In my differential geometry course, the $(1,3)$-riemann curvature tensor $R$ is defined by $$R(X,Y)Z:=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ and the $(0,4)$-riemann curvature tensor Rm by $$Rm(X,Y,\color{red}Z,\color{blue}W):=g(R(X,Y)\color{blue}W,\color{red}Z)$$ In coordinates $Rm_{ij\ell k}=g_{\ell m}R_{ijk}^m$. The symmetries are the following: $$Rm_{ijk\ell}=-Rm_{jik\ell}=-Rm_{ij\ell k}=Rm_{k\ell ij}$$ Then the Ricci tensor of a riemannian manifold $(M,g)$ is the symmetric $(0,2)$-tensor $\text{Rc}\in\Gamma(T^*M\otimes T^*M)$ ${\it defined}$ by: $$\text{Rc}(X,Y)=\sum_{i=1}^nRm(X,e_i,Y,e_i)$$ where $\{e_i\}$ is an orthonormal basis of $T_pM$. It's coordinate representation is $Rc_{ik}=g^{j\ell}Rm_{ijk\ell}$. Then the notes states (without proving it) that $Rc(X,Y)$ is the trace of the map $Z\longrightarrow R(Z,X)Y$.
This implies that (using the symmetries): $$\boxed{Rc_{ik}=g^{j\ell}Rm_{ijk\ell}=-g^{j\ell}Rm_{ij\ell k}=-R_{ijk}^j}$$
Wikipedia defines the tensor $R$ in the same way, while it defines the Ricci tensor $Rc(X,Y)$ directly as the trace of the map $Z\longrightarrow R(Z,X)Y$. Then it states that: $$\boxed{Rc_{ab}=R^c_{acb}}$$
The question is: what's the source of the sign discrepancy between the two boxed formulas? Isn't it true the trace characterization of the Ricci tensor using the definitions in my notes? Or have I made some calculation mistakes in the first boxed formula?
Maybe it's a silly question, but I would be glad of any help.
The notation $R^c_{acb}$ is ambiguous: does it mean $g^{cd}R_{dacb}$, or $g^{cd}R_{adcb}$, or $g^{cd}R_{acdb}$, or $g^{cd}R_{acbd}$? In other words, which index position corresponds to the raised index $c$? The various choices differ by a sign. For that reason, it's important to maintain both the horizontal and vertical positions of indices. The Wikipedia article on Ricci curvature does exactly that: the equation you quoted actually appears there as $$ \text{Ric}_{ab} = R^c{}_{bca} = R^c{}_{acb}, $$ showing that the raised index belongs in the first position. On the other hand, your computation, if you keep track of horizontal index positions, yields $$ Rc_{ik}=g^{j\ell}Rm_{ijk\ell}=-g^{j\ell}Rm_{ij\ell k}=-R_{ij}{}^j{}_{k} $$ Using the symmetries of the curvature tensor, this can be rewritten as $$ -R_{ij}{}^j{}_{k} = -R^j{}_{kij} = R^j{}_{kji}, $$ which matches the Wikipedia formula.