Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor

TL;DR

This paper investigates local obstructions on 2D projective surfaces to admit connections with skew-symmetric Ricci tensors, using polynomial resultants to identify conditions for solutions to the projective Einstein-Weyl equation.

Contribution

It provides the first explicit local obstructions for the existence of skew-symmetric Ricci tensor connections in 2D projective structures, via polynomial resultants.

Findings

Identifies polynomial resultants as obstructions to pEW solutions.

Provides criteria to determine when a projective surface admits a skew-symmetric Ricci tensor connection.

Establishes local conditions for the existence of special projective connections.

Abstract

The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions for projective surfaces to admit such a connection in its projective class. The obstructions are the resultants of polynomial equations that have to be satisfied for there to admit any pEW solution.