Locally conformally flat metrics on surfaces of general type

TL;DR

This paper proves that certain product manifolds, specifically surfaces of genus at least 2 and 1, cannot support locally conformally flat metrics derived from discrete and faithful representations.

Contribution

It establishes a nonexistence theorem for locally conformally flat metrics on product surfaces of general type, expanding understanding of geometric structures on these manifolds.

Findings

No locally conformally flat metrics on _g _h for g   2 and h  1

Locally conformally flat structures are incompatible with the product structure for these surfaces

The result applies to representations that are discrete and faithful.

Abstract

We prove a nonexistence theorem for product type manifolds. In particular we show that the 4-manifold $\Sigma_g\times\Sigma_h$ does not admit any locally conformally flat metric arising from discrete and faithful representations for $g\geq 2$ and $h\geq 1$