Primitive geodesic lengths and (almost) arithmetic progressions

TL;DR

This paper explores the occurrence of arithmetic progressions in primitive geodesic lengths on Riemannian manifolds, showing their rarity in negatively curved metrics but abundance in certain symmetric cases, and proposes a conjecture linking these progressions to arithmeticity.

Contribution

It introduces the concept of almost arithmetic progressions, proves their ubiquity in negatively curved manifolds, and establishes the presence of genuine progressions in non-compact locally symmetric arithmetic manifolds.

Findings

Negatively curved metrics rarely have genuine arithmetic progressions.

All negatively curved, closed manifolds have arbitrarily long almost arithmetic progressions.

Non-compact, locally symmetric arithmetic manifolds contain arbitrarily long genuine arithmetic progressions.

Abstract

In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.