Writing (mixed) tensors

Is the result you are looking for?

\documentclass{article}
\usepackage{amsmath}
\usepackage{tensor}

\begin{document}
$g^{\alpha\beta}g^{\gamma\delta}R_{\alpha\mu\gamma\nu}=R^{\beta\;\delta}_{\;\mu\;\nu}$

$g^{\alpha\beta}g^{\gamma\delta}R_{\alpha\mu\gamma\nu}
    =\tensor{R}{^\beta_\mu^\delta_\nu}$
\end{document}