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Motivated by the ideas and methods used by Naitoh in the consideration ofparallel totally real submanifolds in complex space forms, the author of thepresent paper successfully makes use of the so called Jordan triple and(restricted) structure Lie algebra associated with a given Jordan algebra toestablish a one-to-one correspondence between the set of equivalence classes ofconnected, simply connected and nondegenerate equiaffine symmetrichypersurfaces with a given nonzero affine mean curvature and that of theequivalence classes of semi-simple real Jordan algebras. Then, via the existingclassification theorem of the semi-simple real Jordan algebras with unity, acomplete classification for the nondegenerate and locally equiaffine symmetrichypersurfaces with nonzero affine mean curvatures is established. As an directapplication of the main theorems, we prove at the end of the paper a completeclassification of nondegenerate hypersurfaces with parallel Fubini-Pick formsand nonzero affine mean curvatures.

A classification theorem of nondegenerate equiaffine symmetric hypersurfaces