2014/11/26 by Travis Willse, Willse, Travis
Mathematics · #53A30 #53C29 #58A30 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1411.7172
openalex publication_date 2014/11/26 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28
In his 1910 "Five Variables" paper, Cartan solved the equivalence problem for\nthe geometry of (2, 3, 5) distributions and in doing so demonstrated an\nintimate link between this geometry and the exceptional simple Lie groups of\ntype textrmG2. He claimed to produce a local classification of all such\n(complex) distributions which have infinitesimal symmetry algebra of dimension\nat least 6 (and which satisfy a natural uniformity condition), but in 2013\nDoubrov and Govorov showed that this classification misses a particular\ndistribution bf E. We produce a closed form for the Fefferman-Graham ambient\nmetric widetildeg bf E of the conformal class induced by (a real form\nof) bf E, expanding the small catalogue of known explicit, closed-form\nambient metrics. We show that the holonomy group of widetildeg bf E is\nthe exceptional group textrmG2^* and use that metric to give explicitly a\nprojective structure with normal projective holonomy equal to that group.\n We also present some simple but apparently novel observations about ambient\nmetrics of general left-invariant conformal structures that were used in the\ndetermination of the explicit formula for widetildeg bf E.\n