Nijenhuis Geometry

• Nijenhuis geometry is the study of (1,1)-tensor fields with vanishing torsion that unifies differential geometry, integrable systems, and algebraic deformation theories.
• It provides local classification, normal forms, and structural insights that facilitate the construction and analysis of integrable nonlinear PDEs and geometric flows.
• By integrating concepts such as Poisson-Nijenhuis pairs and left-symmetric algebras, it offers actionable methods for understanding deformation theory and cohomological properties.

Nijenhuis geometry is the study of tensor fields—particularly $(1,1)$-tensor fields—whose Nijenhuis torsion vanishes. This concept unifies and generalizes a wide range of structures in differential geometry, integrable systems, homotopical algebra, and mathematical physics. At its core are operator fields $L: TM \to TM$ on smooth manifolds $M$ satisfying $\mathcal N_L \equiv 0$, where the Nijenhuis torsion is a specific skew-symmetric tensorial measure of non-integrability. Nijenhuis geometry provides a framework for local and global classification, functional calculus, the theory of compatible metrics and Poisson structures, deformation theory, and the systematic construction of integrable nonlinear PDEs and algebraic hierarchies.

1. Foundational Notions: Nijenhuis Torsion and Operators

Let $L: TM \to TM$ be a $(1,1)$-tensor field on a smooth manifold $M$. The Nijenhuis torsion $\mathcal N_L$ is the $(1,2)$-tensor

$$\mathcal N_L(X,Y) = L^2[X,Y] + [LX, LY] - L\Big([LX, Y] + [X, LY]\Big)$$

for all vector fields $X, Y$ (Bolsinov et al., 2019). $L$ is called a Nijenhuis operator if $\mathcal N_L = 0$ everywhere.

Key properties and consequences of the vanishing torsion condition include:

• The eigen-distributions of $L$ are integrable: if $X, Y$ lie in the same generalized eigenspace, then so does $[X, Y]$ (Bolsinov et al., 5 Oct 2024).
• For any analytic function $f$ defined on the spectrum of $L$, $f(L)$ is again Nijenhuis, and such operators commute pointwise (Bolsinov et al., 2019).
• Functional relations among the invariants of $L$ (e.g., the traces of its powers) are determined by the Nijenhuis condition, yielding relations such as $$d(\operatorname{tr} L^k) = k (L^*)^{k-1} d(\operatorname{tr} L)$$ (Bolsinov et al., 2019).

The Nijenhuis property generalizes the classical integrability criteria for almost complex structures ($J^2 = -\operatorname{Id}$ and $\mathcal N_J=0$), almost product structures, and extends to Courant algebroids, Poisson-Nijenhuis pairs, and higher structures (1208.14817, Kosmann-Schwarzbach, 2011, Azimi et al., 2013, Song et al., 28 Mar 2025).

2. Local Structure, Normal Forms, and Splitting Theorem

At a generic point, every Nijenhuis operator admits a local normal form determined by its spectral properties (Bolsinov et al., 2019, Bolsinov et al., 5 Oct 2024, Bolsinov et al., 2020):

• Diagonal case: If $L$ is real-diagonalizable with simple spectrum, there exist coordinates $(y^i)$ such that $L = \operatorname{diag}(f_1(y^1), ..., f_n(y^n))$ (Bolsinov et al., 5 Oct 2024).
• Companion (Jacobson) form: If the differentials of the coefficients $\sigma_k$ of the characteristic polynomial $\chi_L(t)$ are independent, $L$ admits a companion matrix normal form in the $\sigma_k$ coordinates.
• Splitting theorem: If the spectrum at a point splits, there are local coordinates block-diagonalizing $L$ according to the spectral factors, so $L = \operatorname{diag}(L_1, L_2)$ with both $L_i$ Nijenhuis operators of lower dimension (Bolsinov et al., 2019, Bolsinov et al., 2020, Bolsinov et al., 5 Oct 2024).

gl-regular Nijenhuis operators—those whose minimal and characteristic polynomials coincide, i.e., each eigenvalue occurs in a single Jordan block—admit a particularly rich structure. Any gl-regular Nijenhuis operator admits local coordinates in which it is in first or second companion form, determined uniquely by algebraic data, and its symmetry algebra forms a commutative algebra generated by pointwise matrix multiplication (Bolsinov et al., 2020, Bolsinov et al., 2023).

