Non-horizontal submanifolds and coarea formula

Abstract

Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.