Compatibility conditions for the Left Cauchy Green Tensor field in 3-D

The question of local existence of a deformation of a simply connected body whose Left Cauchy Green Tensor matches a prescribed, symmetric, positive definite tensor field is considered. A sufficient condition is deduced after formulation as a problem in Riemannian Geometry. The compatibility condition ends up being surprisingly different from that of compatibility of a Right Cauchy Green Tensor field, a fact that becomes evident after the geometric formulation. The question involves determining conditions for the local existence of solutions to an overdetermined system of Pfaffian PDEs with algebraic constraints that is typically not completely integrable.