Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 25H (Differential Geometry)

For an arclength-parametrized curve define the Frenet frame by $T={\alpha }^{\prime }$, $k=|{\alpha }^{\prime \prime }|>0$, $N=T^{\prime }/k$ and $B=T\times N$. Differentiating the unit lengths gives diagonal frame-derivative coefficients zero. Also $T^{\prime }=kN$ implies $N^{\prime }\cdot T=-k$, while $B^{\prime }\cdot T=-B\cdot T^{\prime }=0$. Define the torsion of a space curve by $\tau =-B^{\prime }\cdot N$. Orthogonality then givesThese are the Frenet formulae. Differentiating $B=T\times N$ also verifies the chosen torsion sign; equivalently $\tau =\det ({\alpha }^{\prime },{\alpha }^{\prime \prime },{\alpha }^{\prime \prime \prime })/|{\alpha }^{\prime }\times {\alpha }^{\prime \prime }{|}^2$.

At a fixed initial parameter choose an orthogonal map $Q$ sending $\tilde{T}$ to $T$, $\tilde{N}$ to $N$ and $\tilde{B}$ to $-B$. Its determinant is $-1$. The transformed curve $Q\tilde{\alpha }$ has Frenet frame $(Q\tilde{T},Q\tilde{N},-Q\tilde{B})$, so its curvature is $k$ and its torsion is $-\tilde{\tau }=\tau$.

Both its frame and the frame of $\alpha$ solve the same linear Frenet system with the same initial data. The uniqueness theorem for a linear ordinary differential system with continuous coefficients makes the frames identical on the whole interval. In particular $Q{\tilde{\alpha }}^{\prime }={\alpha }^{\prime }$, so their difference is a constant vector $v$. ThusThe orientation-reversing rigid motion accounts for the reversal of torsion.

For the positively oriented simple closed boundary, the turning tangent theorem gives $\int k(s)ds=2\pi$. If $L$ is its length, continuity gives a point with $k(s_0)\le 2\pi /L$. The planar isoperimetric inequality gives $L^2\ge 4\pi A(K)$, henceFor the reversed orientation the curvature integral is $-2\pi$, so a point of nonpositive curvature already proves the stated positive upper bound. The region-bounding interpretation is the ordinary Jordan-curve one; self-intersections would require specifying a different notion of enclosed area.