On the number of nontrivial projective transformations of closed manifolds

TL;DR

This paper investigates the structure of projective transformation groups on closed Riemannian manifolds, establishing bounds on their size unless the manifold has constant positive curvature or all transformations are affine.

Contribution

It proves that the quotient of the projective transformation group by the isometry group has at most two elements under general conditions, clarifying the structure of these transformation groups.

Findings

The quotient group has at most two elements unless the metric has constant positive curvature.

If the metric has constant positive curvature, the group can be larger.

All projective transformations are affine transformations unless the manifold has constant positive curvature.

Abstract

We show that for a closed Riemannian manifold the quotient of the group of projective transformations by the group of isometries contains at most two elements unless the metric has constant positive sectional curvature or every projective transformation is an affine transformation.