arXiv research

The paper solves the Integration Problem for principal connections.

In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connect…

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an…

Study integrability of specific geometric structures on odd Courant algebroids.

New definition of Born geometry connects to known geometries.

Study of Tannakian categories for integrable connections on Kaehler manifolds.

Study connects landslide flow to integrable systems for harmonic maps.

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numeri…

Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi-Civita connectio…

This paper explores integrability conditions for generalized metrics and structures on manifolds.

The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable multivector fields in $E$ are equivalent to integrable connections in the bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is applied to the part…

In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.

Study integrability of generalized almost complex structures on S^6.

Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental's work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for the Dubrovin connection. An explicit calculation is given for projective space.

Study of multiplicative connections in Lie groupoids.

Researchers calculate the Ray-Singer Torsion for $S^{1}$ bundles.

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fernández and Gray. This relies on the construction of some G_2 solvmanifolds, whose Levi-Civita connection is known to give a paralle…

Formula connects foliated simplicial volume with group cost.

Integrates Manin pairs to simplify Poisson and symplectic groupoid constructions.

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension $\hat{G}$ of $G$ by $A$, where $G$ is a connected, simply connected Lie group and $A$ is a quotient of its Lie algebra by some discrete subgroup. When $G$ is non-simply connected…

For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $Δ$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2,…

Integrable hierarchies linked to F-manifolds with compatible connection.

Study curvature properties of $G_2$ connections with skew-symmetric torsion.

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^\star$ we obtain a suitably connected integrating symplectic double groupoid $\calS$. As a consequence, the cotangent lift …

We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence …

We show that the integral foliated simplicial volume of a connected compact oriented smooth manifold with a regular foliation by circles vanishes.

We define a 1-cocycle in the space of long knots that is a natural generalization of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalizes the Knizhnik--Zamolodchikov connection. We show that the well-known close relationship between the Kontsevich integral and Vassiliev invariants (via the…

We study the classification of closed, smooth, spin, $1$-connected $7$-manifolds whose integral cohomology ring is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology ring of a closed, smooth, spin, $1$-connected $7$-manifold is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$ or $H^…

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivarian…

Study on SASI-lightlike submanifolds in indefinite Kaehler manifolds.

New connections found for quaternionic and para-quaternionic structures.

We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the semi-direct product associated to an infinitesimal action of one integrable Lie algebroid on another is integrable. This genera…

We introduce an algebra bundle of chord diagrams over the configuration space of N points in the complex plane on which we put the Knizhnik-Zamolodchikov connection. For that particular connection, the holonomy along a loop in the base is shown to be generating the Kontsevich integral for that loop's associated braid.

If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

New connections on symmetric spaces with invariant properties.

We study the types of non-integrable $\mathrm{G}$-structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a $\Spin(7)$-structure play a special role. Any geometry of that type admits a unique c…

The contribution of reducible connections to the U(N) Chern-Simons invariant of a Seifert manifold $M$ can be expressed in some cases in terms of matrix integrals. We show that the U(N) evaluation of the LMO invariant of any rational homology sphere admits a matrix model representation which agrees with the Chern-Simon…

Introduces generalized almost plastic structures on manifolds.

We investigate the integrability of almost complex structures on the twistor space of an almost quaternionic manifold constructed with the help of a quaternionic connection. We show that if there is an integrable structure it is independent on the quaternionic connection. In dimension four, we express the anti-self-dua…

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connect…

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …

Study on flat connections with controlled irregularity.

Study integrable discretizations of cyclic systems with circular coordinate lines.

Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we tu…

This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics--in particular superstring theory--where these naturally appear. Connections with skew-symmetric torsion…

CR embeddings in complex spaces for specific Lie groups.

We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequ…