Hexagonal circle patterns and integrable systems: Patterns with constant angles

Introduction to Möbius Differential Geometry

“…Then 34) I = df, df and II S = − df, dS = I(., A S .) yield the first and second fundamental forms of f as an immersion into Q n κ .…”

“…Then the sphere congruences S i := S + a i f consist of touching spheres that have the same (principal) curvature 36) a i = − S i , K as the hypersurface in 34) Compare this with (1.11): Since f takes values in Q n κ , we have χ i = κ 2 ω i , and its (vectorial) second fundamental form becomes II = II S S − I • (f + κ 2 f ) ⊥ K. 35) Here we use that df, dS ⊥ f, S, K (see §1.7.8). 36) This formula for the curvature of a hypersphere can be checked by assuming that f locally parametrizes the (for that purpose constant) hypersphere in Q n κ : LetS be a constant sphere that is parametrized by f ; thenS = S + a f as f envelopesS, and 0 = d S, df = a I − II S shows that a = − S , K is the (constant) curvature ofS ⊂ Q n κ .…”

“…2.6.3 The surfaces of constant Gauss or mean curvature among the surfaces f t of a family are characterized by the (distinct) zeros 34) of c H and c K , respectively. By (2.31) we have t 2 = (1 + κt 2 )c K (t), so that the Guichard net degenerates exactly at the surfaces of constant mean curvature in the family -and at the infinity boundary of Q 3 κ in the case κ < 0.…”

“…κ . Thus the central sphere congruence of a surface of constant mean curvature in some quadric Q 3 κ of constant curvature has flat normal bundle, 34) that is, it is Ribaucour. 33) Remember from §3.4.8 that Z degenerates exactly at the umbilics of f .…”

“…Conversely, if 34) dω + 1 2 [ω ∧ω] = 0 and g : M 2 → H satisfies (5.15), then there is (locally) a function a : M 2 ⊃ U → H such that g = (v 0 + v ∞ g)a satisfies the linear system dg + ωg = 0.…”