Locally Conformal Symplectic-Haantjes Manifolds

• LCSH manifolds are even-dimensional spaces equipped with a locally conformal symplectic form, a closed Lee form, and a compatible Haantjes algebra, extending traditional symplectic-Haantjes theory.
• They generalize classical structures by replacing closed symplectic forms with conformal ones, thereby enabling the analysis of dissipative Hamiltonian systems and partial integrability.
• Derived from reductions of Jacobi-Haantjes geometry, these manifolds underpin the formulation of Haantjes chains and offer promising avenues for Hamilton-Jacobi separability in non-conservative dynamics.

Locally conformal symplectic-Haantjes manifolds are even-dimensional manifolds equipped simultaneously with a locally conformal symplectic structure and a compatible Haantjes algebra, so that the induced Jacobi structure fits into the broader framework of Jacobi-Haantjes geometry. In the formulation introduced in “Jacobi-Haantjes manifolds, integrability and dissipation” (Azuaje et al., 15 Jul 2025), they constitute the even-dimensional reduction of Jacobi-Haantjes manifolds along characteristic leaves, and they extend symplectic-Haantjes manifolds by replacing the closed symplectic form with a locally conformal symplectic form whose Lee form need not vanish. Their principal role is to provide an intrinsic geometric setting for Haantjes chains, particular involution, and partial integrability of dissipative Hamiltonian systems.

1. Foundational structures

The basic tensorial ingredient is a $(1,1)$-tensor field $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$. Its Nijenhuis torsion is

$$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$

and its Haantjes torsion is

$$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$

A Haantjes operator is a $(1,1)$-tensor with $\mathcal{H}_K\equiv 0$. A Haantjes algebra $(M,\mathscr{H})$ is a family of Haantjes operators closed under $C^\infty(M)$-linear combinations and under composition; if all elements commute, the algebra is Abelian (Azuaje et al., 15 Jul 2025).

The ambient Jacobi structure is a pair $(\Lambda,E)$ satisfying

$$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$

with induced Jacobi bracket

$K:\mathfrak{X}(M)\to\mathfrak{X}(M)$0

A Jacobi manifold is regular when its characteristic distribution

$K:\mathfrak{X}(M)\to\mathfrak{X}(M)$1

has constant rank; its leaves are locally conformal symplectic in even dimension and contact in odd dimension (Datta et al., 2013).

The locally conformal symplectic side consists of an even-dimensional manifold $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$2, a non-degenerate $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$3-form $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$4, and a closed $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$5-form $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$6, the Lee form, such that

$K:\mathfrak{X}(M)\to\mathfrak{X}(M)$7

Equivalently, locally $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$8 and $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$9 for a symplectic form $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$0. The inverse musical map $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$1 associated with $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$2 defines a Jacobi structure by

$$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$3

The Hamiltonian vector field of $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$4 is determined by

$$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$5

and the time evolution satisfies

$$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$6

so the Hamiltonian is generically dissipated rather than conserved (Azuaje et al., 15 Jul 2025). Standard LCS geometry also admits the twisted differential $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$7, with $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$8 because $$\tau_K(X,Y):=K^2[X,Y]+[KX,KY]-K\big([X,KY]+[KX,Y]\big),$$9 (Otiman et al., 2015).

2. Definition of locally conformal symplectic-Haantjes manifolds

A locally conformal symplectic-Haantjes manifold, abbreviated LCSH manifold, is defined as a quadruple $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$0 such that:

1. $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$1 is an LCS manifold: $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$2 is non-degenerate, $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$3 is closed, and

$$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$4

1. $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$5 is a Haantjes algebra of rank $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$6 on $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$7;
2. every $$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$8 is compatible with the LCS structure in the sense that

$$\mathcal{H}_K(X,Y):=K^2\tau_K(X,Y)+\tau_K(KX,KY)-K\big(\tau_K(X,KY)+\tau_K(KX,Y)\big).$$9

Equivalently,

$(1,1)$0

This is the same linear-algebraic compatibility condition used in symplectic-Haantjes geometry, but imposed on a non-closed LCS form rather than on a closed symplectic form (Azuaje et al., 15 Jul 2025).

