Nijenhuis Torsion in Differential Geometry

• Nijenhuis torsion is a tensor measuring the failure of a (1,1)-tensor to generate an integrable structure, central to differential geometry.
• A vanishing torsion defines Nijenhuis operators, ensuring that deformed brackets satisfy the Jacobi identity and support Lie algebroid integration.
• Generalized torsions such as Haantjes torsion enable block-diagonalization and compatibility in Poisson–Nijenhuis and integrable systems.

A Nijenhuis torsion is a tensorial obstruction associated with a (1,1)-tensor on a differentiable manifold, encoding the failure of a candidate endomorphism to generate a compatible integrable structure. Its theory is central in differential geometry, integrable systems, and the structure of Lie algebroids and Courant algebroids, with generalizations to Banach manifolds and higher-rank tensor contexts. The condition of vanishing Nijenhuis torsion defines Nijenhuis operators, which play a unifying role in the geometry of integrable distributions, Lie groupoid integrations, Poisson–Nijenhuis structures, and generalized Haantjes algebras.

1. Definition and Coordinate Expressions

Let $M$ be a smooth (finite- or Banach-dimensional) manifold and $N: TM \to TM$ a (1,1)-tensor field. The Nijenhuis torsion $T_N \in \Omega^2(M, TM)$ is the skew-symmetric, vector-valued 2-form

$$T_N(X, Y) = [N X, N Y] - N([N X, Y] + [X, N Y]) + N^2 [X, Y]$$

for vector fields $X, Y$. This definition is canonical, independent of coordinates or additional structure (Pugliese et al., 2022, Kosmann-Schwarzbach, 2017, Goliński et al., 2024).

In local coordinates $(x^i)$, with $N(\partial_i) = N^j_i(x) \partial_j$,

$$(T_N)^k{}_{ij} = N^\ell_i \,\partial_\ell N^k_j - N^\ell_j \,\partial_\ell N^k_i - N^k_\ell (\partial_i N^\ell_j - \partial_j N^\ell_i)$$

which immediately yields skew-symmetry in $(i, j)$.

For Banach manifolds equipped with local charts, the same definition transcribes verbatim, with $T_N(X, Y)(p)$ given pointwise by the above formula (Grabowska et al., 29 Sep 2025, Goliński et al., 2024).

The torsion can also be characterized as

$T_N = \frac{1}{2} [N, N]^{FN}$

where $[\cdot,\cdot]^{FN}$ is the Frölicher–Nijenhuis bracket on vector-valued forms (Pugliese et al., 2022, Kawai et al., 2016).

2. Nijenhuis Operators and Geometric Properties

A (1,1)-tensor $N$ is called a Nijenhuis operator if $T_N \equiv 0$. This vanishing condition is precisely the obstruction to $N$ inducing a compatible, integrable deformation of the bracket on vector fields. When $T_N = 0$, the following hold (Pugliese et al., 2022, Kosmann-Schwarzbach, 2017):

• The deformed bracket

$[X, Y]_N = [N X, Y] + [X, N Y] - N([X, Y])$

satisfies the Jacobi identity, making $(TM)_N$ a Lie algebroid with anchor $\rho_N = N$.

• Each generalized eigen-distribution of $N$ is involutive (Frobenius integrable); the vanishing torsion is sufficient for integrability of these distributions without reference to the spectrum (Tempesta et al., 2018).
• In the case of an almost complex structure ($N^2 = -\mathrm{Id}$), the vanishing of the torsion is the Newlander–Nirenberg integrability criterion.

Vanishing Nijenhuis torsion constitutes the infinitesimal condition for integrating such a structure globally. In Lie groupoid integration, a Lie algebroid $(TM)_N$ is integrable if and only if it is isomorphic to the Lie algebroid of some Lie groupoid equipped with a multiplicative (1,1)-tensor $\mathcal U$ whose Frölicher–Nijenhuis torsion vanishes. The correspondence between $N$ and $\mathcal U$ establishes the passage from infinitesimal (algebroid) to global (groupoid) structure (Pugliese et al., 2022).

3. Generalizations: Banach Manifolds and Homogeneous Spaces

The Nijenhuis torsion admits a fully parallel extension to the category of Banach manifolds and Banach–Lie groups. Let $G$ be a Banach–Lie group, $K$ a Banach–Lie subgroup, and $M = G/K$. For a suitable bounded operator $N$ on $\mathfrak g = \mathrm{Lie}(G)$ (including "admissibility" with respect to $K$), there is a unique $G$-equivariant (1,1)-tensor $\mathcal N: TM \to TM$ whose torsion is determined by the algebraic relation

$$T_\mathcal{N}(X_v, X_w)(p) = (a_g)_{* \pi(1)} \left( N [v, N w] + N [N v, w] - [N v, N w] - N^2 [v, w] \right)$$

for any projected fields $X_v, X_w$, with the vanishing condition $T_\mathcal{N} \equiv 0$ translating to a Lie-algebraic closure constraint (Goliński et al., 2024). In Banach fibrations, projectability and verticality of torsion precisely control the correspondence between base and total structures (Grabowska et al., 29 Sep 2025).

For homogeneous almost complex structures, induced by bounded operators $J$ with $J^2 = -\mathrm{Id}$, integrability reduces to the property that a certain $Z_+ \subset \mathfrak g^\mathbb{C}$ is a complex Lie subalgebra, echoing finite-dimensional results (Goliński et al., 2024).

