From my class-notes, I have that the coefficient-functions of the riemann-curvature-tensor is $$R^i_{jk\ell} = \partial_j\Gamma^i_{k\ell}-\partial_k\Gamma^{i}_{j\ell}+\Gamma^m_{k\ell}\Gamma^{i}_{jm}-\Gamma^{m}_{j\ell}\Gamma^{i}_{km}.$$ Now, we have $$\operatorname{Ric}_{jk} = R^{i}_{jik} \quad (\text{with ESC}) = \partial_i \Gamma^{i}_{jk}-\partial_j\Gamma^{i}_{ki}+\Gamma^{i}_{ip}\Gamma^{p}_{jk}-\Gamma^{i}_{jp}\Gamma^{p}_{ik}$$ according to wiki. Now, setting $p = m$ in the second expression, and then using the formula from my classnotes, I get that $$R^i_{ijk} = \operatorname{Ric}_{jk}$$ which should not be the case, we should have $$R^{i}_{jik} = \operatorname{Ric}_{jk}.$$ What am I missing?
I am also having a hard time to connect that for $p \in M$, $(M,g)$ SR manifold and $v,w \in T_pM$ we have $$R(v,w) = \operatorname{tr} (\underbrace{u \mapsto R_{u,v}w}_{\in (T_pM)^{1}_{1}}) = \operatorname{tr}_g\underbrace{((u,z) \mapsto R(u,v,w,z)}_{\in (T_pM)^{0}_{2}})$$
since we have an isomorphism $$\mathcal{T}^{1}_{s} \cong L^{s}_{C^{\infty}}(M,\mathfrak{X}(M))$$ where for $A \in L^{s}_{C^{\infty}}(M,\mathfrak{X}(M))$ we have that $$A:\underbrace{\mathfrak{X}(M) \times \ldots \times \mathfrak{X}(M)}_{s} \to \mathfrak{X}(M)$$ and $A$ is $C^{\infty}(M)$-multilinear, where we also get, for $t \in \mathcal{T}^{1}_{s}(M)$ and coefficient functions $t^{i}_{j_1,\ldots,j_s}$ that $$t^{i_1}_{j_1,\ldots,j_s} = dx^{i}(A|_{U}(\partial_{j_1},\ldots,\partial_{j_s}))$$
In our case, I think $A|_{U}$ here would correspond to $A|_{U}(u) = R_{u,v}w$ with $u = \partial_i$ so that $$t^{i}_{i} = dx^iA|_{U}(\partial_i) = dx^i(R_{\partial_i,\partial_j}\partial_k) = \operatorname{Ric}(\partial_j,\partial_k)?$$ I believe this isomorphism can be shown through what in our course is called ”tensor reconstruction lemma” of which I believe ”tensor characterization lemma” is a special case. This last expression seems also not completely right. I would like someone to point out my misconceptions. Thanks in advance.
Edit: Actually, according to some other notes by David Lindemann (source: https://www.math.uni-hamburg.de/home/lindemann/material/dg_lindemann_ss2020_WIP29.pdf) we have $$\operatorname{Ric} = \sum_{i,j = 1}^{n} \operatorname{Ric}_{ij}dx^i \otimes dx^j = \sum_{i,j = 1}^{n} \Big(\sum_{k = 1}^{n} R^k_{kij}\Big) dx^i \otimes dx^j$$ which seems more compatible with what I wrote.
For the first question, there are differnt index conventions for the riemann curvature tensor, so i think that's the problem. For the second question, i dont understand what excatly you are asking, since what you wrote seems just fine to me.