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Abstract. For a regular surface in Euclidean space R 3 , umbilic points are precisely the points where the Gauss and mean curvatures K and H satisfy H 2 = K; moreover, it is wellknown that the only totally umbilic surfaces in R 3 are planes and spheres. But for timelike surfaces in Minkowski space R 1,2 , it is possible to have H 2 = K at a non-umbilic point; we call such points quasi-umbilic, and we give a complete classification of totally quasi-umbilic timelike surfaces in R 1,2 .

Abstract. We prove that second-order Monge-Ampère equations for one function of two variables are connected to the wave equation by a Bäcklund transformation if and only if they are integrable by the method of Darboux at second order.

Abstract. We analyze the geometry of sub-Finsler Engel manifolds, computing a complete set of local invariants for a large class of these manifolds. We derive geodesic equations for regular geodesics and show that in the symmetric case, the rigid curves are local minimizers. We end by illustrating our results with an example.

We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.