Dupin cyclides osculating surfaces (slajdy wykładu na seminarium w

Transkrypt

Dupin cyclides osculating surfaces
Paweł Walczak,
Uniwersytet Łódzki,
Dijon, 25 stycznia 2012
Colaborators:
Remi Langevin (UdeB),
Adam Bartoszek, Szymon Walczak (UŁ)
Paweł Walczak, Uniwersytet Łódzki, Dijon, 25 stycznia 2012
What is extrinsic conformal geometry?
Conformal transformations = transformations preserving angles.
Conformal group = group of conformal transformations;
in R3 (S 3 or H 3 ) = Möbius group Möb3 = 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
T
= (extrinsic geometry) (conformal geometry)
Conformal change of the metric 1
If two surfaces S ans S̃ are conformally equivalent, then their first
fundamental forms g and g̃ are related by:
g̃ = exp(2φ) · g
for some function φ. If so,
(1) their second fundamental forms b and b̃ satisfy
b̃ = exp(φ) · b + ψ · g ,
(2) their shape (Weingarten) operators A and Ã satisfy
Ã = exp(−φ) · A + ψ · I ,
Conformal change of the metric 2
(3) their principal directions are the same while unit principal
vectors Xi and X̃i (i = 1, 2) satisfy
X̃i = exp(−φ) · Xi ,
(4) their principal curvatures ki and k̃i satisfy
k̃i = exp(−φ) · ki + ψ,
therefore
k̃1 − k̃2 = exp(−φ) · (k1 − k2 ).
First conformal invariants
Consequently, the vector fields
ξi = Xi /µ, i = 1, 2,
where µ = (k1 − k2 )/2 are conformally invariant. Conformally
invariant is also their Lie bracket:
1
1
[ξ1 , ξ2 ] = − θ2 · ξ1 − θ1 · ξ2
2
2
and the coefficients θ1 , θ2 (called principal conformal curvatures) in
the above (Darboux, Tresse, 189*).
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
1
(x − y 2 ) + (θ1 x 3 + θ2 y 3 )
2
6
1
(ax 4 + bx 3 y + Ψx 2 y 2 + cx 3 y + dy 4 ) + H.O.T .
24
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: θi = 0 for
some i ∈ {1, 2}. The invariant Ψ is constant along the
characteristic circles.
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.
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 θ1 = θ2 = 0.
=⇒
Dupin cyclides are special canals.
=⇒
Dupin cyclides are conformal images of tori, cylinders and
cones of revolution.
=⇒
On Dupin cyclides Ψ = const.
Integrability condition
Given a surface S, one has the unique map g : S → Möb3 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 Möb3 . Using a matrix representation of Möb3 , one can write
ω = A1 (θ1 , θ2 , Ψ) · ω1 + A2 (θ1 , θ2 , Ψ) · ω2 ,
where (ω1 , ω2 ) is the frame of 1-forms dual to (ξ1 , ξ2 ).
Integrability condition:
1
d ω + [ω, ω] = 0.
2
Fialkov Theorem
Theorem
(Fialkov, 194*) Given 1-forms ω1 , ω2 and functions θ1 , θ2 , Ψ
defined on a simply-connected domain U ⊂ R2 and satisfying the
above integrability condition there, there exists – unique up to a
Möbius transformation – immersion F : U → R3 , S 3 , H 3 such that
the above data appear us the local conformal invariants of the
surface S = F (U).
Dupin necklace
Theorem
(A. B., R. L., P. W.) The osculating spheres Σ2 (t) for the principal
curvature k2 along a characteristic circle C (which is a
parameterized by t line of principal curvature for k1 ) of a canal
surface K have an envelope which is a Dupin cyclide D.
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.
Osculating cyclide 1
Recall the equation of S in the canonical form:
S :z
=
+
1 2
1
(x − y 2 ) + (θ1 x 3 + θ2 y 3 )
2
6
1
4
3
(ax + bx y + Ψx 2 y 2 + cx 3 y + dy 4 ) + H.O.T .
24
Put a cyclide D in the canonical position. Its equation reads as
D:z
=
+
1 2
(x − y 2 ) +
2
1
(3x 4 + ΨD x 2 y 2 − 3y 4 ) + H.O.T .
24
Osculating cyclide 2
=⇒ S and D are tangent of order 3 in the direction y = tx,
where
q
t = − 3 θ1 /θ2 .
For a suitable (unique) value of ΨD , S and D are tangent of order
4 in this direction.
In this case, D is called the osculating cyclide of S at p.
Dupin foliation
Here, we will use the following terminology:
(1) the direction
in Tp S making the angle α such that
p
tg α = − 3 θ1 /θ2 = 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).
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
(6= 0, 6= π/2)?
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. ...
Helcats 1
Example
There exists a natural 1-parameter family of minimal surfaces
connecting the helicoid to the catenoid.
They are called helcats
A helcat
Helcats 2
and are given by the equations
x1 = cosα · sinh s · sin t + sin α · cosh s cos t,
x2 = − cos α · sinh s · cos t + sin α · cosh s · sin t,
x3 = sin α · s + cos α · t,
For them, we have:
θ1 =
q
2(1 − sin α) · sinh s,
θ2 =
q
2(1 + sin α) · sinh s,
cos α
= const .
1 + sin α
In particular, κ = 1 on the helicoid and κ = 0 on the catenoid.
κ := θ1 /θ2 =
Bibliography 1
A. Bartoszek, R. Langevin, P. Walczak, Special canal surfaces
of S 3 , Bull. Braz. Math. Soc. 42 (2011), 301–320.
A. Bartoszek, P. Walczak, Foliations by surfaces of a peculiar
class, Ann. Polon. Math., 94 (2008), 89 – 95.
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), 933 – 959.
G. Darboux, Leçons sur la théorie générale des surfaces,
Guthier-Villars, Paris 1897.
Bibliography 2
A. Fialkov, Conformal differential geometry of a subspace,
Trans. Amer. Math. Soc. 56 (1944), 309 – 433.
R. Garcia, R. Langevin, P. Walczak, Dynamical behaviour of
Darboux curves, preprint, arXiv.0912.3749. (2009).
R. Langevin, P. Walczak, Conformal geometry of foliations,
Geom. Dedicata 132 (2008), 135 – 178.
A. Tresse, Sur les invariants différentiels d’une surface par
rapport aux transformations conformes de l’espace, C.R. Acad.
Sci. Paris 114 (1892), 948 – 950.

Dupin cyclides osculating surfaces (slajdy wykładu na seminarium w

Transkrypt

Podobne dokumenty

Thematic Session

24th CAMERIMAGE OFFICIALLY OPENED!

Conformal geometry of foliations (slajdy wykładu w Tambara

Extrinsic geometric flows - On joint work with Vladimir Rovenski from

1 29 June 2015 Conference 7th International Workshop on Surface

Studia Ecologiae et Bioethicae year: 2011, vol:.9 number: 3, pages