Related papers: The Square of Nijenhuis Tensor and Its Vanishing R…

We express the dual forms of squares of Nijenhuis tensor in terms of the second order component derivatives of the exterior derivative on differential forms and give new vanishing results for the squares of Nijenhuis tensor.

This work is the first, and main, of the series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that should be…

We give several explicit examples of compact manifolds with a $1$-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the $C^0$-norm. The $4$-dimensional examples possess no complex structure, whereas…

We investigate Nijenhuis deformations of $L_\infty$-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie…

In this paper, we generalize some conclusions from the nonnegative irreducible tensor to the nonnegative weakly irreducible tensor and give more properties of eigenvalue problems.

In this note, we state various generalisations of the Nakano vanishing theorem under weak positivity assumptions, and compare them with the known results.

We prove that, on a compact almost complex manifold, the space of almost complex structures whose Nijenhuis tensor has rank at least $k$ at every point is either empty or dense in each path-connected component of the space of almost complex…

We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.

Geometric structures underlying commutative and non commutative integrable dynamics are analyzed. They lead to a new characterization of noncommutative integrability in terms of spectral properties and of Nijenhuis torsion of an invariant…

A tensor -- meaning here a tensor field $\Theta$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of…

Introducing Nijenhuis forms on Lie-infinity algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into a Lie-infinity algebra. Then we…

Exploiting particular features of classical groups, simple constructions are given for the irreducible constituents of the tensor square of the adjoint modules and the leading terms in higher tensor powers. This provides an independent…

We prove that on any symplectic manifold whose symplectic form represents a rational cohomology class there exists a sequence of compatible almost complex structures whose Nijenhuis energy (the $L^2$-norm of the Nijenhuis tensor) tends to…

We write the Kerr-Schild tetrads in terms of the flat space-time tetrads and of a (1,1) tensor $S^\lambda_\mu$. This tensor can be considered as a projection operator, since it transforms (i) flat space-time tetrads into non-flat tetrads,…

A Nijenhuis operator $L$ is a $(1,1)$-tensor field on a smooth manifold $M$ with vanishing Nijenhuis torsion ${ {\mathcal N_L}}$. At each point $x\in M$, the algebraic type of $L(x)$ is characterized by its Jordan normal form. In this…

We are concerned with the tensor equations whose coefficient tensor is an M-tensor. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend…

This is a survey of the spectral theory of tensors.

We discuss an unusual phenomenon in (integral) positive ternary quadratic forms. We also describe an interesting pairing of genera of ternary forms.

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex…

We present existence results for weak solutions to a broad class of degenerate McKean-Vlasov equations with rough coefficients, expanding upon and refining the techniques recently introduced by the third author. Under certain structural…