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

TL;DR

This paper investigates isometric immersions of pseudo-Riemannian manifolds into para-complex projective space, deriving frame equations, and characterizing special surface immersions through harmonicity of Gauss maps.

Contribution

It introduces a framework for studying immersions into para-complex projective space via lifts and frame equations, with specific results for surfaces and special cases.

Findings

Derived frame equations and compatibility conditions for immersions.

Characterized Lagrangian and minimal surface immersions.

Connected surface properties to primitive harmonicity of Gauss maps.

Abstract

In this paper we study isometric immersions $f:M^n \to {\mathbb {C}^{\prime}}\!P^n$ of an $n$-dimensional pseudo-Riemannian manifold $M^n$ into the $n$-dimensional para-complex projective space ${\mathbb {C}^{\prime}}\!P^n$. We study the immersion $f$ by means of a lift $\mathfrak f$ of $f$ into a quadric hypersurface in ${S^{2n+1}_{n+1}}$. 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.