Projective Maps from the Perspective of Elliptic Differential Operators

TL;DR

This paper introduces an analytical framework using elliptic differential operators to study the geometry of projective maps and diffeomorphisms on Riemannian manifolds, linking analysis and geometry.

Contribution

It constructs specific elliptic operators whose kernels characterize projective diffeomorphisms and their groups, providing a novel operator-theoretic approach to projective geometry.

Findings

Characterization of projective diffeomorphisms via elliptic operators

Establishment of a correspondence between analytical and geometric properties

Extension of classical differential geometry results

Abstract

This paper develops an analytical approach to the study of the geometry of projective maps using the theory of elliptic differential operators. We construct two elliptic operators of second and fourth order, whose kernels characterize projective diffeomorphisms between Riemannian manifolds and one-parameter groups of projective diffeomorphisms (transformations) of a Riemannian manifold onto itself, respectively. This approach establishes a natural correspondence between analytical and geometric properties, enabling the study of projective diffeomorphisms via operator-theoretic methods. The proposed framework provides a new understanding of projective structures on Riemannian manifolds and extends classical results in differential geometry.