scholarly journals Left-Invariant Sub-Riemannian Engel Structures: Abnormal Geodesics and Integrability

2018 ◽  
Vol 56 (5) ◽  
pp. 3524-3537 ◽  
Author(s):  
Ivan Beschastnyi ◽  
Alexandr Medvedev
Keyword(s):  
Engel Structures

scholarly journals The Geometry of Marked Contact Engel Structures

2020 ◽  
Author(s):  
Gianni Manno ◽  
Paweł Nurowski ◽  
Katja Sagerschnig
Keyword(s):  
Local Geometry ◽  
Dimensional Manifold ◽  
Twisted Cubic ◽  
Homogeneous Models ◽  
Local Invariants ◽  
Twisted Cubics ◽  
Cubic Structure ◽  
Contact Distribution ◽  
Engel Structures ◽  
Engel Structure

AbstractA contact twisted cubic structure$$({\mathcal M},\mathcal {C},{\varvec{\upgamma }})$$ ( M , C , γ ) is a 5-dimensional manifold $${\mathcal M}$$ M together with a contact distribution $$\mathcal {C}$$ C and a bundle of twisted cubics $${\varvec{\upgamma }}\subset \mathbb {P}(\mathcal {C})$$ γ ⊂ P ( C ) compatible with the conformal symplectic form on $$\mathcal {C}$$ C . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $$\mathrm {G}_2$$ G 2 . In the present paper we equip the contact Engel structure with a smooth section $$\sigma : {\mathcal M}\rightarrow {\varvec{\upgamma }}$$ σ : M → γ , which “marks” a point in each fibre $${\varvec{\upgamma }}_x$$ γ x . We study the local geometry of the resulting structures $$({\mathcal M},\mathcal {C},{\varvec{\upgamma }}, \sigma )$$ ( M , C , γ , σ ) , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $${\mathcal M}$$ M by curves whose tangent directions are everywhere contained in $${\varvec{\upgamma }}$$ γ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $$\ge 6$$ ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

scholarly journals Notes on open book decompositions for Engel structures

2018 ◽  
Vol 18 (7) ◽  
pp. 4275-4303 ◽  
Author(s):  
Vincent Colin ◽  
Francisco Presas ◽  
Thomas Vogel
Keyword(s):  
Open Book ◽  
Open Book Decompositions ◽  
Engel Structures

scholarly journals Existence h-principle for Engel structures

2017 ◽  
Vol 210 (2) ◽  
pp. 417-451 ◽  
Author(s):  
Roger Casals ◽  
José Luis Pérez ◽  
Álvaro del Pino ◽  
Francisco Presas
Keyword(s):  
Engel Structures

scholarly journals Engel structures on complex surfaces

2021 ◽  
Vol 200 (3) ◽  
pp. 983-997
Author(s):  
N. Pia ◽  
G. Placini
Keyword(s):  
Complex Surfaces ◽  
Engel Structures

scholarly journals Riemannian Properties of Engel Structures

2020 ◽  
Author(s):  
Nicola Pia
Keyword(s):  
Contact Structures ◽  
Totally Geodesic ◽  
Engel Structures ◽  
Engel Structure

Abstract 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 the splitting $TM=\mathcal{D}\oplus \mathcal{R}$ orthogonal and such that $\mathcal{D}$ is totally geodesic then there exists another Reeb distribution, which is integrable. We introduce the notion of K-Engel structures in analogy with K-contact structures, and we classify the topology of K-Engel manifolds. As natural consequences of these methods, we provide a construction that is the analogue of the Boothby–Wang construction in the contact setting, and we give a notion of contact filling for an Engel structure.

A construction of Engel structures

2005 ◽  
Vol 340 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Thomas Vogel
Keyword(s):  
Engel Structures

scholarly journals Holomorphic Engel structures

2013 ◽  
Vol 27 (1) ◽  
pp. 327-344 ◽  
Author(s):  
Francisco Presas ◽  
Luis E. Solá Conde
Keyword(s):  
Engel Structures
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