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Non-Euclidean elasticity for rods and almost isometric embeddings of geodesic tubes

Milan Kroemer, Stefan Müller

TL;DR

This work extends non-Euclidean elasticity to Riemannian settings by establishing a Γ-convergence framework for thin geodesic tubes around a curve. The authors derive a compactness result and identify the Γ-limit of the scaled elastic energy as a quadratic form depending on the difference between pullback curvature tensors along the domain and target geodesics, with explicit structure through G and T terms. They show that the limiting energy minimizes a curvature-difference functional, decomposed into parabolic, cross-sectional, and normal contributions, and prove that the limit is unique up to a natural equivalence. These results generalize Mora–Müller’s Euclidean rod theory and Maor–Shachar’s embedding energy to general Riemannian manifolds, providing a precise, curvature-driven measure of near-isometric embeddings of geodesic tubes with potential applications in geometric elasticity and biomechanics.

Abstract

We consider a geodesic $γ$ of length $2L$ in an oriented Riemannian manifold $(\mathcal M, g)$ and a thin tube $Ω^*_h$ around $γ$ of radius $h$. We study an 'elastic' energy per unit volume $E_h(u)$ of maps $u$ from $Ω^*_h$ into another oriented Riemannian manifold $(\tilde {\mathcal M},\tilde g)$. The energy $E_h$ is based on the squared distance of the differentials $du$ from the set of orientation preserving linear maps between the corresponding tangent spaces. We prove a compactness result for sequences of maps $u_h$ for which $h^{-4} E_h(u_h)$ remains bounded and we study the $Γ$-Limit of $h^{-4} E_h(u_h)$ as $h \to 0$ with respect to a suitable notion of convergence for $u_h$ that involves certain blow-ups in the radial direction. This $Γ$-convergence result ge\-ne\-ra\-lizes work by Mora and Müller on the limiting energy of thin rods in the Euclidean setting. We also obtain an expression for the minimum of the limiting energy as a specific quadratic functional in the difference of the pullbacks of the curvature tensors of $\mathcal M$ and $\tilde{\mathcal M}$ along the curves $γ$ and $u \circ γ$, respectively, thus answering a question by Maor and Shachar, J. Elasticity 134 (2019), pp. 149--173.

Non-Euclidean elasticity for rods and almost isometric embeddings of geodesic tubes

TL;DR

This work extends non-Euclidean elasticity to Riemannian settings by establishing a Γ-convergence framework for thin geodesic tubes around a curve. The authors derive a compactness result and identify the Γ-limit of the scaled elastic energy as a quadratic form depending on the difference between pullback curvature tensors along the domain and target geodesics, with explicit structure through G and T terms. They show that the limiting energy minimizes a curvature-difference functional, decomposed into parabolic, cross-sectional, and normal contributions, and prove that the limit is unique up to a natural equivalence. These results generalize Mora–Müller’s Euclidean rod theory and Maor–Shachar’s embedding energy to general Riemannian manifolds, providing a precise, curvature-driven measure of near-isometric embeddings of geodesic tubes with potential applications in geometric elasticity and biomechanics.

Abstract

We consider a geodesic of length in an oriented Riemannian manifold and a thin tube around of radius . We study an 'elastic' energy per unit volume of maps from into another oriented Riemannian manifold . The energy is based on the squared distance of the differentials from the set of orientation preserving linear maps between the corresponding tangent spaces. We prove a compactness result for sequences of maps for which remains bounded and we study the -Limit of as with respect to a suitable notion of convergence for that involves certain blow-ups in the radial direction. This -convergence result ge\-ne\-ra\-lizes work by Mora and Müller on the limiting energy of thin rods in the Euclidean setting. We also obtain an expression for the minimum of the limiting energy as a specific quadratic functional in the difference of the pullbacks of the curvature tensors of and along the curves and , respectively, thus answering a question by Maor and Shachar, J. Elasticity 134 (2019), pp. 149--173.

Paper Structure

This paper contains 12 sections, 24 theorems, 382 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fermi coordinates around a geodesic in $\mathcal{M}$
  4. Lift of maps from a small geodesic tube in $\mathcal{M}$
  5. Estimates for the center of mass in normal coordinates
  6. Setup
  7. Main theorem
  8. Convergence of the energy
  9. Minimization of the limiting functional
  10. Uniqueness of the limit in $\mathcal{X}^{\mathcal{G}}/\sim$
  11. Lipschitz approximation on thin tubes
  12. Fermi coordinates

Key Result

Lemma 3

Let $(\mathcal{M}, g)$ be a smooth Riemannian manifold. Given a framed unit speed geodesic $(\gamma, \underline{\nu})$ on $[-L,L]$ in $\mathcal{M}$ there exists $\rho >0$ and $C>0$ such that the Fermi coordinates $\psi_{\gamma, \underline{\nu}}$ are defined in $\Omega_\rho = (-L, L) \times B_\rho$ a and with $\overline{\mathcal{R}}_{jkil}(x_1) = \psi_{\gamma,\underline\nu}^*\mathcal{R}(x_1,0)(e_j

Theorems & Definitions (58)

  • Remark 1
  • Remark 2
  • Lemma 3: Manasse1963,Gray2004
  • proof : Proof of Lemma \ref{['lem:metric-taylor-expansion']}
  • Lemma 4
  • Lemma 5
  • proof
  • proof : Proof of Lemma \ref{['lem:existence-lift_new']}
  • Definition 6: Riemannian center of mass Karcher1977
  • Remark 7
  • ...and 48 more