---
title: 'Jacobi–Haantjes Manifolds: Unified Integrability'
url: https://www.emergentmind.com/topics/jacobi-haantjes-manifolds
type: topic
---

# Jacobi–Haantjes Manifolds: Unified Integrability

Searching arXiv for the specified paper and closely related Haantjes-geometry work.
Jacobi-Haantjes manifolds are a class of geometric structures introduced to formulate the integrability of conservative and dissipative Hamiltonian systems within a single tensorial framework. In the formulation proposed by R. Azuaje and P. Tempesta, the Jacobi data $(M,\Lambda,E)$ encode the Hamiltonian dynamics, while Haantjes operators organize compatible recursion-type structures and the associated chains of first integrals or particular integrals. The theory includes an extended version, adapted to the Jacobi setting, and admits as reductions contact-Haantjes and locally conformal symplectic-Haantjes manifolds. In this setting, complete integrability of contact Hamiltonian systems and partial integrability on invariant submanifolds are expressed through Haantjes chains and their extended analogues [2507.11715].

## 1. Jacobi geometry and Haantjes structures

A Jacobi manifold is a triple $(M,\Lambda,E)$, where $M$ is a smooth manifold, $\Lambda$ is a smooth bivector field, and $E$ is a smooth vector field such that
$$
[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0.
$$
The pair $(\Lambda,E)$ is a Jacobi structure. It induces on $C^\infty(M)$ the Jacobi bracket
$$
\{f,g\}=\Lambda(df,dg)+fEg-gEf,
$$
and for each $f\in C^\infty(M)$ the Hamiltonian vector field is
$$
X_f=\Lambda^\sharp(df)-fE.
$$
Along the flow of $X_H$, observables evolve according to
$$
\frac{d}{dt}f=X_Hf=\{f,H\}-fEH.
$$
The characteristic distribution generated by the Hamiltonian vector fields $\{X_f\}$ is Stefan-Sussmann integrable, and each leaf inherits a transitive Jacobi structure, as stated through Kirillov’s theorem [2507.11715].

The Haantjes-theoretic side of the construction starts from a $(1,1)$-tensor $K$ on $M$. Its Nijenhuis torsion $\tau_K$ and Haantjes tensor $H_K$ are
$$
\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K([X,KY]+[KX,Y]),
$$
$$
H_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).
$$
A Haantjes operator is a tensor $K$ with $H_K\equiv 0$. A Haantjes algebra is a pair $(M,\mathfrak{h})$ in which $\mathfrak{h}$ is a set of Haantjes operators forming a free $C^\infty(M)$-module and a ring under composition. If the operators commute pairwise, one has an Abelian Haantjes algebra; in that case there exist Haantjes coordinates in which all $K\in\mathfrak{h}$ are simultaneously block-diagonal, and diagonal if semisimple. In the symplectic-Haantjes background, for a symplectic form $\omega$ with bundle isomorphism $\Omega:=\omega^\flat$, the algebraic compatibility is
$$
\Omega K=K^T\Omega,\qquad \forall K\in\mathfrak{h}.
$$

## 2. Extended Jacobi-Haantjes manifolds and their reduction

To combine Jacobi geometry and Haantjes geometry, the theory introduces extended operators acting on the $C^\infty(M)$-module $\mathfrak{X}(M)\times C^\infty(M)$. Given a quadruple $(K,Y,\gamma,k)$, the associated extended operator is
$$
\mathcal{K}(X,f):=(KX+fY,\gamma(X)+kf).
$$
Its transpose $\mathcal{K}^T$ acts on $\Omega^1(M)\times C^\infty(M)$ through
$$
(\mathcal{K}^T(\alpha,f))(X,h):=(\alpha,f)(\mathcal{K}(X,h)).
$$
The Jacobi sharp map is defined by
$$
(\Lambda,E)^\sharp:T^*M\times \mathbb{R}\to TM\times \mathbb{R},\qquad
(\Lambda,E)^\sharp(\alpha,f):=(\Lambda^\sharp(\alpha)+fE,-\alpha(E)).
$$
Extended Nijenhuis and Haantjes torsions are defined by exact analogues using the Lie algebra structure on $\mathfrak{X}(M)\times C^\infty(M)$ [2507.11715].

