There are many kinds of ovarian tumors and tumorlike conditions that produce estrogen or androgen. Magnetic resonance imaging can demonstrate not only ovarian tumors but also an enlarged uterus with a thick endometrium, even in cases of a clinically latent excess of estrogen. These clinical and indirect imaging findings can aid in the differential diagnosis of ovarian tumors. Granulosa cell tumor and thecoma are well-known estrogen-producing tumors. In pediatric or postmenopausal patients, they manifest as precocious pseudopuberty or postmenopausal bleeding, respectively. Conversely, Sertoli-Leydig cell tumor is representative of hormone-producing tumors that cause virilization. However, there are other functioning ovarian tumors besides the sex cord-stromal tumors. It is well known that metastatic ovarian tumors often have androgen-producing stroma and that mucinous cystadenoma sometimes produces estrogens. Most other ovarian tumors can produce sexual hormones in their stroma. In addition, some endocrinologic abnormalities (eg, polycystic ovary syndrome) also cause virilization.

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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