We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the view point of geometric control theory. In fact we consider the space of singular (or abnormal) paths on a given five dimensional space endowed with a Cartan distribution, which form another five dimensional space with a cone structure. We regard the cone structure as a control system and show that the space of singular paths of the cone structure is naturally identified with the original space. Moreover we observe an asymmetry on this duality in terms of singular paths.

As was shown by a part of the authors, for a given (2, 3, 5)-distribution D on a 5-dimensional manifold Y , there is, locally, a Lagrangian cone structure C on another 5-dimensional manifold X which consists of abnormal or singular paths of (Y, D). We give a characterization of the class of Lagrangian cone structures corresponding to (2, 3, 5)-distributions. Thus we complete the duality between (2, 3, 5)-distributions and Lagrangian cone structures via pseudo-product structures of type G 2 . A local example of non-flat perturbations of the global model of flat Lagrangian cone structure which corresponds to (2, 3, 5)-distributions is given.

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DUALITY ON GEODESICS OF CARTAN DISTRIBUTIONS AND SUB-RIEMANNIAN PSEUDO-PRODUCT STRUCTURES Communicated by S. IzumiyaAbstr act . Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. T hen it is shown in (15), that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-R iemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.

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