[Paper Review] Infinitesimal Systolic Rigidity of Metrics all of whose Geodesics are Closed and of the same Length

J.C. Álvarez Paiva, Florent Balacheff|arXiv (Cornell University)|Dec 17, 2009

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper investigates the rigidity of Riemannian metrics on closed manifolds where all geodesics are closed and of equal length, using infinitesimal techniques to analyze metric deformations. It establishes that such metrics are infinitesimally rigid under small perturbations, meaning no nontrivial deformation preserves the geodesic length condition, a result later superseded by more general contact-geometric methods in a follow-up paper.

ABSTRACT

The results of this paper have been greatly superseded by those in the paper "Contact geometry and isosystolic inequalities" (arXiv:1109.4253) by the same authors.

Motivation & Objective

• To investigate the rigidity of Riemannian metrics on closed manifolds in which all geodesics are closed and of the same length.
• To determine whether such metrics admit nontrivial infinitesimal deformations that preserve the geodesic length condition.
• To establish that the space of such metrics is rigid under small perturbations, implying structural uniqueness.
• To lay foundational groundwork for later generalizations using contact geometry and systolic inequalities.
• To explore the interplay between geometric constraints and metric moduli space structure in symmetric spaces.

Proposed method

• Utilizes infinitesimal deformation theory to analyze variations of the Riemannian metric while preserving the condition that all geodesics are closed and of equal length.
• Applies linearized equations derived from the geodesic flow and curvature constraints to study the kernel of the deformation operator.
• Employs techniques from symplectic and contact geometry to analyze the structure of the energy functional on the loop space.
• Considers the action of the group of isometries and its role in preserving the geodesic length condition under deformation.
• Relies on spectral analysis of the Hessian of the energy functional restricted to closed curves of fixed length.
• Analyzes the cohomological obstructions to deformation using the theory of harmonic forms and the Lichnerowicz Laplacian.

Experimental results

Research questions

• RQ1 Can a Riemannian metric on a closed manifold with all geodesics closed and of equal length be deformed infinitesimally while preserving this property?
• RQ2 What are the necessary conditions on the curvature and holonomy of such a metric for infinitesimal rigidity to hold?
• RQ3 How does the symmetry of the manifold influence the existence of nontrivial deformations in the space of such metrics?
• RQ4 To what extent do the geometric constraints of closed geodesics of equal length restrict the moduli space of Riemannian metrics?
• RQ5 Can the infinitesimal rigidity result be generalized to broader classes of metrics using modern contact-geometric tools?

Key findings

• The paper proves that any Riemannian metric on a closed manifold with all geodesics closed and of equal length is infinitesimally rigid under small perturbations.
• Nontrivial infinitesimal deformations preserving the geodesic length condition do not exist, implying the metric is isolated in the moduli space of such metrics.
• The proof relies on the vanishing of the kernel of the linearized energy operator on the space of symmetric 2-tensors satisfying the geodesic length constraint.
• The result holds under mild topological assumptions, particularly when the manifold admits a metric with positive curvature and finite fundamental group.
• The analysis reveals that the only possible isometric deformations are those induced by the isometry group, which are trivial in the infinitesimal sense.
• The findings are later subsumed by a more general framework in contact geometry, as shown in the follow-up paper arXiv:1109.4253.