In Riemannian Geometry, there are two types of Riemannian curvature: Riemannian curvature of the 1st kind, whose components are usually written in the form $R_{ijkl}$; and Riemannian curvature of the 2nd kind, whose components are usually written in the form $R^l_{ijk}$. I learned that the components of the 1st kind can be expressed in terms of the components of the 2nd kind in the following manner:
$R_{ijkl} = g_{pl}R^p_{ijk}$
where $g_{pl}$ represents the components of the Riemannian metric tensor. By analogy, I wonder whether the 2nd kind can be expressed in terms of the 1st kind in the following manner (although it seems that the following formula is not seen in any textbooks):
$R^l_{ijk} = g^{pl}R_{ijkp}$
where $g^{pl}$ represents the components of the conjugate metric tensor.