Calculus of Variations and Geometric Measure Theory

V. Magnani

Non-horizontal submanifolds and coarea formula

created by magnani on 22 Oct 2007
modified on 21 Jan 2010


Published Paper

Inserted: 22 oct 2007
Last Updated: 21 jan 2010

Journal: J. Anal. Math.
Volume: 106
Pages: 95-127
Year: 2008


Let $k$ be a positive integer and let $m$ be the dimension of the horizontal subspace of a stratified group. Under the condition that $k$ is less than or equal to $m$, 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.

Keywords: stratified groups, Hausdorff dimension and Hausdorff measure, transverse submanifolds, area formula and coarea formula