An extended Haantjes algebra is a set $\mathfrak{h}$ of extended operators $\mathcal{K}$ such that $H(\mathcal{K})\equiv 0$ and the set is closed under $C^\infty(M)$-linear combinations and composition. An extended Jacobi-Haantjes manifold of class $m$ is then a quadruple $(M,\Lambda,E,\mathfrak{h})$ where $(M,\Lambda,E)$ is a Jacobi manifold and $\mathfrak{h}$ is an extended Haantjes algebra of rank $m$ satisfying the compatibility condition
$$
\mathcal{K}\circ(\Lambda,E)^\sharp=(\Lambda,E)^\sharp\circ \mathcal{K}^T,\qquad \forall \mathcal{K}\in\mathfrak{h}.
$$

The corresponding chain concept is likewise extended. A function $H$ generates an extended Haantjes chain if there exists a distinguished basis $\{\mathcal{K}_1,\dots,\mathcal{K}_m\}$ of $\mathfrak{h}$ and functions $H_i$ such that
$$
(dH_i,H_i)=\mathcal{K}_i^T(dH,H),\qquad i=1,\dots,m.
$$
If $\mathcal{K}_i=(K_i,Y_i,\gamma_i,k_i)$, this expands to
$$
dH_i=K_i^TdH+H\gamma_i,\qquad H_i=Y_iH+Hk_i.
$$
The main involution theorem states that if $(M,\Lambda,E,\mathfrak{h})$ is an extended Jacobi-Haantjes manifold and $\mathcal{C}$ is an extended Abelian Haantjes chain generated by $H$, then
$$
\{H_i,H\}=0,\qquad \{H_i,H_j\}=0.
$$

The reduced notion is a Jacobi-Haantjes manifold of class $m$, defined as a quadruple $(M,\Lambda,E,\mathfrak{h})$ in which $\mathfrak{h}$ is now a Haantjes algebra of $(1,1)$-operators on $TM$ satisfying
$$
K\Lambda=\Lambda K^T,\qquad \forall K\in\mathfrak{h}.
$$
This is the Jacobi analogue of the symplectic compatibility condition and reduces to a Poisson-Haantjes manifold when $E=0$. It is also the reduction of the extended theory when $Y=\gamma=k=0$.

## 3. Haantjes chains, particular involution, and partial integrability

On a Haantjes algebra $(M,\mathfrak{h})$ with basis $\{K_1,\dots,K_m\}$, a function $H$ generates a Haantjes chain if
$$
d(K_\alpha^TdH)=0,\qquad \alpha=1,\dots,m,
$$
so that locally one has exact potentials $H_\alpha$ satisfying
$$
dH_\alpha=K_\alpha^TdH.
$$
The geometric characterization used in the theory is formulated through the codistribution
$$
\mathcal{D}_H^\circ:=\operatorname{Span}\{K_1^TdH,\dots,K_m^TdH\}.
$$
A function $H$ generates a chain if and only if $\mathcal{D}_H^\circ$, equivalently its annihilator distribution, is Frobenius integrable [2507.11715].

On an Abelian Jacobi-Haantjes manifold, the chain potentials satisfy not exact involution in general but the identity
$$
\{H_\alpha,H_\beta\}=H_\alpha EH_\beta-H_\beta EH_\alpha.
$$
This is termed “particular involution.” The same framework leads to the notion of particular integrals: functions $f_1,\dots,f_k$ are particular integrals for $(M,\Lambda,E,H)$ if
$$
\{f_i,H\}=\sum_{j=1}^k a_j^if_j
$$
for suitable functions $a_j^i$. The common zero level set
$$
M_f:=\{x\in M: f_1=\cdots=f_k=0\}
$$
is then invariant under $X_H$. If the $f_i$ are functionally independent and also satisfy
$$
\{f_i,f_j\}=\sum_{\ell=1}^k a_\ell^{ij}f_\ell,
$$
the dynamics reduces on $M_f$, with the number of degrees of freedom reduced by $k$, and the reduced system may be integrable by quadratures on $M_f$.

Within the Jacobi-Haantjes setting, the chain potentials satisfy
$$
\frac{d}{dt}H_\alpha=-HEH_\alpha,
$$
so they are particular integrals. In the Poisson case $E=0$, the identity above reduces to the standard involutivity relation $\{H_\alpha,H_\beta\}=0$. Since the characteristic distribution of a Jacobi manifold is integrable and its leaves are contact or locally conformal symplectic, this suggests that Jacobi-Haantjes chains provide a mechanism for organizing integrability either globally or leafwise, depending on the geometry of the characteristic foliation.

