  The isotropic Hencky strain energy appears naturally as a distance measure of
the deformation gradient to the set SO(n) of rigid rotations in the canonical
left-invariant Riemannian metric on the general linear group GL(n). Objectivity
requires the Riemannian metric to be left-GL(n)-invariant, isotropy requires
the Riemannian metric to be right-O(n)-invariant. The latter two conditions are
satisfied for a three-parameter family of Riemannian metrics on the tangent
space of GL(n). Surprisingly, the final result is basically independent of the
chosen parameters. In deriving the result, geodesics on GL(n) have to be
parametrized and a novel minimization problem, involving the matrix logarithm
for non-symmetric arguments, has to be solved.
