H-type structures

Posted on September 17, 2019 by Fabrice Baudoin

In two works in collaboration with Erlend Grong, Gianmarco Molino and Luca Rizzi, we introduced the notion of H-type structure:

  1. H-type foliations
  2. Comparison theorems on H-type sub-Riemannian manifolds

Consider a triple (\mathbb{M}, \mathcal{H}, g) where \mathbb{M} is a manifold, \mathcal{H} a constant rank sub-bundle of the tangent bundle of T \mathbb{M} and g a Riemannian metric. We say that \mathcal{H} is of H-type if for every smooth vector field Z which is unit and orthogonal to \mathcal{H}, the map J_Z : \mathcal{H} \to \mathcal{H} defined by the relation

g(J_Z X, Y)=g (Z,[Y,X]), \quad X,Y \in \Gamma (\mathcal{H})

satisfies J^2_Z=-\mathbf{Id}_\mathcal{H}. H-type structures are therefore natural generalizations to the Riemannian manifold setting of the H-type groups introduced by A. Kaplan. The sub-Riemannian manifold (\mathcal{H}, g_\mathcal{H}) and its intrinsic geometry is then of special interest and is studied in details in our papers. In the first paper, we prove various classifications theorems and several necessary and sufficient conditions for the existence of H-type structures on a Riemannian manifold. In the second paper, we focus on developing a comparison geometry for the sub-Riemannian manifold (\mathcal{H}, g_\mathcal{H}). In particular, we obtain several sharp sub-Hessian and sub-Laplacian comparison theorems.

The following video is a talk given at the University of Potsdam in March 2019 explaining results from the first paper:

Talk H-type foliations

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