Q: What is the Riemann curvature tensor of the second kind? 
What is the  Riemann curvature tensor of the second kind?

I have tried to look on-line but I cant find a given expression. Maybe I have come across it but in a different form or under a different name?
How is it different to the Riemann curvature tensor of the first kind which I know is $R_{ijkl} =\frac{\partial \Gamma_{jli}}{\partial x^k}-\frac{\partial \Gamma_{jki}}{\partial x^l} + \Gamma _{ilr} \Gamma_{jk} ^r -\Gamma _{ikr} \Gamma^r _{jl}$

A: The Riemann curvature tensor of second is closely associated to the Riemann curvature tensor since
\begin{equation}
R_{ijkl}=R^{m}_{ijk} g_{ml}
\end{equation}
where $g_{ml}$ is the metric tensor. In local coordinates, the Riemann curvature tensor can be expressed using the Christoffel symbols of the manifoild's Levi_Civita connection. This is:
\begin{equation}
R^{l}_{ijk}  =\frac{\partial \Gamma^{l}_{ik}}{\partial x^j}-\frac{\partial \Gamma^{l}_{jk}}{\partial x^i} + \Gamma^{m}_{ik} \Gamma^{l}_{mj} -\Gamma^{m} _{jk} \Gamma^{l}_{mi}
\end{equation}
Since $R_{1212}=det(L_{ij})$, where L is the second fundamental form, the Gaussian curcature can be written as:
\begin{equation}
K=\frac{R_{1212}}{det(g_{ij})}
\end{equation}