The definition is obtained by reduction from Jacobi-Haantjes geometry. A Jacobi-Haantjes manifold of class $(1,1)$1 is a quadruple $(1,1)$2 where $(1,1)$3 is Jacobi, $(1,1)$4 is a Haantjes algebra of rank $(1,1)$5, and every $(1,1)$6 satisfies

$(1,1)$7

When $(1,1)$8 is induced by an LCS pair $(1,1)$9, condition $\mathcal{H}_K\equiv 0$0 is equivalent to $\mathcal{H}_K\equiv 0$1, so LCSH manifolds are precisely the LCS-induced instances of Jacobi-Haantjes manifolds (Azuaje et al., 15 Jul 2025).

The compatibility condition is conformally stable in the local LCS sense. If locally $\mathcal{H}_K\equiv 0$2, then

$\mathcal{H}_K\equiv 0$3

because multiplication by a function does not alter the symmetry property. This implies that, locally, an LCSH manifold is indistinguishable from a symplectic-Haantjes manifold up to conformal rescaling of the $\mathcal{H}_K\equiv 0$4-form (Azuaje et al., 15 Jul 2025).

3. Geometric position within Jacobi, contact, and symplectic-Haantjes geometry

LCSH manifolds occupy the even-dimensional branch of a hierarchy organized by Jacobi geometry. In a regular Jacobi manifold, characteristic leaves are locally conformal symplectic when they are even-dimensional and contact when they are odd-dimensional (Datta et al., 2013). The Jacobi-Haantjes framework refines this by attaching a compatible Haantjes algebra to the Jacobi data. Restricting a Jacobi-Haantjes structure to a characteristic leaf yields either a contact-Haantjes manifold in odd dimension or an LCSH manifold in even dimension (Azuaje et al., 15 Jul 2025).

This placement has two important consequences. First, LCSH manifolds are not ad hoc extensions of symplectic-Haantjes manifolds; they arise naturally as reductions of the general Jacobi-Haantjes structure. Second, their dynamics inherits the Jacobi interpretation of dissipation: the Hamiltonian is generically not conserved, in contrast with the Poisson or symplectic case (Azuaje et al., 15 Jul 2025).

The relation with symplectic-Haantjes geometry is exact when the Lee form vanishes. If $\mathcal{H}_K\equiv 0$5, then $\mathcal{H}_K\equiv 0$6, so $\mathcal{H}_K\equiv 0$7 is symplectic, and the LCSH compatibility condition becomes the defining compatibility for a symplectic-Haantjes, or $\mathcal{H}_K\equiv 0$8, manifold: $\mathcal{H}_K\equiv 0$9 Hence

$(M,\mathscr{H})$0

(Azuaje et al., 15 Jul 2025). The symplectic-Haantjes framework itself was developed as a tensorial setting for Liouville-Arnold integrability, with Abelian Haantjes algebras and Darboux-Haantjes coordinates as central ingredients (Tempesta et al., 2014).

A further link is provided by Poissonization. The Jacobi-Haantjes construction on $(M,\mathscr{H})$1 induces an invertible Poisson-Haantjes structure, hence an $(M,\mathscr{H})$2 manifold, on $(M,\mathscr{H})$3. In particular, for an even-dimensional Jacobi-Haantjes manifold with invertible $(M,\mathscr{H})$4, one recovers a symplectic-Haantjes manifold on the Poissonized space (Azuaje et al., 15 Jul 2025). This shows that LCSH geometry is simultaneously a reduction of Jacobi-Haantjes geometry and a conformal generalization of symplectic-Haantjes geometry.

Local normal-form theory for LCS manifolds strengthens this picture. The Darboux-Weinstein theorem in the LCS setting states that, locally, LCS forms are conformally equivalent to symplectic Darboux forms, and near compact submanifolds one obtains conformal equivalence up to a smooth factor provided the Lee forms agree along the submanifold (Otiman et al., 2015). This local conformal equivalence suggests that many constructions familiar in $(M,\mathscr{H})$5 geometry should admit LCSH analogues after replacing exact symplectic normal forms by conformal ones.

