sub-Riemannian Geometry

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I’m attending the conference Microlocal and Global Analysis, Interactions with Geometry in Potsdam this week; it’s being held at the University of Potsdam in the Neues Palais, which is literally a palace (in typical German math department style, cf. the Department of Mathematics at Leibniz Universitat, which is in a castle.)

Neues Palais, an historic palace and run-of-the-mill German university.

The conference is dedicated to Professor Bert-Wolfgang Schulze on the occasion of his 75th birthday; he created this conference (which has been an annual event for some years) and has made significant contributions to the field.

The schedule is very busy, and already halfway through the second day there have been a number of good talks related to my research interests. In particular, I was impressed by Francesco Bei, “Degenerating Hermitian metrics and spectral convergence” in which he gave strong results of the spectrum of the Hodge-Kodaira Laplacian on Hermitian complex spaces, and Gerardo Mendoza, “Singular foliations by tori” in which he classified closed manifolds foliated by Killing fields in a manner analogous to the classification of line bundles by their Chern classes. I’m looking forward to the rest of the conference, I’ll update here as it goes!

I should mention, I was able to present a poster on my joint research on H-type foliations with Fabrice Baudoin, Erlend Grong, and Luca Rizzi. My thanks to the conference for this opportunity!

Neues Palais, as seen from the air

Update, Thursday 7 March:

The fourth day of the conference has wrapped up, and it’s continued to present very good speakers. On Wednesday Irina Markina talked about the Cauchy-Szegö kernel for the quaternionic Seigel uppee half space, which was very interesting and had a surprising number of connections to my recent studies in Clifford algebras, and Wolfram Bauer presented his work (joint with Irina Markina) on ultrahyperbolic operators on pseudo H type groups, which is a different generalization than the H type foliations I’ve recently been involved with. Both were excellent talks. I would be remiss not to mention that Fabrice Baudoin presented our paper (with Erlend Grong and Luca Rizzi) on H type foliations. It was well presented and well received. It was a very good day for topics close to my interests!

On Thursday there were two talks on index theory that caught my attention; Maxim Braverman spoke about Callias type operators and index theory on noncompact manifolds, and Paolo Piazza presented work about K-theory classes and their properties on a type of singular manifold. Georges Habib also talked about the Bochner formula for Riemannian foliations, which I found very familiar.

Outside Krongut Bornstedt

There was a dinner on Wednesday night at the Krongut Bornstedt, a brewery that has served the conference for many years. The beer was as great as I remember from last year, and I had a great conversation with several very good speakers from the conference!

I had the opportunity to present The Horizontal Einstein Property for H-type Foliations at the Journées sous-Riemanniennes last Fall, which was recorded. Here’s a link to the video on YouTube! Thanks again to Luca Rizzi for the invitation and the great conference.

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.