On strain measures and the geodesic distance to 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 [12], is to select a Riemannian metric on 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-GL_n-invariant and right-O_n-invariant. We proceed to investigate alternative distance functions on 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 GL_n. Lastly, we investigate strain measures induced by inverse-invariant distances.