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

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.

💡 Research Summary

The paper originates from lecture notes on “Metric projective geometry” delivered at a summer school and a workshop in 2014‑2015. Its purpose is twofold: first, to show how projectively invariant objects can be employed to give short, transparent proofs of several classical results in differential geometry; second, to prove a new theorem concerning the index of the affine transformation group inside the full projective transformation group on a complete Riemannian manifold with non‑constant curvature.

The author begins by recalling the definition of a projective structure on an n‑dimensional manifold M: a smooth family F of unparameterised curves such that through any point and any direction there is exactly one curve of the family, and there exists a torsion‑free affine connection ∇ whose geodesics (after suitable re‑parameterisation) coincide with the curves of F. Two connections ∇ and (\bar∇) are said to be projectively equivalent if they have the same unparameterised geodesics. The classical Levi‑Civita theorem (1896) is restated: (\bar∇_X Y = ∇_X Y + φ(Y)X + φ(X)Y) for a unique 1‑form φ. This shows that the space of connections in a given projective class is an infinite‑dimensional affine space parametrised by φ.

In dimension two the author works out the local description explicitly. A torsion‑free connection has six independent Christoffel symbols; after quotienting by the two‑dimensional freedom coming from φ, a projective structure is encoded by four functions. These four functions appear as the coefficients (K_0,\dots,K_3) of a third‑degree ODE \