[Paper Review] Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

Robert J. Martin, Patrizio Neff|arXiv (Cornell University)|Sep 27, 2014

Advanced Differential Geometry Research13 references3 citations

TL;DR

This paper provides a self-contained, accessible derivation of geodesic curves and distance on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics using only calculus of variations and classical analysis. The key result is an explicit parametrization of geodesics and a global characterization of the geodesic distance as the norm of the matrix logarithm of the right Cauchy-Green tensor, which links directly to the Hencky strain energy in nonlinear elasticity.

ABSTRACT

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated.

Motivation & Objective

• To provide a simplified, accessible derivation of geodesic curves and distance on GL(n) without relying on advanced differential geometry.
• To establish the global existence and uniqueness of length-minimizing geodesics for left-invariant, right-O(n)-invariant Riemannian metrics on GL(n).
• To explicitly parametrize geodesic curves using solutions to the geodesic equation derived from energy minimization.
• To connect the geodesic distance to the logarithmic strain tensor and demonstrate its relevance in nonlinear elasticity.
• To lay the foundation for computing geodesic distances in special cases and to identify open problems in the geometry of GL(n) and SO(n) under these metrics.

Proposed method

• Derives the geodesic equation for general left-invariant, right-O(n)-invariant Riemannian metrics on GL(n) via the calculus of variations on the energy functional.
• Uses the matrix exponential and logarithm to parametrize geodesic curves in terms of symmetric positive definite matrices and orthogonal transformations.
• Applies the polar decomposition $ F = R U $ with $ R o ext{SO}(n) $, $ U = ext{Sym}^+(n) $, and shows that the geodesic distance to SO(n) depends only on $ ext{dev} ext{log}(U) $ and $ ext{tr} ext{log}(U) $.
• Characterizes the geodesic distance as $ ext{dist}_{ ext{geod}}(F, ext{SO}(n)) = ig\| ext{log} ig( ext{sym}(F^T F)^{1/2} ig) igig floor_{ ext{metric}} $, using a weighted Frobenius norm.
• Proves global existence and uniqueness of energy minimizers (length-minimizing geodesics) using classical analysis and properties of the matrix logarithm.
• Establishes that the geodesic distance to SO(n) equals the Hencky strain energy, linking the Riemannian geometry to hyperelasticity theory.

Experimental results

Research questions

• RQ1 Can the geodesic curves and distance on GL(n) be explicitly parametrized for left-invariant, right-O(n)-invariant Riemannian metrics using only elementary calculus of variations?
• RQ2 What is the global structure of length-minimizing geodesics on GL(n) under such metrics, and do they always exist?
• RQ3 How does the geodesic distance to the special orthogonal group SO(n) relate to the logarithmic strain tensor in nonlinear elasticity?
• RQ4 Is the geodesic distance between two matrices in GL⁺(n) computable in closed form, or are there special cases where it can be explicitly evaluated?
• RQ5 What are the geometric properties of GL(n) and SO(n) under these metrics, such as curvature or non-local distance behavior?

Key findings

• The geodesic distance from any $ F o ext{SO}(n) $ is given by $ igig floor ext{log} ig( ext{sym}(F^T F)^{1/2} ig) igig floor_{ ext{metric}} $, where the norm is defined by the isotropic inner product with parameters $ u, u_c, u $.
• The geodesic distance to SO(n) coincides with the Hencky strain energy $ W(F) = u ig floor ext{dev} ext{log} ig( ext{sym}(F^T F)^{1/2} ig) ig floor^2 + rac{ u_c}{n} ig[ ext{tr} ext{log} ig( ext{sym}(F^T F)^{1/2} ig) ig]^2 $.
• The parametrization of geodesic curves is explicitly given via the solution $ U(t) $ to the geodesic equation, which evolves as $ U(t) = ext{exp}(tM) $ for some $ M o ext{Sym}(n) $, with $ M $ determined by initial conditions.
• Length-minimizing geodesics exist globally on GL⁺(n) for these metrics, and their existence is guaranteed by the convexity of the energy functional and compactness arguments.
• The result holds globally for all $ F o ext{GL}^+(n) $, even for large $ ig floor F - oldsymbol{ ext{1}} ig floor $, confirming that the geodesic distance formula (5.20) is valid without local restrictions.
• The paper establishes that the geodesic distance to SO(n) is independent of the rotation part $ R $ in the polar decomposition $ F = R U $, depending only on the symmetric part $ U $.