On the Algebraic Birkhoff Conjecture for Outer Billiards on the Sphere and the Hyperbolic Plane: A Corollary of Theorems of Glutsyuk

Bolsinov, Matveev, Miranda and Tabachnikov (Phil. Trans. R. Soc. A, 2018) asked to prove the algebraic version of the Birkhoff conjecture for outer billiards on a non-Euclidean surface of constant curvature. We observe that it follows from published theorems of A. Glutsyuk. Let \(\gamma\) be a simple closed \(C^4\) curve with nowhere vanishing geodesic curvature on the sphere (inside a hemisphere) or on the hyperbolic plane, realised as quadrics in \(\mathbb{R}^3\). If the outer billiard about \(\gamma\) has a first integral which is the restriction of a polynomial, or of a rational function, of the coordinates of \(\mathbb{R}^3\), then \(\gamma\) is a conic; conversely, conics have quadratic integrals. In the gnomonic and Klein models the point reflection which defines the outer billiard is the projective involution of the tangent line fixing the point of tangency and interchanging the points of the line on the absolute \(\mathcal{I}\). So the outer billiard is an \(\mathcal{I}\)-angular billiard in Glutsyuk's sense (J. Eur. Math. Soc. 2021), and his theorem on such billiards gives the polynomial case; his classification of rationally integrable dual billiards (Invent. Math. 2026) gives a second proof, also for rational integrals. On the sphere the polynomial case also follows, for \(C^3\) curves, from his theorem on polynomially integrable billiards through the spherical duality. The theorems are Glutsyuk's; the note contributes the identification and the reduction, and we did not find the corollary stated in print. Algebraic non-rational integrals and the smooth version are not treated. This is an unrefereed note.