I learn some Einstein's notation and apply it to my knowledge on Differential Geometry. For instance, we can use his notation on the geodesic curve second order differential equality $\ddot{c}^k + \Gamma^k_{ij} \dot{c}^i \dot{c}^j=0$ such that it becomes format $\ddot{c} + \Gamma(c, \dot{c}) = 0$ on manifold $\mathcal{X}$ by use of below identities:
\begin{align*} (\dot{c} \dot{c}^\intercal)_{ij} &= \dot{c}^i \, \dot{c}^j \\ (\Gamma^k)_{ij} &= \Gamma^k_{ij} \\ \Gamma^k_{ij} x^i x^j &= \sum\limits_{i, j} (\Gamma^k \circ \dot{c} \dot{c}^\intercal)_{ij} \\ & = 1^\intercal (\Gamma^k \circ \dot{c} \dot{c}^\intercal) 1 \\ & = \mbox{tr}(\Gamma^k \dot{c} \dot{c}^\intercal) \end{align*}
The $k$-rowise element $\Gamma^k(c, \dot{c})$ of here-called Christoffel vector $\Gamma(c, \dot{c})$ is given by trace $\mbox{tr}(\Gamma^k \, \dot{c} \dot{c}^\intercal)$ of product $\Gamma^k \, \dot{c} \dot{c}^\intercal$.
My question begins the geodesic equation constrained to some tangent subspace $U \subseteq T_{c(t)} \mathcal{X}$ such that velocity $\dot{c} \in T_{c(t)} \mathcal{X}$ and spanned by linear product $B_\ell p^\ell$, for anew velocities $p \in U$. By naive substitution, we get the equality: $B^k_\ell \dot{p}^\ell + \dot{B}^k_\ell p^\ell + \Gamma^k(c, B \, p)=0$. Since tensor $B$ may not be square, usually with more rows than columns, we multiply by metric tensor $g_{m k}$ and also dual-space tensor $(B^\intercal)_m^n$, which brings to design:
$$(B^\intercal)_m^n \, g_{m k} B^k_\ell \dot{p}^\ell + (B^\intercal)_m^n \, g_{m k} \dot{B}^k_\ell p^\ell + (B^\intercal)_m^n \, g_{m k} \Gamma^k(c, B \, p)=0$$
Let us call the resulting constrained metric $(B^\intercal)_m^n \, g_{m k} B^k_\ell$ as metric $h_{n\ell}$ and constrained Christoffel vector element $I^n(c, p)$ as map $h^{n\ell} \left((B^\intercal)_m^n \, g_{m k} \dot{B}^k_\ell p^\ell + (B^\intercal)_m^n \, g_{m k} \Gamma^k(c, B \, p)\right)$. I am sure, there is some more compact form, which I am not investigating in this text.
Are my algebraic correct according to Differential Geometry formalism? If so, is there some canonical notation to $(B^\intercal)_m^n$.