New advances in the modern field of Contact Geometry

Institute of Mathematical Sciences (ICMAT)

The development of contact geometry is motivated by the fact that in quite a few spaces there is more than one contact structure, thereby posing the question of their classification. In 2015 the group led by ICMAT researcher Fran Presas published a paper in the journal “Annals of Mathematics”, which showed that at least one contact structure existed in every 5 dimensional manifold satisfying an obvious necessary condition. This work was a wake-up call that a few months later led to the proof of the existence of contact structures in any odd-dimensional spaces satisfying the obvious necessary topological condition. Now, Rogel Casals, José Luis Pérez, Álvaro del Pino, and Fran Presas have published an article entitled “Existence h-principle for Engel structures” in Inventiones Mathematicae, that contributes the first theorem on the existence of Engel structures in manifolds of dimension 4.

Contact structures are non-integrable distributions (a continuous selection of “admissible” directions at each point in space) defined in odd-dimensional geometric spaces; they are constructed in spaces or manifolds of dimension 2n+1 and select a linear subspace of 2n directions at each point. For instance, in a manifold of dimension 3, a plane is chosen at each point; one example is the mechanical system modelling a skateboard, which has three degrees of freedom: the two coordinates of its position on the plane, and the angle at which its axis is pointing. The distribution of “admissible” motions at each point is given by the line defined by the axis (one degree of freedom) and the angle (a second degree of freedom). The fact that it can reach any point in the space (moving by admissible motions from one triplet of coordinates to another) is a measure of the non-integrability of the distribution. The analogous objects in dimension 4 are Engel structures and are non-integrable 2-dimensional distributions.

Non-integrable distributions are related to control theory, an interdisciplinary field of engineering and computational mathematics dealing with the behaviour of dynamical systems (those systems that evolve with time, such as the motion of a skateboard or a particle in a fluid) under a series of “controls” that correspond to the admissible directions. The theory studies geometric properties of the set of admissible trajectories and seeks to optimize observables such as the time spent and the energy consumed. Furthermore, non-integrable distributions represent classes of mechanical systems, and thus are closely related to symplectic geometry, which is the field of geometry arising from the study of Newton’s equations under the formalism of differential geometry.

It is for those reasons that non-integrable distributions are placed at the intersection of symplectic geometry, the abstract formalism of mechanical systems, and control theory, which studies the pathways and efficiency of the admissible trajectories of the distributions. A further important actor in play is the h-principle, a set of topological-geometric tools used for determining the existence and subsequent classification of certain classes of geometric structures. The h-principle or homotopy principle emerged in the 1960s for the study of geometric structures that are locally equivalent or isomorphic, but which globally may not be so. This field of research found its initial inspiration in theorems of isometric Euclidean embeddings with low regularity by John Nash (USA, 1928) and was extended to the theory of differentiable immersions by S. Smale (USA, 1930). Mikhail Gromov (Russia, 1943) formulated the general principle underlying these apparently different results. In his book “Partial Differential Relations”, Gromov employed the h-principle to find solutions to many types of partial differential equations.

Contact structures are also related to the theory of relativity and Hamiltonian dynamical systems. Their history goes back to Élie Cartan (France, 1869), who was the first person to study them locally and compare them with other classes of distributions.

This research work took off as an independent field 40 years ago, although it was not until the 1980s that a leap forward was made in their understanding, specifically in the case of 3-dimensional spaces. Since then, interest in this area has continued to grow. Progress in the remaining dimensions began in 2015, when the group led by ICMAT researcher Fran Presas published a paper in the journal “Annals of Mathematics”, which showed that at least one contact structure existed in every 5 dimensional manifold satisfying an obvious necessary condition. This work was a wake-up call that a few months later led to the proof of the existence of contact structures in any odd-dimensional spaces satisfying the obvious necessary topological condition.

What motivated the development of contact geometry is that in quite a few spaces there is more than one contact structure, thereby posing the question of their classification. With that in mind, in the 1980s Gromov and Andreas Floer (Germany, 1956) developed the theory of pseudo-holomorphic curves and their associated invariants, a tool enabling different distributions to be distinguished. The zoology or classification of contact structures is currently a highly active research field because of the large number of examples found in these geometric spaces.

This year, together with Roger Casals, José Luis Pérez and Álvaro del Pino, Fran Presas has published an article entitled “Existence h-principle for Engel structures” in Inventiones Mathematicae. This work contributes the first theorem on the existence of Engel structures in manifolds of dimension 4. These types of geometric spaces possess special characteristics and constitute the less understood class of topologically stable distributions, a concept introduced by E. Cartan, and the study of these authors has applications in general relativity. The first pointer towards a possible application of the h-principle in Engel geometry was provided by T. Vogel in 2009, although it was not until 2017 when Roger Casals, José Luis Pérez, Álvaro del Pino and Fran Presas proved a totally general existence result by employing the homotopy principle. At present, the classification of Engel structures is still an emerging field. As in the case of contact geometry, maturity in the area would be reached if examples of spaces with at least two different Engel structures were to be found.

References:

Roger Casals, Dishant M. Pancholi, Francisco Presas. Almost contact 5-manifolds are contact. Annals of Mathematics. Pages 429-490 from Volume 182 (2015), Issue 2.

Casals, R., Pérez, J.L., del Pino, Á. et al. Existence h-principle for Engel structures. Invent. math. 210: 417 (2017).