Submanifolds of exponential families

Exponential family with ±1 connection plays an important role in information geometry. Amari proved that a submanifold M of an exponential family S is exponential if and only if M is a ∇1autoparallel submanifold. We show that if all ∇1auto parallel proper submanifolds of a ±1− flat statistical manifold S are exponential then S is an exponential family. Also shown that the submanifold of a parameterized model S which is an exponential family is a ∇1 autoparallel submanifold.