  On the probability simplex, we can consider the standard information
geometric structure with the e- and m-affine connections mutually dual with
respect to the Fisher metric. The geometry naturally defines submanifolds
simultaneously autoparallel for the both affine connections, which we call {\em
doubly autoparallel submanifolds}.
  In this note we discuss their several interesting common properties. Further,
we algebraically characterize doubly autoparallel submanifolds on the
probability simplex and give their classification.
