## Prof. Dr. Wolfram Bauer

### Project leader

Leibniz-Universität Hannover

E-mail: bauer(at)math.uni-hannover.de

Telephone: +49 511 762-2361

## Project

**6**Spectral Analysis of Sub-Riemannian Structures

## Publications within SPP2026

Pseudo H-type Lie groups \(G_{r,s}\) of signature (r,s) are defined via a module action of the Clifford algebra \(C\ell_{r,s}\) on a vector space V≅\(\mathbb{R}^{2n}\). They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let \(\mathcal{N}_{r,s}\) denote the Lie algebra corresponding to \(G_{r,s}\). A choice of left-invariant vector fields [\(X_1, \ldots, X_{2n}\)] which generate a complement of the center of \(\mathcal{N}_{r,s}\) gives rise to a second order operator

\(\Delta_{r,s}:=\big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ldots +X_{2n}^2 \big{)}\)

which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of \(\Delta_{r,s}\) in the case r=0, s>0 and study their properties. In the case of r>0 we prove that \(\Delta_{r,s}\) admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of \(\Delta_{r,s}\) and the existence of a fundamental solution in the space of Schwartz distributions.

**Related project(s):****6**Spectral Analysis of Sub-Riemannian Structures

We construct a codimension 3completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a (4*n *+3)-dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere *S*7.

Journal | Appl. Anal. 96 (2017), 2390–2407. |

Link to preprint version |

**Related project(s):****6**Spectral Analysis of Sub-Riemannian Structures