This paper deals with one of the fundamental problems of the mathematical theory of inhemogeneities in simple elastic material bodies, namely the availability of local homogeneous configurations. Utilizing the original differential geometric approach of W. Noll and C.-C. Wang, adopted for our particular needs and restricted to the hyperelastic case, we develop a system of partial differential equations for the locally homogeneous material configurations. The existence and the form of solutions of this system are discussed by means of examples.