This week I’m in Hanover, DE at the Workshop in Analysis and PDE.  It’s been very enjoyable (it’s worth noting that the conference is hosted in the Leibniz Universitat Hanover, which is literally a castle!) The talks have been diverse and interesting; a collection of sub-Riemannian talks by Davide Barilari, Erlend Grong, and Fabrice Baudoin were of particular interest to me.

Update: I’m just leaving Grenoble, where I presented at the conference Journées sous-Riemanniennes at the Institut Fourier. It’s been a great week in a beautiful city! The conference was particularly good, as it specialized in sub-Riemannian geometry and gave me the chance to see a number of different perspectives on the subject. The talks by Dario Prandi on Weyl’s law and Francesco Boarotto on regular abnormal curves especially stood out for me. The talks were video recorded, and will be uploaded online once they’ve been edited; I’ll link here once they are.

Grenoble itself was a wonderful place to visit; it is nestled in the mountains of southeastern France. I got the chance to explore the city some and to climb to La Basitille, an old fort perched on one of the mountains on the edge of town. The food was exceptional, and I finally got the chance to practice my French! I definitely plan to return next October.

So, I’m feeling inspired to outline what I’ve been working on and some things that I’d like to do in the near future.  First, ongoing work: I’m presenting The Horizontal Einstein property for H-type Foliations both here and in Grenoble this upcoming week.  The H-type sub-Riemannian manifolds are the primary object of interest; they were introduced by Baudoin and Kim in 2016, and seem to be an ideal class of sub-Riemannian manifolds on which to attempt to recover many Riemannian results.

Defining for a sub-Riemannian manifold with metric complement \((\mathbb{M}, \mathcal{H}, g)\) a map \(J \colon \mathcal{V} \rightarrow \operatorname{End}^-(\mathcal{H})\) by

\[\langle J_Z X, Y \rangle_\mathcal{H} = \langle Z, T(X,Y) \rangle_\mathcal{V}\]

where \(T\) is the torsion tensor of the Hladky-Bott connection, we say that \((\mathbb{M},\mathcal{H},g)\) is an H-type sub-Riemannian manifold if

1. \(\mathcal{V}\) is integrable,
2. For all \(X,Y \in \Gamma^\infty(\mathcal{H}), Z \in \Gamma^\infty(\mathcal{V})\), \[\langle J_ZX, J_ZY \rangle_\mathcal{H} = \|Z\|^2_\mathcal{V} \langle X,Y \rangle_\mathcal{H} \]

The second property is essential, and induces a Clifford structure over \(\mathcal{H}\). This class of spaces in remarkably broad, while still allowing for a number of strong results (which are forthcoming in a paper with F. Baudoin, E. Grong, and L. Rizzi.)

I’m also still working to continue the material posted earlier in this blog.  I’d like to gather the results for various connections on foliated manifolds and relate them to connections on sub-Riemannian spaces; the Hladky-Bott connection seems to be particularly well adapted to this setting.

There are also a few new directions I’ve been thinking about, particularly the Hamiltonian approach on sub-Riemannian manifolds, as well as heat kernels (leading towards index theory) on sub-Riemannian manifolds.  I’ll post updates on my progress in these directions going forward.