Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds

Abstract

In this article, we study the infinitesimal isometries on tangent sphere bundles over orientable three-dimensional Riemannian manifolds. Focusing on the vector fields which do not preserve fibers, we show the existence of lifts to the bundles of orthonormal frames. These lifts enable us to analyze the infinitesimal isometries by the symmetry of principal fiber bundles. We prove that the tangent sphere bundle admits a non-fiber-preserving infinitesimal isometry if and only if the base manifold has the same constant sectional curvatures as the fibers have. As an application, we classify the infinitesimal isometries on tangent sphere bundles for the three dimensional case.