I will use a local coordinate formalism here, since this is related to research in general relativity, and my supervisor only tolerates local coordinate formalisms. Plus the research papers I base my research on also use local coordinates.
If $M$ is an $n$ dimensional $C^\infty$ manifold, with local coordinates $(x^\mu)=(x^1,...,x^n)$, and $F:\Sigma\rightarrow M$ is a $C^\infty$ embedding of an $n-1$ dimensional smooth manifold $\Sigma$ into $M$, where $\Sigma$ is described by local coordinates $(\xi^i)=(\xi^1,...,\xi^{n-1})$, then "covariant" tensor fields $A_{\mu_1,...,\mu_k}$ may be naturally pulled back to $\Sigma$ by $$ A_{i_1...i_k}=e^{\mu_1}_{i_1}...e^{\mu_k}_{i_k}A_{\mu_1...\mu_k}, $$ where $e^\mu_i=\partial x^\mu/\partial \xi^i$ is both the matrix of the tangent map and the components of the (pushforward of the) coordinate frame $\partial/\partial\xi^i$ in the system $x^\mu$. Likewise, if $T^{i_1...i_k}$ is a "contravariant" tensor field on $\Sigma$, then it may be pushed forward to $M$ by $$ T^{\mu_1...\mu_k}=e^{\mu_1}_{i_1}...e^{\mu_k}_{i_k}T^{i_1...i_k}. $$
On the other hand, $F$ as an embedding is a diffeomorphism onto its image, thus it should be possible to pull back those contravariant tensor fields defined along the image of $\Sigma$ in $M$ that are "tangent" to $\Sigma$ where tangency can be defined as having vanishing contractions with the one-form $n_\mu$, where the $n_\mu$ is given by $n_\mu\sim\partial_\mu\phi$ where $\phi$ is a scalar field whose level set locally generates $F(\Sigma)$.
One possible construction I see to give this pullback explicitly is to construct a local foliation of $M$ of which (an open subset of) $F(\Sigma)$ is a leaf and set new coordinates $(x^{\mu'})=(\xi^1,...,\xi^{n-1},\phi)$, where the scalar field $\phi$ is fired up in such a way that the leaves of the foliation correspond to different level sets of $\phi$. Then, if $T^{\mu_1...\mu_k}$ is a contravariant tensor field along $F(\Sigma)$, the pullback $T^{i_1...i_k}$ can be defined as transforming to the slice chart $(x^{\mu'})$ and restricting the range of the the indices $\mu'$ to $1,...,n-1$, since the $\phi$-components will be zero anyways.
With this said, I can present my problem. If a Lorentzian metric $g$ is given on $M$, and the induced metric on $\Sigma$, $\gamma=F^*g$, $\gamma_{ij}=e^\mu_ie^\nu_jg_{\mu\nu}$ is nondegenerate, then the induced connection on $\Sigma$ is usually given as $$ D_iT^{i_1...i_k}_{j_1...j_l}=e^\mu_i e_{\mu_1}^{i_1}...e^{\nu_1}_{j_1}...\nabla_\mu T^{\mu_1...\mu_k}_{\nu_1...\nu_l} ,$$ where greek indices are raised/lowered by $g$ and latin indices are raised/lowered by $\gamma$, and $T$ is tangent to the hypersurface. It can be easily seen that this connection on $\Sigma$ is precisely the Levi-Civita connection of $\gamma$.
If $\gamma$ is degenerate, then it is usually said that there is no unique induced connection on $\Sigma$, however the pullback of the covariant derivative on $M$ can be defined the same way as above, except all the latin indices will be covariant incides, because $\gamma^{ij}$ doesn't exist to raise indices.
So if we are given a tensor on $\Sigma$, we can express it in terms of greek indices (covariant latin indices can be made covariant greek indices the same way contravariant greeks can be made latin I detailed in the early part of my post), then we can calculate its covariant derivative in $M$, then project it down into $\Sigma$ with covariant indices, essentially the process is as such: $$ T^{i_1...i_k}_{j_1...j_l}\rightarrow T^{\mu_1...\mu_k}_{\nu_1...\nu_l}\rightarrow e^\mu_i\nabla_\mu T^{\mu_1...\mu_k}_{\nu_1...\nu_l}\rightarrow D_iT_{i_1...i_kj_1...j_l}.$$
Admittedly, I don't know if it is possible to give any connection coefficients on $\Sigma$, which describes this process.
On the other hand, consider a contravariant vector field $A^\mu$ that is tangent to $\Sigma$. Then $$ D_iA_j=e^\mu_i e_{\nu j}\nabla_\mu A^\nu=e^\mu_i(\nabla_\mu(e_{\nu j}A^\nu)-A^\nu\nabla_\mu e_{\nu j})= \\ =\partial_i A_j-A^\nu e^\mu_i\nabla_\mu e_{\nu j}=\partial_i A_j-A^ke^\nu_k e^\mu_i\nabla_\mu e_{\nu j}=\partial_i A_j-A^k\Gamma_{k,ij}, $$ and I haven't checked but pretty sure the $\Gamma_{k,ij}$s are given by the usual formula of the Christoffel symbols of the first kind for $\gamma_{ij}$.
I guess, my question is, what can we say about induced covariant derivatives on null hypersurfaces? The general relativity literature seems to be adamant that it is ill-defined, but it seems to me it can be defined in certain cases. What further causes confusion in me is that it appears to me that one CAN in fact raise indices on $\Sigma$. Take $T_{i_1...i_k}$ on $\Sigma$, push it forward to $T_{\mu_1...\mu_k}$ by the procedure given in my post, then raise indices with $g^{\mu\nu}$, then pull back $T^{\mu_1...\mu_k}$ to $T^{i_1...i_k}$ once again with the process using the slice chart.
Thus, it seems to me that there actually is a way to define induced covariant derivatives on $\Sigma$, but obtaining it is not "nice". Is my conclusion correct? I am very confused, any clarification is much appreciated.
