On strain measures and the geodesic distance to $\text{SO}_n$ in the general linear group

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-$\text{GL}_n$-invariant and right-$\text{O}_n$-invariant. We proceed to investigate alternative distance functions on $\text{GL}_n$, and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on $\text{GL}_n$. Lastly, we investigate strain measures induced by inverse-invariant distances.