Dupin cyclides osculating surfaces

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1 Paweł Walczak, Uniwersytet Łódzki, Dijon, 25 stycznia 2012 Colaborators: Remi Langevin (UdeB), Adam Bartoszek, Szymon Walczak (UŁ)

2 What is extrinsic conformal geometry? Conformal transformations = transformations preserving angles. Conformal group = group of conformal transformations; in R 3 (S 3 or H 3 ) = Möbius group Möb 3 = the group generated by all isometries and inversions Conformal geometry= theory of invariants of the Möbius group Extrinsic geometry (of surfaces) = theory of invariants of the second fundamental form (principal curvatures, principal directions and foliations, lines of curvature) etc. = extrinsic conformal geometry = (extrinsic geometry) (conformal geometry)

3 Conformal change of the metric 1 If two surfaces S ans S are conformally equivalent, then their first fundamental forms g and g are related by: g = exp(2φ) g for some function φ. If so, (1) their second fundamental forms b and b satisfy b = exp(φ) b + ψ g, (2) their shape (Weingarten) operators A and Ã satisfy Ã = exp( φ) A + ψ I,

4 Conformal change of the metric 2 (3) their principal directions are the same while unit principal vectors X i and X i (i = 1, 2) satisfy X i = exp( φ) X i, (4) their principal curvatures k i and k i satisfy k i = exp( φ) k i + ψ, therefore k 1 k 2 = exp( φ) (k 1 k 2 ).

5 First conformal invariants Consequently, the vector fields ξ i = X i /µ, i = 1, 2, where µ = (k 1 k 2 )/2 are conformally invariant. Conformally invariant is also their Lie bracket: [ξ 1, ξ 2 ] = 1 2 θ 2 ξ θ 1 ξ 2 and the coefficients θ 1, θ 2 (called principal conformal curvatures) in the above (Darboux, Tresse, 189*).

6 Canonical position Problem Are the invariants we described above sufficient to determine a surface up to a Möbius transformation? Answer: NO. Proof. Any surface (at a non-umbilical point) can be mapped by a unique Möbius transformation to the one given locally by z = 1 2 (x 2 y 2 ) (θ 1x 3 + θ 2 y 3 ) (ax 4 + bx 3 y + Ψx 2 y 2 + cx 3 y + dy 4 ) + H.O.T.

7 Canal surfaces Canal surfaces = envelopes of 1-parameter families of spheres. Proposition (A. B., R. L., P. W.,20**) Canal surfaces can be characterized by vanishing of one of their conformal principal curvatures: θ i = 0 for some i {1, 2}. The invariant Ψ is constant along the characteristic circles.

8 Special canals A canal surface is special if its nontrivial conformal principal curvature is constant along its characteristic circles. Proposition (A. B., R. L., P. W., 20**) Special canal surfaces are conformal images of surfaces of revolution, cylinders and cones over planar (spherical) curves. These three classes are characterized, respectively, by Ψ < 2, Ψ = 2, Ψ > 2.

9 Dupin cyclides Dupin cyclides = canal surfaces in two ways (there are two 1-parameter families of spheres enveloped by such a surface). = On Dupin cyclides θ 1 = θ 2 = 0. = Dupin cyclides are special canals. = Dupin cyclides are conformal images of tori, cylinders and cones of revolution. = On Dupin cyclides Ψ = const.

10 Integrability condition Given a surface S, one has the unique map g : S Möb 3 such that g(p), p S, maps S to the canonical position (at p). Let ω = g 1 dg. ω is the 1-form on S with values in the Lie algebra of the Möbius group Möb 3. Using a matrix representation of Möb 3, one can write ω = A 1 (θ 1, θ 2, Ψ) ω 1 + A 2 (θ 1, θ 2, Ψ) ω 2, where (ω 1, ω 2 ) is the frame of 1-forms dual to (ξ 1, ξ 2 ). Integrability condition: dω + 1 [ω, ω] = 0. 2

11 Fialkov Theorem Theorem (Fialkov, 194*) Given 1-forms ω 1, ω 2 and functions θ 1, θ 2, Ψ defined on a simply-connected domain U R 2 and satisfying the above integrability condition there, there exists unique up to a Möbius transformation immersion F : U R 3, S 3, H 3 such that the above data appear us the local conformal invariants of the surface S = F (U).