4. Higher Nijenhuis Torsions and Polarizations

Tempesta–Tondo and Reyes Nozaleda developed generalizations—higher-level (or generalized) Nijenhuis torsions $T_N^{(m)}$—which yield a graded hierarchy of tensorial obstructions: $$T_N^{(m)}(X, Y) = \sum_{p, q = 0}^{m} (-1)^{p+q} \binom{m}{p} \binom{m}{q} N^{p+q} [N^{m-p}X, N^{m-q}Y]$$ with $T_N^{(1)}$ the classical torsion and $T_N^{(2)}$ the Haantjes torsion. The vanishing $T_N^{(m)} = 0$ ensures progressively stronger integrability: all generalized eigen-distributions of $N$ and their sums are involutive, and there exist local coordinates in which $N$ is block-diagonal (Tempesta et al., 2018, Nozaleda et al., 2022, Tempesta et al., 2022).

Polarization of these torsions—specifically, the Frölicher–Nijenhuis bracket as the polarization of the classical torsion, and analogous higher-bracket polarizations—organize the module-theoretic structure of operator fields and their compatibility in Haantjes modules (Tempesta et al., 2022).

5. Applications: Integrability, Geometry, and Singularities

Nijenhuis torsion arises in multiple geometric and analytic contexts:

• Integrable Systems: Vanishing Nijenhuis torsion of recursion operators enables the construction of integrable hierarchies via Lenard–Magri chains and ensures commutativity of hydrodynamic flows (Lorenzoni et al., 2022).
• Block-Diagonalization: Families of commuting generalized Nijenhuis operators with vanishing high-level torsion can be simultaneously block-diagonalized in suitable local charts, generalizing classical results for diagonalizable operators (Nozaleda et al., 2022).
• Singularity Theory: The local classification of singularities, especially in dimension two (normal forms, degeneracy loci, canonical types), is controlled by the smooth extendability of certain analytic expressions involving the torsion (Akpan, 14 Mar 2025, Bolsinov et al., 2020, Konyaev, 2019).
• Poisson–Nijenhuis and Courant algebroids: Vanishing of the Nijenhuis torsion is central to the compatibility of Poisson and Nijenhuis structures, construction of hierarchies in algebroid and Courant algebra settings, and deformation of Courant structures (Antunes et al., 2012, Kosmann-Schwarzbach, 2011, Aldi et al., 2023).

6. Relationship with the Frölicher–Nijenhuis Bracket and Higher-Structure Analogs

The Nijenhuis torsion arises as half the Frölicher–Nijenhuis bracket $[N, N]^{FN}$ for (1,1)-tensors. This bracket structure underlies the construction of bidifferential bicomplexes (Frölicher–Nijenhuis bicomplexes) whose exactness is equivalent to the vanishing torsion condition. The extension to vector-valued k-forms—including applications to the intrinsic torsion of $G_2$ and Spin(7) structures in Riemannian geometry—shows that vanishing Frölicher–Nijenhuis brackets characterize torsion-free special holonomy structures and realize classical classifications (Fernández–Gray, Fernández) in higher dimensions (Kawai et al., 2016).

Similarly, for Courant algebroids, the shifted Courant–Nijenhuis torsion is the maximal tensorial integrability condition that applies to general skew-symmetric endomorphisms, refining the conventional torsion concept (Aldi et al., 2023).

7. Examples and Normal Forms

The general theory is illuminated by explicit normal forms and concrete models:

• Trivial and invertible Nijenhuis operators lead to the abelian tangent bundle Lie algebroid and standard complex or product structures.
• Pre-Lie algebra structures produce linear Nijenhuis operators, with vanishing torsion equivalent to left-symmetry.
• Singularities in two dimensions display fold, cusp, and higher degeneracies, fully classified by analytic invariants of the torsion smoothness (Akpan, 14 Mar 2025, Bolsinov et al., 2020).
• Cyclic and higher-level Haantjes algebras show the effectiveness of generalized torsion theory in generating and block-diagonalizing families of operators (Nozaleda et al., 2022, Tempesta et al., 2018).

• "Integrating Nijenhuis Structures," Pugliese–Sparano–Vitagliano (Pugliese et al., 2022)
• "Nijenhuis operators on Banach homogeneous spaces," Goliński–Larotonda–Tumpach (Goliński et al., 2024)
• "Polarization of generalized Nijenhuis torsions," Tempesta–Tondo (Tempesta et al., 2022)
• "Singularities of two-dimensional Nijenhuis operators," (Akpan, 14 Mar 2025)
• "Higher Haantjes Brackets and Integrability," Tempesta–Tondo (Tempesta et al., 2018)
• "Nijenhuis geometry of parallel tensors," (Derdzinski et al., 2024)
• "A note on the shifted Courant-Nijenhuis torsion," Aldi–da Silva–Grandini (Aldi et al., 2023)
• "Nijenhuis operators on Banach fibration," (Grabowska et al., 29 Sep 2025)
• "Nijenhuis and Compatible Tensors on Lie and Courant algebroids," Antunes–Nunes da Costa (Antunes et al., 2012)
• "Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators," (Konyaev, 2019)
• "Nijnehuis Geometry III: gl-regular Nijenhuis operators," Bolsinov–Konyaev–Matveev (Bolsinov et al., 2020)
• "Frölicher-Nijenhuis bracket and geometry of $G_2$-and ${\rm Spin}(7)$-manifolds," Kawai–Lê–Schwachhöfer (Kawai et al., 2016)
• "Generalized Nijenhuis Torsions and block-diagonalization of operator fields," Reyes Nozaleda–Tempesta–Tondo (Nozaleda et al., 2022)