4. Haantjes chains, particular involution, and partial integrability

The principal integrability mechanism in LCSH geometry is the Haantjes chain. Given a Haantjes algebra $(M,\mathscr{H})$6 with distinguished basis $(M,\mathscr{H})$7, a function $(M,\mathscr{H})$8 generates a Haantjes chain of closed $(M,\mathscr{H})$9-forms when

$C^\infty(M)$0

equivalently, when there exist potentials $C^\infty(M)$1 such that

$C^\infty(M)$2

In symplectic-Haantjes geometry, such chains encode commuting first integrals and, in the maximal-rank Abelian case, underlie the Liouville-Haantjes theorem (Tempesta et al., 2014).

In Jacobi-Haantjes geometry the involutivity relation is modified by the Reeb-like field $C^\infty(M)$3. For an Abelian Jacobi-Haantjes manifold, if $C^\infty(M)$4 are the potentials of a Haantjes chain generated by $C^\infty(M)$5, then

$C^\infty(M)$6

When $C^\infty(M)$7, this reduces to ordinary involution; in the general Jacobi case, the functions are in particular involution rather than in Poisson involution (Azuaje et al., 15 Jul 2025).

The LCSH specialization introduces an additional compatibility with the Lee field. Let $C^\infty(M)$8 be the vector field determined by the Lee form. If $C^\infty(M)$9 generates a Haantjes chain with potentials $(\Lambda,E)$0, the LCS Hamiltonian dynamics yields

$(\Lambda,E)$1

Using the Jacobi-Haantjes relation, this becomes

$(\Lambda,E)$2

which is equivalent to

$(\Lambda,E)$3

Under this hypothesis, Theorem 6.1 of (Azuaje et al., 15 Jul 2025) states that for an LCSH manifold,

$(\Lambda,E)$4

or, equivalently,

$(\Lambda,E)$5

Thus the potentials of the chain are in particular involution in the Jacobi sense, now with an explicit geometric condition involving the Lee vector field.

The integrability content of $(\Lambda,E)$6 is not Liouville integrability in the conservative sense. Rather, the functions $(\Lambda,E)$7 define families of particular integrals whose common zero sets are invariant submanifolds, and the restricted dynamics on those submanifolds has reduced degrees of freedom (Azuaje et al., 15 Jul 2025). This is the even-dimensional dissipative counterpart of the partial-separability phenomena studied in symplectic-Haantjes geometry, where non-semisimple or non-maximal-rank structures yield block-separated Hamilton-Jacobi equations in Darboux-Haantjes coordinates (Reyes et al., 2023).

A common misconception is that the presence of an LCS form merely perturbs conservative Haantjes theory by a conformal factor. The LCSH theorem shows otherwise: the Lee field enters the involution relations explicitly, and the relevant notion is particular involution adapted to dissipative Jacobi dynamics, not ordinary Poisson commutativity (Azuaje et al., 15 Jul 2025).

5. Local models, coordinates, and construction patterns

LCSH manifolds inherit their local structure from LCS geometry. Since an LCS form is locally conformal to a symplectic form, one may locally write

$(\Lambda,E)$8

with $(\Lambda,E)$9 (Azuaje et al., 15 Jul 2025). The Darboux-Weinstein theorem for LCS manifolds further implies that locally an LCS form can be represented as

$$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$0

with $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$1 in a suitable chart (Otiman et al., 2015). This local conformal Darboux picture is the natural environment for importing symplectic-Haantjes constructions into the LCS setting.

