Geodesics on Grushin spaces

Abstract

We consider higher-dimensional generalizations of the $\alpha$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using the calculus of generalized trigonometric functions, we obtain explicit conjugate times for geodesics starting at a Riemannian point. We propose a conjectured cut time $\tau=\min\{\tau_j\}$ obtained as the minimum of several candidates, each deriving from the symmetries of components of the Hamiltonian flow. We prove that it provides a lower bound on the first conjugate time, a key step in the extended Hadamard technique. In the three-dimensional case, we combine this method with a density argument to establish the conjecture and obtain the full optimal synthesis.