Anisotropic Geodesics on Matrix Lie Groups and Geometric Strain Measures in Nonlinear Elasticity

Geometric formulations of nonlinear elasticity interpret strain through distances on the deformation group GL^{+}(n). In the isotropic setting, a left-GL(n)-invariant and right-O(n)-invariant Riemannian metric leads to explicit geodesic formulas and recovers the logarithmic strain tensor through the geodesic distance from a deformation gradient to the rotation group. The anisotropic case is substantially more delicate: the material symmetry group is generally a proper subgroup K \subset O(n), the corresponding metric has fewer invariances, and the geodesic equations no longer admit the same straightforward reduction.

In this talk, we study left-invariant and right-K-invariant Riemannian metrics on matrix Lie groups, with particular emphasis on GL^{+}(n). We derive the nonlinear evolution equations governing the body velocity and examine how the algebraic structure of the material symmetry group influences their integrability. We compare the anisotropic problem with the classical isotropic case, identify the geometric mechanisms responsible for the loss of explicit solvability, and discuss symmetry reductions and special classes for which additional structure remains available.

The resulting framework provides a systematic way to define anisotropic strain measures as geodesic distances adapted to material symmetry. It therefore connects Riemannian geometry on Lie groups, the representation theory of symmetry groups, and the constitutive modelling of anisotropic elastic materials. We conclude with open problems concerning the explicit integration of the geodesic equations, the classification of solvable anisotropic symmetry classes, and the extension of the construction to plasticity and more general geometric theories of continua.