Extrinsic geometry and linear differential equations of SL(3)-type

Abstract: As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry of submanifolds in flag varieties and systems of linear PDEs for a class of special interest associated with the adjoint representation of SL(3). It may be seen as a contact generalization of the classical description of surfaces in P^3 in terms of two linear PDEs of second order.

We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE's of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE's called contact Cayley's surface and one which has SL(2) symmetry.

Joint work with Tohru Morimoto.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

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Geometry of differential equations seminar