Osculating curves Étienne Ghys CNRS- ENS Lyon « Geometry and the Imagination » Bill Thurston’s 60 th birthday Princeton, June 2007

Partially joint work with S. Tabachnikov and V.Timorin Thanks to J. Leys for his help with animations.

Osculating circles

P.G. TAIT

« When the curvature of a plane curve continuously increases, no two of the osculating circles can intersect each other ».

How can a curve be tangent to a foliation without being contained in a single leaf ? Dis-proof ?

The foliation is not smooth ! It’s only Hölder…

Observations (Zeghib) : Foliations by lines are Lipschitz regular Foliations by circles are …nothing special !

Proof of Tait’s theorem dist(P,Q) ≥ dist(P’,Q’)+dist(Q,Q’)

The modulus of the annulus defined by two osculating circles is a conformal invariant. This defines some kind of conformal length. Möbius transformations of the Riemann sphere send circles to circles and preserve the order of contact. Hence osculating circles are mapped to osculating circles by Möbius transformations. The infinitesimal annulus between s and s + ds has modulus Simple proofs of the four vertex theorem

Taylor polynomials Proposition : If does not vanish on [a,b], then the graphs of the polynomials are two by two disjoint.

Proof :

At a generic point on a curve, there is a unique solution of the ODE which intersects the curve on a (2k+1) multiple point. Osculating solution. Every differential equation of odd order yields a similar local result. Two infinitesimally close osculating solutions only intersect locally on a (2k) multiple point. So, locally, they don’t intersect each other.

Examples Polynomials of degree 2k Schwarzian derivative : Möbius functions (ax+b)/(cx+d) Circles

Let f : [a,b]  RP 1 be a local diffeomorphism. Assume that the schwarzian derivative of f does not vanish on the interval [a,b]. For each x in [a,b] consider the Möbius map which « osculates » f at x to the order 3. Then the graphs of these Möbius maps in RP 1  RP 1 are 2 by 2 disjoint. For every diffeomorphism of RP 1, the schwarzian derivative vanishes at least four times. Four vertex theorem for diffeomorphisms

Monge’s equation Conics depend on 5 parameters, they intersect on at most 4 points Special points are called sextactic points

If an arc in the plane does not contain sextactic points, then the osculating conics don’t intersect. Two « consecutive » osculating conics intersect locally on a multiple point of order 4. Two conics intersect in at most 4 points ! Osculating conics

Are these results optimal ? Yes : Vogt (1920) For conics the answer is no in general.

Proposition : There is a family of partial orderings  t on the space of non singular conics in the real projective plane, depending on a parameter t  [0,1], such that: 1)  t is projectively invariant. 2)C  0 C’ simply means that the interior of C is contained in the interior of C’. 3)If s ≤ t and C  t C’ then C  s C’. 4)These are the only invariant « reasonable » orderings on the space of conics. 5)On a curve with no sextactic points, osculating conics are nested with respect to the ordering  1. 6)Conversely, given C  1 C’, and a pair of points C,C’, then there is a curve with no sextactic point connecting the points.

Suppose C and C’ are defined by 3  3 symmetric matrices A and A’ of signature -, +, +. Normalize so that det(A)=det(A’). Then one says that C  1 C’ if A’- A is definite positive. Example : Definition of C  1 C’ C  0 C’ C  1 C’

Halphen’s equation Cubic curves

In degree d, algebraic curves depend on d(d+3)/2 parameters and they intersect in d 2 points. For ovals in cubics (d=3), still OK: they intersect in an even number of points which is at most 9=d 2, hence at most 8= d(d+3)/2-1. If d 2 > d(d+3)/2-1, i.e. d ≥3, the fact that osculating curves don’t interect locally (at generic points) is still true but this does not imply anymore that they don’t intersect.

Assume d is even and d(d+3)/2 is odd (so d = 4k+2). Consider the space X d of non singular real algebraic curves of degree d, equipped with a base point. Every smooth curve with no d-tactic point defines a curve in X d that we can orient in such a way that the interiors of the osculating curves are locally increasing. The tangent vectors to these curves in generate a convex cone field in X d. Question : does this cone field define an ordering in X d ? Higher degree

Sextics (d=6) depend on d(d+3)/2= 27 parameters and they intersect two by two in at most 36 points. Does there exist a closed immersed curve in the plane with no 6-tactic point ? If yes, what kind of dynamics ?

(Local) orderings on Lie groups Theorem (Vinberg). A simple Lie algebra contains a non trivial convex cone invariant under the adjoint representation iff the associated symmetric space G/K is a hermitian symmetric space. Examples : Poincaré disk Siegel domain Complex hyperbolic ball

Theorem (Vinberg) : If G/K is hermitian, there is a unique maximal (resp. minimal) invariant convex cone in the Lie algebra. Example : For the symplectic group, there is only one invariant convex cone (up to sign). For the group of isometries of the complex hyperbolic ball, there is a one parameter family of convex invariant cones. These cones yield local orderings in Lie groups defined by invariant cone fields on the Lie group. Do they define global orderings?

Theorem (Vinberg) : The minimal cone C min always defines an invariant ordering on the universal cover of the Lie group. Theorem (Oshanskii) : Full description of those invariant cones which yield a global ordering on the universal cover : those which are contained in a explicit invariant cone C ord. For the symplectic group C min = C ord = C max For the complex hyperbolic ball, C min = C ord ≠ C max

Invariant cone-fields on homogenous spaces G/H ? Which homogeneous spaces are orderable ?

Eliashberg-Kim-Polterovich : cone structure on the group of contact diffeomorphisms of the 3- sphere (for instance). Isometries of the complex ball induce contact diffeos on the 3-sphere. E-K-P’s cone field induces on a convex cone which is the maximal one, hence does not define an ordering. This cone structure does not define an ordering on the universal cover of the group : there is a positive loop in the group which is homotopic to a point.

Does there exist a convex cone in the Lie algebra of contact vector fields of the standard contact 3-sphere which defines a global ordering of the (universal cover) of the contactomorphisms group ? The orthogonal for some « Killing form » still to be defined ? Generalize Vinberg-Olshanskii’s work to infinite dimensional Lie groups ?