3. Singularities, Scalar Type Points, and Left-Symmetric Algebras

A special focus of Nijenhuis geometry is the local and global analysis of singular points where the algebraic type of $L$ is non-generic. Notable types include:

• Scalar type points: $L(p) = \lambda \operatorname{Id}$. The linearization at $p$ is governed by a left-symmetric algebra (LSA) structure (also called pre-Lie) on $T_pM$. The structure constants $a^i_{jk}$ of the linear part of $L$ at $p$ satisfy the LSA identities (Konyaev, 2019). The local linearization problem reduces to the classification of degenerate and non-degenerate LSAs.
• gl-regular singular strata: Normal forms for gl-regular but non-diagonalizable $L$ (e.g., single Jordan block, collision loci) have been exhaustively classified in low dimensions, and the corresponding global topological obstructions for such Nijenhuis structures have been identified (Bolsinov et al., 2020).
• Poisson-Nijenhuis singularities: For compatible Poisson-Nijenhuis pairs, a complete local classification of nondegenerate singular points is available, with explicit normal forms using companion and block matrices (Bolsinov et al., 2020).

These singularity theories are not only of intrinsic geometric importance but are fundamental for understanding the structure of integrable systems, the nature of separation variables, and the topology of eigen-distributions.

4. Nijenhuis Geometry and Integrable Systems

Nijenhuis operators serve as recursion operators in both finite- and infinite-dimensional integrable systems:

• Hydrodynamic-type PDEs: A system $u^i_t = L^i{}_j(u) u^j_x$ is integrable (possesses infinitely many commuting flows and conservation laws, admits generalized hodograph reduction) when $L$ is a Nijenhuis operator (Bolsinov et al., 5 Oct 2024, Bolsinov et al., 2020, Bolsinov et al., 2023, Lorenzoni et al., 2022).
• Poisson-Nijenhuis and bi-Hamiltonian hierarchies: Compatible Poisson structures $P_1, P_2$ yield a Nijenhuis recursion operator $R = P_2 P_1^{-1}$; the torsion-free condition guarantees the Magri–Lenard scheme and hierarchical integrability (Bolsinov et al., 5 Oct 2024, Bolsinov et al., 2020).
• Frobenius and F-manifold theory: The Frölicher–Nijenhuis bicomplex $(d, d_L)$, with $d_L$ defined via $L$, systematically produces hierarchies of commuting flows and, in the semisimple case, connects to Lauricella functions and Frobenius manifold flat coordinates (Lorenzoni et al., 2022).
• Conservation laws and symmetries: For gl-regular $L$, symmetries and conservation laws form commutative algebras. Every (generic) conservation law is generated by a single law under action by symmetry operators (Bolsinov et al., 2023).
• Integrable non-autonomous PDEs: The bi-differential graded algebra associated to a pair of commuting Nijenhuis operators controls families of integrable, non-autonomous, nonlinear matrix PDEs via the Frölicher–Nijenhuis bracket and binary Darboux transformations (Müller-Hoissen, 2 Sep 2024).

A broad program, as in (Bolsinov et al., 5 Oct 2024), seeks to classify all integrable hydrodynamic-type flows and geometric Poisson brackets in terms of Nijenhuis data.

