Higher Order Schwarzians for Geodesic Flows, Moment Sequences, and the Radius of Adapted Complexifications

TL;DR

This paper introduces higher order Schwarzians related to geodesic flows on 2D manifolds, characterizes the existence of adapted complex structures via Hankel matrices and differential inequalities, and explores their properties and applications.

Contribution

It develops a novel framework linking higher order Schwarzians, Hankel matrices, and complex structures on tangent bundles, extending curvature inequalities and providing new characterizations.

Findings

Characterization of adapted complex structures via infinite Hankel matrices.

Differential inequalities involving higher order Schwarzians and curvature.

Analysis of Schwarzians for imaginary radii and moment sequence representations.

Abstract

In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical theorem of Loewner on analytic continuation we are able to characterize the existence of the adapted complex structure induced by g on the set T^RM of vectors in TM of length up to R, equivalently for M compact, to the existence of a Grauert tube of radius R in terms of infinite Hankel matrices involving these Schwarzian functions. The basic characterization so obtained can be expressed as a sequence of differential inequalities of increasing order polynomial in the covariant derivatives of the Gauss curvature on M and in {\pi}/R that should be regarded as the higher order versions of a curvature inequality by L. Lempert and R. Sz\"oke. The second part of the paper, sections 7 through 11, includes a discussion of the rank of the infinite Hankel matrix of the Schwarzians from part 1 and of new Schwarzians defined now for purely imaginary radius, as well as some computations and examples. A characterization of the existence of the adapted structure on T^RM in terms of moment sequences with parameters R and v in TM is also noted.