Higher Haantjes Brackets and Integrability

We propose a new, infinite class of brackets generalizing the Frölicher–Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We also prove that the vanishing of a higher-level Nijenhuis torsion of an operator field is a sufficient condition for the integrability of its eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.

Maurer-Cartan equation in the DGLA of graded derivations

Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d * and i * components of D. Under the hypothesis that IdT ( M ) + Ψ is invertible we prove that Ψ = b ( Ψ ) = - 1 2 _ ( I d T M + Ψ ) - 1 ∘ [ Ψ , Ψ ] ℱ 𝒩 {\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}} , where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions e Ψ = ℒ Ψ +𝒥b ( Ψ ) of the Maurer-Cartan equation according to their type: e Ψ is of finite type r if there exists r∈ 𝒩 such that Ψr∘ [Ψ, Ψ]𝒡𝒩 = 0 and r is minimal with this property, where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂TM of codimension k ⩾ 1 is integrable if and only if the canonical solution e Ψ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1.

Bundle Prolongations and Connections

A synthetic introduction to the fundamental notions of differential geometry, including tangent, vertical and jet prolongations of fibered manifolds; the Frölicher-Nijenhuis bracket; connections of fibered manifolds and, in particular, linear connections of vector bundles and tangent bundles; the covariant differential of vector-valued forms as a generalisation of the standard covariant derivative.

Frölicher–Nijenhuis cohomology on G2- and Spin(7)-manifolds

In this paper, we show that a parallel differential form [Formula: see text] of even degree on a Riemannian manifold allows to define a natural differential both on [Formula: see text] and [Formula: see text], defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel [Formula: see text]-form on a [Formula: see text]- and [Formula: see text]-manifold, respectively. We calculate the cohomology groups of [Formula: see text] and give a partial description of the cohomology of [Formula: see text].

A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.

Covariant Lie derivatives and Frölicher–Nijenhuis bracket on Lie algebroids

We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher–Nijenhuis bracket on Lie algebroids.

Some applications of the Mangiarotti–Modugno formula for tangent-valued forms

First, we present a classical approach to the general connections on arbitrary fibered manifolds. Then we compare this approach with the use of the Frölicher–Nijenhuis bracket by Mangiarotti and Modugno [Graded Lie algebras and connections on a fibered space, J. Math. Pures Appl. 63 (1984) 111–120]. Finally, we demonstrate that the latter viewpoint is very efficient in the theory of torsions of connections on Weil bundles.