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We study the local equivalence problem for real-analytic (C ω ) hypersurfaces M 5 ⊂ C 3 which, in some holomorphic coordinates (z 1 , z 2 , w) ∈ C 3 with w = u + iv, are rigid in the sense that their graphing functions:are independent of v. Specifically, we study the group Hol rigid (M ) of rigid local biholomorphic transformations of the form:where a ∈ R\{0} and D(f1,f2) D(z1,z2) = 0, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate {e}-structure, we find exactly two primary invariants I 0 and V 0 , which we express explicitly in terms of the 5-jet of the graphing function F of M . The identical vanishing 0 ≡ I 0 (J 5 F ) ≡ V 0 (J 5 F ) then provides a necessary and sufficient condition for M to be locally rigidly-biholomorphic to the known model hypersurface:We establish that dim Hol rigid (M ) 7 = dim Hol rigid (M LC ) always. If one of these two primary invariants I 0 ≡ 0 or V 0 ≡ 0 does not vanish identically, then on either of the two Zariski-open sets {p ∈ M : I 0 (p) = 0} or {p ∈ M : V 0 (p) = 0}, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain 5-dimensional {e}-structure on M , that is, we get an invariant absolute parallelism on M 5 . Hence dim Hol rigid (M ) drops from 7 to 5, illustrating the gap phenomenon.
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