SubRiemannian Geodesics in the Grushin Plane

• DOI: 10.1007/s12220-011-9215-y
• Publication Date: 2011-02-08T17:18:45Z
• Authors: Der-Chen Chang; Yutian Li
• Keywords: Totally geodesic

We obtain all the subRiemannian geodesics induced by the Grushin operators \(\Delta_{k}=\frac{1}{2}(\frac{\partial^{2}}{\partial x^{2}}+x^{2k}\frac{\partial^{2}}{\partial y^{2}})\) in ℝ2 where k=1,2,… . We show that the y-axis is the canonical submanifold whose tangent space recovers the missing direction.