Geometric structures and submanifolds

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As a joint research with Kaoru Suizu, we obtained fundamental theorem for Lagrange surfaces in the Riemann product of round 2-spheres. Namely, we showed that for minimal Lagrange surfaces in $S^2\times S^2$, Gauss and Codazzi equations are sufficient and necessary condition far the existence of such minimal Lagrange immersions. Also for Lagrange surfaces in $S^2\times S^2$, second fundamental tensor and the angle function, which we introcuced are invariant for congruent.Next we investigated Lagrange submanifolds in complex projective spaces , which is obtained as a 1-parameter family of totally geodesic, totally real (n-1)-dimensional submanifolds $RP^<{n-1}>$. Such submanifolds axe obtained from curves in the moduli space op $RP^<{n-1}>$ in $CP^n$. For a curve in the moduli space, we showed that the corresponding $n$-dimensional submanifold is Lagrangian if only if the curve is horizon with respect to the natural fibration from the moduli space to $CP^n$. Then we showed that if such a Lagrange submanifold is minimal, then is total geode-sic. More we investigated the condition for which such Lagrange submanifolds to be Hamiltonian minimal.