12 Dupin necklace Theorem (A. B., R. L., P. W.) The osculating spheres Σ 2 (t) for the principal curvature k 2 along a characteristic circle C (which is a parameterized by t line of principal curvature for k 1 ) of a canal surface K have an envelope which is a Dupin cyclide D.

13 A problem The above privides motivation for the following Problem Given a generic point p of a surface S, find a Dupin cyclide D osculating S at p and determine the direction of highest order of tangency of D and S at p.

14 Osculating cyclide 1 Recall the equation of S in the canonical form: S : z = 1 2 (x 2 y 2 ) (θ 1x 3 + θ 2 y 3 ) (ax 4 + bx 3 y + Ψx 2 y 2 + cx 3 y + dy 4 ) + H.O.T. Put a cyclide D in the canonical position. Its equation reads as D : z = 1 2 (x 2 y 2 ) (3x 4 + Ψ D x 2 y 2 3y 4 ) + H.O.T.

15 Osculating cyclide 2 = S and D are tangent of order 3 in the direction y = tx, where t = 3 θ 1 /θ 2. For a suitable (unique) value of Ψ D, S and D are tangent of order 4 in this direction. In this case, D is called the osculating cyclide of S at p.

16 Dupin foliation Here, we will use the following terminology: (1) the direction in T p S making the angle α such that tg α = 3 θ 1 /θ 2 = Dupin direction, (2) the distribution (line field) on S built of straight lines in Dupin directions = Dupin line field, (3) the foliation determined by the Dupin line field = Dupin foliation, (4) leaves of the Dupin foliation = Dupin lines (and, perhaps, so on).

17 An example Example On canal surfaces (different from Dupin cyclides), the Dupin foliation coincides with one of the foliations by lines of curvature. Problem Do there exist surfaces for which the Dupin direction is constant ( 0, π/2)?

18 Main Theorem (of today) Theorem (A. B., P. W., Sz. W., 201*) Given a foliation F on a convex planar domain U making non-zero angle with the direction of one of the coordinate lines, there exist surfaces S on which the coordinate lines correspond to the lines of curvature while the leaves of F correspond to the Dupin lines; the family of such surfaces is parametrized (up to Möbius transformations) by pairs of two functions: one of two variables and another one of one variable. Proof....

19 Helcats 1 Example There exists a natural 1-parameter family of minimal surfaces connecting the helicoid to the catenoid. They are called helcats

20 A helcat

21 Helcats 2 and are given by the equations x 1 = cosα sinh s sin t + sin α cosh s cos t, x 2 = cos α sinh s cos t + sin α cosh s sin t, x 3 = sin α s + cos α t, For them, we have: θ 1 = 2(1 sin α) sinh s, θ 2 = κ := θ 1 /θ 2 = 2(1 + sin α) sinh s, cos α 1 + sin α = const. In particular, κ = 1 on the helicoid and κ = 0 on the catenoid.

22 Bibliography 1 A. Bartoszek, R. Langevin, P. Walczak, Special canal surfaces of S 3, Bull. Braz. Math. Soc. 42 (2011), A. Bartoszek, P. Walczak, Foliations by surfaces of a peculiar class, Ann. Polon. Math., 94 (2008), A. Bartoszek, P. Walczak, Sz. Walczak, Dupin cyclides osculating surfaces, in preparation. G. Cairns, R. W. Sharpe and L. Webb. Conformal invariants for curves in three dimensional space forms, Rocky Mountain J. Math. 24 (1994), G. Darboux, Leçons sur la théorie générale des surfaces, Guthier-Villars, Paris 1897.

23 Bibliography 2 A. Fialkov, Conformal differential geometry of a subspace, Trans. Amer. Math. Soc. 56 (1944), R. Garcia, R. Langevin, P. Walczak, Dynamical behaviour of Darboux curves, preprint, arxiv (2009). R. Langevin, P. Walczak, Conformal geometry of foliations, Geom. Dedicata 132 (2008), A. Tresse, Sur les invariants différentiels d une surface par rapport aux transformations conformes de l espace, C.R. Acad. Sci. Paris 114 (1892),

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