5. Operadic and Homotopical Aspects

Nijenhuis geometry extends beyond classical differential geometry into homotopy and deformation theory:

• The operad $\mathfrak{NjL}$ encodes the structure of Nijenhuis Lie algebras (pairs $(\mathfrak{g}, N)$ with $N$ Nijenhuis); its minimal model $\mathfrak{NjL}_\infty$ governs homotopy Nijenhuis Lie algebras (Song et al., 28 Mar 2025).
• The controlling $L_\infty$-algebra encodes simultaneous deformations of the Lie bracket and the Nijenhuis operator, giving rise to a novel cohomology theory for Nijenhuis Lie algebras and their geometric analogs—Nijenhuis Lie algebroids.
• For geometric Nijenhuis structures, the Poincaré lemma holds in the Frölicher–Nijenhuis complex for diagonal Nijenhuis operators, confirming a vanishing cohomology conjecture (Song et al., 28 Mar 2025).

In the broader context, the Richardson–Nijenhuis bracket organizes the deformation theory of $L_\infty$-algebras, Lie $n$-algebras, and multisymplectic structures, arranging all classical and higher Nijenhuis structures in a unified framework (Azimi, 2013, Azimi et al., 2013).

6. Nijenhuis-Type Tensors and Integrability Criteria for Arbitrary Tensors

Nijenhuis geometry also elucidates the integrability (in the sense of admitting local torsion-free parallelizations) of quadratic forms, bivectors, and higher tensors:

• For various tensor types $(p,q)$, integrability is characterized by algebraic constancy together with the vanishing of a Nijenhuis-type torsion $D(\Theta)$, a first-order quasilinear operator generalizing the classical Nijenhuis torsion (Derdzinski et al., 5 Jul 2024).
• In the $(1,1)$-tensor case, vanishing torsion (plus algebraic constancy conditions) is a necessary and sufficient local integrability criterion in generic algebraic types.
• The complex-diagonalizable and nilpotent $(1,1)$-tensors admit especially sharp criteria for integrability in terms of the vanishing of $D(\Theta)$ and involutivity of associated distributions.

This generalized viewpoint provides a unifying perspective on classical results (e.g., Darboux’s theorem for closed forms, integrability of almost complex structures) and extends projectively to the setting of parallelizations and $G$-structures.

7. Global Aspects, Obstructions, and Open Problems

Global classification of Nijenhuis structures intertwines with the topology of the underlying manifold:

• Compactness and the constancy of complex eigenvalues, topological restrictions for gl-regular Nijenhuis operators, and obstructions on closed surfaces and higher-genus manifolds have all been established in precise terms for various cases (Bolsinov et al., 2019, Bolsinov et al., 2020).
• Open problems remain in the classification of singularities and stability, degeneracies of left-symmetric algebras, explicit construction of new global Frobenius manifolds, and the geometric characterization of nontrivial projective and Killing tensor equations in higher order (Bolsinov et al., 5 Oct 2024).

Table: Core Types of Nijenhuis Structures and Associated Frameworks

Type	Defining Structure	Reference
Classical Nijenhuis operator	$(1,1)$-tensor $L:TM \to TM$ with $\mathcal N_L=0$	(Bolsinov et al., 2019)
Nijenhuis Lie algebra	$(\mathfrak g, N)$ with $N:\mathfrak g \to \mathfrak g$, $T_N=0$	(Song et al., 28 Mar 2025)
Courant-Nijenhuis structure	Skew-symmetric $N$ on a Courant algebroid with vanishing torsion	(Kosmann-Schwarzbach, 2011)
Poisson-Nijenhuis structure	Compatible pair $(P,N)$ with $[P,P]=0$, $N$ Nijenhuis, compatibility	(Bolsinov et al., 2020)
$L_\infty$-Nijenhuis form	Degree-0 vector-valued form $N$ with $[N,[N,\mu]] = [K,\mu]$	(Azimi et al., 2013)
Homotopy Nijenhuis Lie algebra	Algebra over the dg operad $\mathfrak{NjL}_\infty$	(Song et al., 28 Mar 2025)

Nijenhuis geometry integrates local analytic and algebraic techniques, global topological analysis, deformation and cohomological methods, and the construction of new classes of integrable PDEs. It is central to current research at the intersection of differential geometry, algebraic deformation theory, and the mathematical theory of integrable systems.