External fields as intrinsic geometry

There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one. We shall also recall some previous incidences of the same phenomena involving bosonic field theories. It is known that some such theories on the commutative geometry of space-time can be re-expressed as abelian-gauge theory in an appropriate noncommutative geometry. The noncommutative structure can be considered as containing extra modes all of whose dynamics are given by the one abelian action.