A geometry where everything is better than nice (Q2832843)

The authors consider the following metric in the unit disk: NEWLINE\[NEWLINEg=\frac{4}{1-r^2}(dx^2+dy^2)\text{ where }r^2=x^2+y^2.NEWLINE\]NEWLINE Let \(c(t)=(1-a)e^{i\theta(t)}+ae^{-i\theta(t)}\), \(0<a<1\). The authors show:NEWLINENEWLINE1. The curve \(c(t)\) is a geodesic for the metric \(g\), parameterized by arc length.NEWLINENEWLINE2. The closed geodesics (i.e. keep rolling the generating circle of the hypocycloid until it closes up) have length \(4\pi\sqrt n\), and the number of geometrically distinct geodesics of length \(4\pi\sqrt n\) is given by the arithmetic function \(\psi(n)\).NEWLINENEWLINEThe function \(\psi(n)\) counts the number of different ways that the integer \(n\) may be written as a product \(n=pq\), with \(p\leq q\), \((p,q)=1\). Values of this function are tabulated in sequence A007875 in the online encyclopedia of integer sequences.NEWLINENEWLINE3. The eigenvalues \(\lambda_n\) are precisely the positive integers \(n=1,2,3,\dots\). The eigenfunctions are polynomials. The dimension of the eigenspace for eigenvalue \(n\) is the number of divisors of \(n\).NEWLINENEWLINE4. The spectral function is precisely the square of the Riemann zeta function.NEWLINENEWLINE5. Unlike the standard vibrating membrane whose eigenfrequencies are proportional to zeros of \(J_0\), for each eigenfrequency \(\omega_n=\sqrt n\) of \(w_{tt}+\Delta w=0\), all the higher harmonics \(m\omega_n(2<m\in Z_+)\) are also eigenfrequencies.NEWLINENEWLINEThe authors also consider acoustic imaging and combinatorics.