The construction template identified for LCSH manifolds is explicit, although the foundational paper does not supply a fully worked coordinate example. One begins with an LCS manifold $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$2, for instance with local expression

$$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$3

One then chooses Haantjes operators $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$4 on $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$5 such that $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$6, $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$7, and, if one wishes to apply the LCSH chain theorem, $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$8. In local Darboux-type coordinates $$[\Lambda,\Lambda]_{SN}=2E\wedge\Lambda,\qquad [\Lambda,E]_{SN}=0,$$9, one may mimic the standard $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$00 construction by taking diagonal operators

$K:\mathfrak{X}(M)\to\mathfrak{X}(M)$01

which are Haantjes and compatible with $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$02 (Azuaje et al., 15 Jul 2025). The fact that diagonal operators are Haantjes is standard in the symplectic-Haantjes literature (Tempesta et al., 2014).

Given such operators, a Hamiltonian $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$03 is tested for the existence of a Haantjes chain

$K:\mathfrak{X}(M)\to\mathfrak{X}(M)$04

and the additional condition $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$05 then guarantees the Jacobi-type involution relation $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$06 (Azuaje et al., 15 Jul 2025). The invariant submanifolds determined by these particular integrals are the natural candidates for reduced conservative dynamics.

The absence of explicit LCSH examples in the founding paper is itself a significant datum. Unlike the contact-Haantjes and $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$07 cases, Section 6 of (Azuaje et al., 15 Jul 2025) does not provide a fully worked LCSH manifold with explicit coordinates, forms, and operators. This indicates that the theory is currently structural rather than example-driven. A plausible implication is that the first phase of the subject is devoted to establishing the Jacobi-Haantjes framework and the LCS reduction, while systematic model-building remains open.

6. Research context, flexibility, and prospective developments

The emergence of LCSH manifolds combines three strands of research. The first is classical LCS and Jacobi geometry, in which even-dimensional transitive Jacobi leaves are locally conformal symplectic and the Lee class controls the conformal obstruction (Datta et al., 2013). The second is symplectic-Haantjes geometry, where Abelian Haantjes algebras, Darboux-Haantjes coordinates, and Haantjes chains give a coordinate-free formulation of conservative integrability and separability [(Tempesta et al., 2014); (Nozaleda et al., 2020)]. The third is the extension of Haantjes methods to partial separability, non-semisimple structures, and non-maximal-rank algebras (Reyes et al., 2023).

On open manifolds, the supply of LCS or leafwise LCS backgrounds is large. An $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$08-principle holds for locally conformal symplectic foliations and for regular Jacobi structures on open manifolds, so formal data can often be homotoped to genuine leafwise LCS or Jacobi structures with prescribed Lee class (Datta et al., 2013). This suggests that, on open manifolds, the principal rigidity of an LCSH structure should come from the Haantjes side—vanishing Haantjes torsion, algebra closure, and compatibility with the LCS and Lee data—rather than from the existence of the underlying LCS background.

The Hamilton-Jacobi direction is especially prominent. The LCS Hamiltonian formalism admits a time-dependent Hamilton-Jacobi theory based on the twisted differential $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$09, LCS cotangent models, and Lagrangian sections satisfying $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$10 (Ragnisco et al., 2021). The Jacobi-Haantjes paper identifies the LCSH case as particularly promising for a future Hamilton-Jacobi theory in the generalized Haantjes framework (Azuaje et al., 15 Jul 2025). This suggests an eventual synthesis between LCS Hamilton-Jacobi equations and Haantjes-based separability, although that synthesis is not yet developed in the available articles.

The present conceptual status of LCSH manifolds can therefore be summarized in three points. First, they are rigorously defined as reductions of Jacobi-Haantjes manifolds and as conformal generalizations of $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$11 manifolds (Azuaje et al., 15 Jul 2025). Second, their core integrability statement is the production of families of functions in particular involution under the Lee-field constraint $K:\mathfrak{X}(M)\to\mathfrak{X}(M)$12, furnishing the algebraic basis for partial integrability of dissipative Hamiltonian systems (Azuaje et al., 15 Jul 2025). Third, many local symplectic techniques remain available after conformal localization, but explicit examples, global classification, and a full Hamilton-Jacobi-separation theory for LCSH manifolds remain undeveloped.