Families of Sections of Quadrics and Classical Geometries*

To the memory of my father Georgii Sergeevich Khovanskii In the note we describe all Riemannian metrics on nondegenerate quadrics whose geodesics are plane curves. It is shown that any such metric is of constant curvature and hence locally de¯nes one of the classical geometries on the quadric. The present note is the continuation of [5], where the problem of recti¯cation of circles was solved. This practical problem was posed by G. S. Khovanskii in connection with his work on the transformation of nomograms of adjusted points into compass nomograms [1{4]). 1. Quadric points of a projective surface. A point A on a germ of a real regular surface in a real projective space is said to be quadric if there is a quadric that anomalously closely approximates the surface at this point, i.e., the distance from a point B on the surface to the quadric is an in¯nitesimal of order four with respect to the distance from A to B. We say that an a±ne coordinate system x, y, z in an a±ne neighborhood of the projective space is adapted to a surface at a point A if the origin coincides with this point and the plane z = 0 is tangent to the surface at A. Suppose that a local equation of the surface in a neighborhood of A in some adapted coordinate system has the form z = f(x; y) = B2(x; y)+K3(x; y)+: : : , where B2 and K3 are the quadratic and the cubic terms of the Taylor series of f at the origin, respectively