Half-dimensional immersions into the para-complex projective space and Ruh-Vilms type theorems

In this paper we study isometric immersions $f:M^n\to {\mathbb{C}}^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}{P}^n$ of an $n$-dimensional pseudo-Riemannian manifold $M^n$ into the $n$-dimensional para-complex projective space ${\mathbb{C}}^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}{P}^n$. We study the immersion $f$ by means of a lift $\mathfrak{f}$ of $f$ into a quadric hypersurface in $S_{n+1}^{2n+1}$. We find the frame equations and compatibility conditions. We specialize these results to dimension $n=2$ and a definite metric on $M^2$ in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.