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

$\left|\begin{array}{cccc}x& x^{\prime }& x^{″}& x^{‴}\\ x_1& {x_1}^{\prime }& {x_1}^{″}& {x_1}^{‴}\\ x_2& {x_2}^{\prime }& {x_2}^{″}& {x_2}^{‴}\\ x_3& {x_3}^{\prime }& {x_3}^{″}& {x_3}^{‴}\\ \end{array}\right|=0.$

Generically, this equation can be put into the form

$x^{‴}+A{x}^{″}+B{x}^{\prime }+Cx=0$

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

$x^{‴}+Rx=0.$

The invariant $P=(f^{\prime }{)}^{2}R$ is the Laguerre–Forsyth invariant.

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

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), 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