine connections, then $A \doteq \nabla'-\nabla$ is $C^\infty(M)$-bilinear.
This relates to your situation like this: if $(M,\nabla)$ is an affine manifold and $(U,\varphi)$ is a chart on $M$, we can push the connection $\nabla$ to a connection $\varphi_\ast \nabla$ on the open subset $\varphi[U]$ of $\mathbb{R}^n$, where $n=\dim M$. This will differ from the standard flat connection ${\rm D}$ (given by directional derivatives) by a tensor. This tensor is $\Gamma$ and I have seen people call it "the Christoffel tensor of $\nabla$ relative to $(U,\varphi)$". More precisely, we have $\varphi_\ast \nabla = {\rm D} + \Gamma$, so $$(\nabla_{\dot{\gamma}(t)}\dot{\gamma})(t) = 0 \iff [{\rm D}_{(\varphi\circ\gamma)^{\boldsymbol \cdot}(t)}(\varphi\circ \gamma)^{\boldsymbol \cdot}](t) + \Gamma_{\varphi(\gamma(t))}((\varphi\circ\gamma)^{\boldsymbol\cdot}(t),(\varphi\circ\gamma)^{\boldsymbol\cdot}(t))=0. $$Here, we have that $$[{\rm D}_{(\varphi\circ\gamma)^{\boldsymbol \cdot}(t)}(\varphi\circ \gamma)^{\boldsymbol \cdot}](t) = (\varphi\circ\gamma)^{\boldsymbol \cdot\boldsymbol\cdot}(t)\quad\mbox{and}\quad \Gamma_{\varphi(p)}(v,w) = \sum_{i,j,k=1}^n \Gamma_{ij}^k(p) v^iv^je_k$$for all $p \in U$, $v,w \in T_{\varphi(p)} \varphi[U] = \mathbb{R}^n$, where $(e_1,\ldots, e_n)$ is the standard basis of $\mathbb{R}^n$.
