Stripe Structures and the Berry-Phase Connection: Concept of Geometric Energy

Electronic states of an $e_g$ electron are calculated in the system composed of two MnO$_6$ octahedra with the inclusion of the Berry phase acquired by parallel transport. Based on this calculation, a comment is made on the controversy between ``Wigner-crystal'' and ``paired-stripe'' models for the the insulating charge-ordered manganese oxides.