Projectively invariant objects and the index of the group of affine transformations in the group of projective transformations

TL;DR

This paper explores projectively invariant objects in differential geometry, providing new proofs of known results and establishing that on certain Riemannian manifolds, the index of affine transformations within projective transformations is at most two.

Contribution

It introduces new proofs for existing theorems and proves a novel bound on the index of affine transformation groups in the context of complete Riemannian manifolds.

Findings

On a complete Riemannian manifold of nonconstant curvature, the index of the affine group in the projective group is at most two.

Provides new, simplified proofs for classical results in metric projective geometry.

Establishes a bound on the group index that was previously unknown.

Abstract

The paper is grown from the lecture course "Metric projective geometry" which I hold at the summer school "Finsler geometry with applications" at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements, and also prove the following new statement: on a complete Riemannian manifold of nonconstant curvature the index of the group of affine transformations in the group of projective transformations is at most two.