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 [math]\displaystyle{ p:\mathbf P^1\to\mathbf P^2 }[/math] is a three-times continuously differentiable immersion of the projective line into the projective plane, with homogeneous coordinates given by [math]\displaystyle{ p(t)=(x_1(t),x_2(t),x_3(t)) }[/math] then associated to p is the third-order ordinary differential equation

[math]\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. }[/math]

Generically, this equation can be put into the form

[math]\displaystyle{ x'''+Ax''+Bx'+Cx = 0 }[/math]

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

[math]\displaystyle{ x''' + Rx = 0. }[/math]

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

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

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), 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, http://projecteuclid.org/download/pdf_1/euclid.nmj/1114649292