Affine surfaces which are K\"ahler, para-K\"ahler, or nilpotent K\"ahler

TL;DR

This paper investigates the geometric structures of affine surfaces, focusing on conditions for the existence of special parallel tensors related to Kähler and para-Kähler geometries, with implications for Bach flat neutral signature extensions.

Contribution

It characterizes when affine surfaces admit parallel trace-free (1,1)-tensors based on Ricci tensor recurrence, linking affine geometry to Kähler and para-Kähler structures.

Findings

Existence of parallel trace-free tensors is characterized by Ricci tensor recurrence.

Provides conditions for affine surfaces to be Kähler, para-Kähler, or nilpotent Kähler.

Connects affine surface geometry with neutral signature Riemannian extensions.

Abstract

Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.