Integrable geodesic flows and super polytropic gas equations

The polytropic gas equations are shown to be the geodesic flows with respect to an L² metric on the semidirect product space Diff(S¹)⊙C^∞(S¹), where Diff(S¹) is the group of orientation preserving diffeomorphisms of the circle. We also show that the N=1 supersymmetric polytropic gas equation constitute an integrable geodesic flow on the extended Neveu-Schwarz space. Recently other kinds of supersymmetrizations have been studied vigorously in connection with superstring theory and are called supersymmetric-B (SUSY-B) extension. In this paper we also show that the SUSY-B extension of the polytropic gas equation form a geodesic flow on the extension of the Neveu-Schwarz space.