Symmetries of (2, 3, 5)-distributions and associated Legendrian cone structures

• Published in: Annals of global analysis and geometry Vol. 67; no. 3; p. 18
• Main Authors: Hwang, Jun-Muk, The, Dennis
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2025; Springer Nature B.V
• Subjects: Analysis; Classification; Differential Geometry; Geometry; Global Analysis and Analysis on Manifolds; Lie groups; Mathematical Physics; Mathematics; Mathematics and Statistics; Symmetry; 53A55; 58J70; (2, 3, 5)-distribution; Secondary 32L25; 34C41; Primary 58A30; Legendrian cone structure; Symmetry
• ISSN: 0232-704X; 1572-9060
• DOI: 10.1007/s10455-025-09992-1

We exploit a natural correspondence between holomorphic (2, 3, 5)-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5 to present a new perspective in the study of symmetries of (2, 3, 5)-distributions. This leads to a number of new results in this classical subject, including an unexpected relation between the multiply-transitive families of models having 7- and 6-dimensional symmetries, and a one-to-one correspondence between equivalence classes of nontransitive (2, 3, 5)-distributions with 6-dimensional symmetries and nonhomogeneous nondegenerate Legendrian curves in P 3 . An ingredient for establishing the former is an explicit classification of homogeneous nondegenerate Legendrian curves in P 3 , which we present. Moreover, our approach gives a new perspective on exceptionality of the 3 : 1 ratio for two 2-spheres rolling on each other without twisting or slipping.