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

2024/05/20 by Josef F. Dorfmeister, Dorfmeister, Josef F., Roland Hildebrand +3 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2405.11771

openalex publication_date 2024/05/20 · openalex created_date 2024/05/22 · openalex updated_date 2026/07/28

Abstract

In this paper we study isometric immersions f:Mⁿ → \mathbb C^′ Pⁿ of an n-dimensional pseudo-Riemannian manifold Mⁿ into the n-dimensional para-complex projective space \mathbb C^′ Pⁿ. We study the immersion f by means of a lift \mathfrak f of f into a quadric hypersurface in S²ⁿ⁺¹_n+1. We find the frame equations and compatibility conditions. We specialize these results to dimension n = 2 and a definite metric on M² 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.

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