Abstract
Abstract Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry.
Highlights
In general Cauchy–Riemann (CR) dimension n 1, the classification of locally homogeneous real hypersurfaces M2n+1 ⊂ Cn+1 is a vast, infinite problem
We show that specializes to the known quartic expression in the semi-integrable LC (SILC) case
We will use the notation dim(symILC(g; e, f)) to denote the integrable LC (ILC) symmetry dimension of the unique left-invariant ILC structure on any Lie group G with Lie algebra g determined by the data (g; e, f)
Summary
In general Cauchy–Riemann (CR) dimension n 1, the classification of locally homogeneous real hypersurfaces M2n+1 ⊂ Cn+1 (up to local biholomorphisms) is a vast, infinite problem. Local Lie groups are analytic, so homogeneous M2n+1 ⊂ Cn+1 may be assumed from the outset to be real analytic (Cω). By Lie’s infinitesimalization principle [15], the group Hol(M) of local biholomorphic transformations of Cn+1 stabilizing M is better viewed as the real Lie algebra: hol(M) := X =. Any real affine symmetry S = Ak x + bk ∂xk (summation assumed on 1 ≤ k, ≤ n + 1) of S has “complexification” X = Scr = Ak z + bk ∂zk in hol(MS ). An affinely homogeneous base yields a holomorphically homogeneous tube
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