# Nicola Pia's research while affiliated with Institut für Therapieforschung München and other places

## Publications (8)

Article
We classify complex surfaces $$(M,\,J)$$ admitting Engel structures $${\mathcal {D}}$$ which are complex line bundles. Namely, we prove that this happens if and only if $$(M,\,J)$$ has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges [7]. We also study associated Engel defining forms and...
Article
This paper is about geometric and Riemannian properties of Engel structures. A choice of defining forms for an Engel structure $\mathcal{D}$ determines a distribution $\mathcal{R}$ transverse to $\mathcal{D}$ called the Reeb distribution. We study conditions that ensure integrability of $\mathcal{R}$. For example, if we have a metric $g$ that makes...
Preprint
We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges. We also study associated Engel defining forms and define a uniq...
Preprint
Let $\mathcal{E}^3\subset TM^n$ be a smooth $3$-distribution on a smooth manifold of dimension $n$ and $\mathcal{W}\subset\mathcal{E}$ a line field such that $[\mathcal{W},\mathcal{E}]\subset\mathcal{E}$. Under some orientability hypothesis, we give a necessary condition for the existence of a plane field $\mathcal{D}^2$ such that $\mathcal{W}\subs... Preprint This paper is about geometric and Riemannian properties of Engel structures, i.e. maximally non-integrable$2$-plane fields on$4$-manifolds. Two$1$-forms$\alpha$and$\beta$are called Engel defining forms if$\mathcal{D}=\ker\alpha\cap\ker\beta$is an Engel structure and$\mathcal{E}=\ker\alpha$is its associated even contact structure, i.e.$\...
Article
A holomorphic Engel structure determines a flag of distributions $$\mathcal {W}\subset \mathcal {D}\subset {\mathcal {E}}$$. We construct examples of Engel structures on $$\mathbf {C}^4$$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $$\mathbf {C}$$. We also construct two infinite families of pair...
Article
A holomorphic Engel structure determines a flag of distributions $\mathcal{W}\subset \mathcal{D}\subset \mathcal{E}$. We construct examples of Engel structures on $\mathbf{C}^4$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $\mathbf{C}$. We also construct two infinite families of pairwise non-isom...

## Citations

... For them, Question 1.4 was answered positively by McDuff [27], proving that they are (topologically) much more flexible than contact structures. However, interesting questions about them from a geometric perspective remain open [30]. 1.4.4. ...
... Two 1-forms α and β are said to be Engel defining forms for a given Engel structure D if D = ker α ∩ ker β and E = [D, D] = ker α. A pair of defining forms determines a complementary distribution R = T, R called the Reeb distribution (see [17]). The conformal class of α is uniquely determined by D, whereas in general the conformal class of β is not. ...
... In Zhao's work J-Engel structures appear under the name of complex Engel structures. We prefer to use the former name in order not to create confusion with holomorphic Engel structures [6,19], which are the analogue of Engel structures in the holomorphic category. ...