## 4. Contact-Haantjes manifolds and dissipative contact dynamics

A contact manifold is a pair $(M^{2n+1},\theta)$ with
$$
\theta\wedge(d\theta)^n\neq 0.
$$
The Reeb vector field $R$ is characterized by
$$
R\lrcorner \theta =1,\qquad R\lrcorner d\theta=0.
$$
With the bundle map
$$
\flat:\mathfrak{X}(M)\to \Omega^1(M),\qquad \flat(V)=V\lrcorner d\theta +(V\lrcorner \theta)\theta,
$$
and inverse $\sharp$, the associated Jacobi structure is
$$
\Lambda(\alpha,\beta)=d\theta(\sharp\alpha,\sharp\beta),\qquad E=\sharp\theta=R.
$$
In Darboux coordinates $(q^i,p_i,z)$,
$$
\theta=dz-p_i\,dq^i,\qquad
\Lambda=(\partial_{q^i}+p_i\partial_z)\wedge \partial_{p_i},\qquad
E=\partial_z.
$$
The contact Hamiltonian vector field $X_f$ satisfies
$$
\theta(X_f)=-f,\qquad X_f\lrcorner d\theta=df-(Rf)\theta,
$$
and the evolution law becomes
$$
\frac{d}{dt}f=\{f,H\}-fRH,\qquad \frac{d}{dt}H=-HRH.
$$
Thus $H$ is a dissipated quantity [2507.11715].

A contact-Haantjes manifold is a triple $(M,\theta,\mathfrak{h})$ in which $\mathfrak{h}$ is a Haantjes algebra of $(1,1)$-operators $K$ satisfying, for all $K\in\mathfrak{h}$,
$$
d\theta(KX,Y)=d\theta(X,KY),\qquad \theta(KX)\theta(Y)=\theta(X)\theta(KY).
$$
These identities follow from the Jacobi-Haantjes compatibility $K\Lambda=\Lambda K^T$ under the contact identifications, and it is convenient to assume
$$
\sharp K^T=K\sharp,
\qquad\text{equivalently}\qquad
\flat K=K^T\flat.
$$
A key auxiliary condition used in the theory is
$$
\theta(KX_f)=-f\,\theta(KR),\qquad \forall f\in C^\infty(M),
$$
which in particular implies $\theta(KX_H)=-H\,\theta(KR)$.

Under that condition, if $\{H_i\}$ are the potentials of a Haantjes chain generated by $H$, then
$$
\{H_i,H_j\}=H_iRH_j-H_jRH_i.
$$
Two special subclasses are singled out. In contact-Haantjes manifolds of the first kind, one has
$$
d\theta(KX,Y)=d\theta(X,KY),\qquad \theta(KX_f)=0,\ \forall f,
$$
and therefore
$$
RH_i=0,\qquad \{H_i,H_j\}=0,\qquad \{H_i,H\}=H_iRH.
$$
In those of the second kind, the condition
$$
\theta(KX_f)=-f\,\theta(KR)
$$
is imposed for all functions homogeneous of degree $0$ in the momenta in Darboux coordinates; when $H$ and the $H_i$ are degree-$0$ homogeneous, one gets
$$
\{H_i,H_j\}=H_iRH_j-H_jRH_i,\qquad \{H_i,H\}=-HRH_i.
$$
These formulas make explicit how dissipation deforms involution in contact Hamiltonian systems.

## 5. Locally conformal symplectic-Haantjes manifolds

A locally conformal symplectic manifold is a triple $(M^{2n},\Omega,\eta)$ where $\Omega$ is a nondegenerate $2$-form and $\eta$ is a closed $1$-form satisfying
$$
d\Omega=\eta\wedge \Omega.
$$
The associated Jacobi structure is
$$
\Lambda(\alpha,\beta)=\Omega(\sharp\alpha,\sharp\beta),\qquad E=\sharp\eta.
$$
For $f\in C^\infty(M)$, the Hamiltonian vector field is determined by
$$
X_f\lrcorner \Omega = df-f\eta.
$$
The Jacobi bracket can be written as
$$
\{f,g\}=\Omega(X_f,X_g)=X_gf-f\eta(X_g),
$$
and the evolution along $X_H$ is
$$
\frac{d}{dt}f=\{f,H\}+f\eta(X_H),\qquad \frac{d}{dt}H=H\eta(X_H).
$$
This is the even-dimensional Jacobi counterpart of the contact case [2507.11715].

