On binary differential equations

Abstract

We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to congruences of lines in Euclidean 3-space and allows us to introduce a new one. The new BDE yields a new singular surface in the Euclidean 3-space associated to a congruence of lines. We determine the generic local configurations of the above BDEs on congruences.

FAMILIES OF SURFACES IN ℝ4

“…As in the proof of Theorem 2.1 in [4], we consider in the projectivized tangent bundle to M , P(T M), the setM which consists of the asymptotic directions at non-umbilics and all directions at umbilic points. Since the surface M is generic, the parabolic set is either smooth or has Morse singularities at umbilic points, and hence the surfaceM is smooth [6], and the boundary of M h , the closure of the hyperbolic region of M , is a union of circles. The projection π :M → M h is a smooth 2-fold covering away from the parabolic set and umbilics.…”

“…The bivalued asymptotic field on M h lifts to a smooth line field ξ onM . For a generic surface M this field has one or three zeros on each exceptional fibre [6], and well-folded saddles/nodes/foci on the lift of the parabolic set. The result now follows using Poincaré's Theorem.…”

Self-adjoint operators on surfaces with singular metrics

“…The whole exceptional fibre (0, 0) × RP 1 is an integral curve of ξ. The surface N is regular along the exceptional fibre if and only if the discriminant function δ of the BDE has a Morse singularity ( [7]). If j 1 (a, b, c) = (a 1 x + a 2 y, b 1 x + b 2 y, c 1 x + c 2 y), then the singularities of ξ on the exceptional fibre are given by the roots of the cubic…”

“…The eigenvalues of the linear part of ξ at a singularity are −φ (p) and α 1 (p) (see [7] for details), where…”

“…If this is the case, the topological models of the BDE are completely determined by the singularity type of the discriminant (A + 1 or A − 1 ), the number (1 or 3) and the type (saddle or node) of the singularities of ξ (see [7] for the analytic case and Remark 2 in [23] for the smooth case). We label such BDEs Morse Type 2 BDEs (MT2).…”