An elementary approach to tangent space variation on Riemannian submanifolds

We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidean geometry. We also show that using the tangent variation result one can prove a new structural property of local feature size function. This structural property is a generalization of a result of Giesen and Wagner [GW04, Lem. 7].