A locally conformal symplectic-Haantjes manifold is a quadruple $(M,\Omega,\eta,\mathfrak{h})$ in which $\mathfrak{h}$ is a Haantjes algebra on $TM$ satisfying
$$
\Omega(KX,Y)=\Omega(X,KY),\qquad \forall K\in\mathfrak{h}.
$$
If, in addition,
$$
\eta(KE)=0,\qquad \forall K\in\mathfrak{h},
$$
then the potentials $\{H_i\}$ of a Haantjes chain obey
$$
\{H_i,H_j\}=H_iEH_j-H_jEH_i.
$$
Equivalently, in Jacobi notation with $E=\sharp\eta$, the same particular-involution identity as in the general Jacobi-Haantjes case is recovered. This places locally conformal symplectic geometry within the same Haantjes-based integrability scheme as contact geometry, but in even dimension.

## 6. Conservative limit, Poissonization, and model examples

When $E=0$, a Jacobi-Haantjes manifold becomes a Poisson-Haantjes manifold $(M,P,\mathfrak{h})$ with $P=\Lambda$ and
$$
KP=PK^T.
$$
If $P$ is invertible, this is an $\omega\mathcal{H}$ manifold $(M,\omega,\mathfrak{h})$ with $\omega=P^{-1}$ and
$$
\Omega K=K^T\Omega.
$$
In that conservative limit, the Haantjes-chain potentials satisfy the ordinary involutivity relation
$$
\{H_i,H_j\}=0,
$$
and the symplectic-Haantjes machinery, including Darboux-Haantjes coordinates, is recovered [2507.11715].

The relation between Jacobi and Poisson geometry is made explicit by Poissonization. Given $(M,\Lambda,E)$, on $M\times \mathbb{R}$ with coordinate $t$ one defines
$$
\widetilde{P}=e^{-t}\big(\Lambda+\partial_t\wedge E\big).
$$
Then the Jacobi and Poisson brackets are related by
$$
\{f,g\}_{Jacobi}=e^t\{\widetilde{f},\widetilde{g}\}_{\widetilde{P}}\big|_{t=0},
$$
where $\widetilde{f},\widetilde{g}$ are the liftings. A Jacobi-Haantjes structure on $M$ induces an invertible Poisson-Haantjes, equivalently $\omega\mathcal{H}$, structure on $M\times \mathbb{R}$.

A basic contact-integrable example is given on the $3$-dimensional contact manifold with Darboux coordinates $(q,p,z)$, contact form
$$
\theta=dz-p\,dq,
$$
and Hamiltonian
$$
H(q,p,z)=p-z.
$$
An extended Abelian Haantjes chain of length $2$ is constructed by taking $\mathcal{K}_1$ to be the extended identity with
$$
K=\mathrm{Id},\qquad k=1,\qquad Y=0,\qquad \gamma=0,
$$
and $\mathcal{K}_2$ with
$$
K=\mathrm{Id},\qquad Y=p\,\frac{\partial}{\partial p},\qquad \gamma=0,\qquad k=0.
$$
The corresponding potentials are
$$
H_1=H,\qquad H_2=p.
$$
They are independent dissipated quantities and satisfy
$$
\{H_1,H_2\}=0,\qquad \{H_\alpha,H\}=0
$$
in the extended Jacobi-Haantjes setting.

The paper also constructs explicit families of Haantjes operators on $5$-dimensional contact manifolds, represented by matrices $F_1$, $F_2$, and $F_3$, which are compatible or quasi-compatible with the contact structure and generate Abelian or non-Abelian Haantjes algebras depending on the choice of arbitrary functions. Together with the regularity assumptions that all tensors are smooth, that $\Omega$ is nondegenerate and $\eta$ is closed in the locally conformal symplectic case, and that $\theta\wedge(d\theta)^n\neq 0$ in the contact case, these constructions exhibit the nontriviality of the framework. The overall picture is that extended Jacobi-Haantjes manifolds govern complete integrability of contact Hamiltonian systems through extended Haantjes chains, whereas Jacobi-Haantjes manifolds encode partial integrability through particular integrals and invariant submanifolds, thereby unifying conservative and dissipative dynamics within a single Haantjes-based formalism.

Source: https://www.emergentmind.com/topics/jacobi-haantjes-manifolds