Laguerre–Forsyth invariant

In projective geometry, the Laguerre–Forsyth invariant is a cubic differential that is an invariant of a projective plane curve. It is named for Edmond Laguerre and Andrew Forsyth, the latter of whom analyzed the invariant in an influential book on ordinary differential equations.

Suppose that p : P 1 → P 2 {\displaystyle p:\mathbf {P} ^{1}\to \mathbf {P} ^{2}} is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by p ( t ) = ( x 1 ( t ) , x 2 ( t ) , x 3 ( t ) ) {\displaystyle p(t)=(x_{1}(t),x_{2}(t),x_{3}(t))} then associated to p is the third-order ordinary differential equation

| x x ′ x ″ x ‴ x 1 x 1 ′ x 1 ″ x 1 ‴ x 2 x 2 ′ x 2 ″ x 2 ‴ x 3 x 3 ′ x 3 ″ x 3 ‴ | = 0. {\displaystyle \left|{\begin{matrix}x&x'&x''&x'''\\x_{1}&x_{1}'&x_{1}''&x_{1}'''\\x_{2}&x_{2}'&x_{2}''&x_{2}'''\\x_{3}&x_{3}'&x_{3}''&x_{3}'''\\\end{matrix}}\right|=0.}

Generically, this equation can be put into the form

x ‴ + A x ″ + B x ′ + C x = 0 {\displaystyle x'''+Ax''+Bx'+Cx=0}

where A , B , C {\displaystyle A,B,C} are rational functions of the components of p and its derivatives. After a change of variables of the form t → f ( t ) , x → g ( t ) − 1 x {\displaystyle t\to f(t),x\to g(t)^{-1}x} , this equation can be further reduced to an equation without first or second derivative terms

x ‴ + R x = 0. {\displaystyle x'''+Rx=0.}

The invariant P = ( f ′ ) 2 R {\displaystyle P=(f')^{2}R} is the Laguerre–Forsyth invariant.

A key property of P is that the cubic differential P(dt)3 is invariant under the automorphism group P G L ( 2 , R ) {\displaystyle PGL(2,\mathbf {R} )} of the projective line. More precisely, it is invariant under t → a t + b c t + d {\displaystyle t\to {\frac {at+b}{ct+d}}} , d t → a d − b c ( c t + d ) 2 d t {\displaystyle dt\to {\frac {ad-bc}{(ct+d)^{2}}}dt} , and x → C ( c t + d ) − 2 x {\displaystyle x\to C(ct+d)^{-2}x} .

The invariant P vanishes identically if (and only if) the curve is a conic section. Points where P vanishes are called the sextactic points of the curve. It is a theorem of Herglotz and Radon that every closed strictly convex curve has at least six sextactic points. This result has been extended to a variety of optimal minima for simple closed (but not necessarily convex) curves by Thorbergsson & Umehara (2002), depending on the curve's homotopy class in the projective plane.

References

  • Sasaki, Shigeo (1999), Projective differential geometry and linear homogeneous differential equations
  • Thorbergsson, G; Umehara, M (2002), "Sextactic points on a simple closed curve", Nagoya Mathematical Journal, 167 (4): 55–94