The Cayley Cubic and Differential Equations
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The Cayley Cubic and Differential Equations Wojciech Krynski ´ 1 · Omid Makhmali2 Received: 3 May 2020 / Accepted: 19 September 2020 © Mathematica Josephina, Inc. 2020
Abstract We define Cayley structures as a field of Cayley’s ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations and the theory of integrable systems. In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., halfflatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair. After solving the equivalence problem we obtain the fundamental invariants, find the local generality of several classes of Cayley structures and give examples. Keywords Causal geometry · Conformal geometry · Path geometry · Integrable systems · Half-flatness · Lax pair · Cayley’s ruled cubic Mathematics Subject Classification 53C10 · 53A20 · 53A30 · 53B15 · 53B25 · 58A15 · 37K10
1 Introduction The main purpose of this article is to demonstrate a link between the theory of integrable systems and a class of 4-dimensional causal structures which is established via an extension of the twistor theory. Roughly speaking, causal structures are defined as the field of light cones arising from the conformal class of a pseudo-Finsler metric. We assume an additional condition of half-flatness for causal structures which is
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Omid Makhmali [email protected] Wojciech Kry´nski [email protected]
1
´ Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-656 Warszawa, Poland
2
Institute of Mathematics and Statistics, Masaryk University, Kotláˇrská 2, 61137 Brno, Czech Republic
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W. Krynski ´ , O. Makhmali
a generalization of the notion of half-flatness or self-duality for indefinite conformal structures in dimension four. We consider causal structures referred to as Cayley structures for which the light cone in each tangent space is the cone over the projective surface known as Cayley’s ruled cubic. As will be explained, Cayley structures display the main features of general half-flat causal structures and moreover stand out among them due to many interesting properties of Cayley’s ruled cubic compared to other projective surfaces. The content of the paper is divided into a study of the geometry of Cayley structures, finding suitable twistorial constructions, and analyzing the corresponding dispersionless Lax pair system. Our results, when compared with the well-known case of indefinite conformal structures, establish a rich interplay among twistor theory, integrable systems and half-flat causal structures in dimension four. To give some perspective, recall that Cayley’s ruled cubic is a projective surface in P3 expressed as 1 2 3 0 3 3 1 2 3 (1.1) 3 (y ) + y y y − y y y = 0, where [y 0 : y 1 : y 2 : y 3 ] are homogeneous coordinates for P3 . Among projective surfaces
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