A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact Hamiltonian systems

In this paper we study the coisotropic reduction in diﬀerent types of dynamics according to the geometry of the corresponding phase space. The relevance of the coisotropic reduction is motivated by the fact that these dynamics can always be interpreted as Lagrangian or Legendrian submanifolds.


INTRODUCTION
momentum mapping when in the presence of symmetries such as the so-called coisotropic reduction [1,4,9,26]. Another relevant example, in the quantitative aspects, is the development of geometric integrators that respect geometric aspects and prove to be more efficient than the traditional ones (see for instance [28,31]).
Regarding the reduction in the presence of symmetries, the most relevant result is the socalled Marsden-Weinstein symplectic reduction theorem [27] (a preliminary version can be found in Meyer [30]), using the momentum mapping, a natural extension of the classical linear and angular momentum. The reduced manifold is obtained using a regular value of the momentum mapping and the corresponding isotropy group, and the dynamics is projected to this reduced manifold, gaining for integration a smaller number of degrees of freedom. This theorem has been extended to many other contexts: cosymplectic, contact, and more general settings (see [2,3,7,10,19,24,29,42] and the references therein).
On the other hand, Lagrangian submanifolds play a crucial role, since it is easy to check that the image of a Hamiltonian vector field X H in a symplectic manifold (M, ω) can be interpreted as a Lagrangian submanifold of the symplectic manifold (T M, ω c ), where ω c is the complete or tangent lift of ω to the tangent bundle T M. This result has its equivalent in Lagrangian mechanics, and has led to the so-called Tulczyjew triples, which elegantly relate the different Lagrangian submanifolds that appear in Lagrangian and Hamiltonian descriptions of mechanics via the Legendre transformation [9,37,38]. These constructions have been extended to other scenarios, including the Tulczyjew triple [8, 14, 16-18, 21, 22, 42]. Lagrangian submanifolds are also relevant to develop the so-called Hamilton-Jacobi theory since they provide the geometric setting for solutions of the Hamilton-Jacobi problem (see [15] for a recent topical review on the subject). In this sense, we follow the Weinstein's creed: "Everything is a Lagrangian submanifold" [40].
As we said, the coisotropic reduction works when we give a coisotropic submanifold N of a symplectic manifold (M, ω) and we consider (if it is well defined) the quotient manifold N/(T N) ⊥ , where (T N) ⊥ is the symplectic complement of T N. Being involutive, this distribution along N defines a foliation. The corresponding leaf space is again symplectic with a reduced symplectic form of the one given in M. If in addition we have a Lagrangian submanifold L with clean intersection with N, then L ∩ N projects into a Lagrangian submanifold of the quotient (see [1,40]). Coisotropic reduction can be combined with symplectic reduction to develop a reduction procedure for the Hamilton-Jacobi equation when we are in presence of symmetries (see [11]).
Coisotropic reduction has been extended to the field of contact manifolds (with the interest of being in a dissipative context) [7,36], but it has not been studied in sufficient detail in the case of cosymplectic manifolds nor in that of co-contact manifolds, the latter the natural settings to study time-dependent Hamiltonian contact systems [6,12].
The objectives of this paper are twofold. On the one hand, to develop in detail the coisotropic reduction in the case of cosymplectic manifolds and those of co-contact, covering a gap in the literature. On the other hand, to present a survey that brings together in one place the different cases that appear in the study of Hamiltonian systems.

SYMPLECTIC VECTOR SPACES
The paper is structured as follows. Sections 2 and 3 are devoted to recall the main ingredients concerning symplectic Hamiltonian systems and the classical coisotropic reduction procedure. In order to go to the cosymplectic setting, we recall some general notions in Poisson structures (Section 4) and then we consider the case of coisotropic reduction in the cosymplectic setting in Section 5 (remember that this is the scenario to develop time dependent Hamiltonian systems). Contact manifolds require a more general notion that Poisson structures, indeed, they are examples of Jacobi structures, so that we give some fundamental notions in Section 6. The coisotropic reduction scheme developed in contact manifolds is the subject of Section 7, which is very different to the cosymplectic case since we are in presence of dissipative systems. To combine dissipative systems with Hamiltonians depending also on time, we consider cocontact manifolds in Section 8, and develop there the corresponding coisotropic reduction procedure. Finally, we discuss a recent generalization of contact and cosymplectic systems called stable Hamiltonian systems in Section 9.

Definition 2.1 (Symplectic vector space). A symplectic vector space is a pair (V, ω)
where V is a finite dimensional vector space and ω is a non-degenerate 2-form, where non-degenerary means that the map is an isomorphism. ω will be called a symplectic form.
For every non-degenerate 2-form on V there exist a basis (x i , y i ) such that ω = x i ∧ y i , where (x i , y i ) is the dual basis. This implies that a symplectic vector space is necessarily of even dimension, say 2n. Definition 2.2 (ω-orthogonal). Let W ⊆ V be a subspace of V . We define its ωorthogonal complement as Note that W ⊥ω = Ker(i * ♭ ω ) where i : W ֒→ V is the natural inclusion. Using the nondegeneracy of ω, this implies that dim W ⊥ω = dim V − dim W , which will result useful along this paper.
The antisymmetry of ω gives rise to a wide variety of situations. In particular, we say that W ⊆ V is: ii) Coisotropic if W ⊥ω ⊆ W (if W is coisotropic, necessarily dim W ≥ n); A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems iii) Lagrangian if W is isotropic and has an isotropic complement. (if W is Lagrangian, A subspace W is Lagrangian if and only if W = W ⊥ω . This implies that Lagrangian subspaces are the isotropic subspaces of maximal dimension and the coisotropic subspaces of minimal dimension.
It can be easily checked that the symplectic complement has the following properties:

Coisotropic reduction in symplectic geometry
is a manifold and ω is a closed 2-form such that (T q M, ω q ) is a symplectic vector space, for every q ∈ M. As in the linear case, for the existence of such form, M needs to have even dimension 2n.
Every manifold is locally isomorphic, that is, there exists a set of canonical coordinates around each point: There exist a coordinate system around q, (q i , p i ) such that ω = dq i ∧ dp i .. These coordinates are called Darboux coordinates.
This non-degenerate form induces a bundle isomorphism between the tangent and cotangent bundles of M point-wise, namely Definition 3.2 (Hamiltonian vector field). Given H ∈ C ∞ (M), we define the Hamiltonian vector field of H as The definitions of the different cases of subspaces given in the linear case can be extended point-wise to submanifolds N ֒→ M. Consequently, we say that N ֒→ M is: iii) Lagrangian if N is isotropic and there is a isotropic subbundle (where we understand isotropic point-wise) E ⊆ T M|N such that T M = T N ⊕ E (here ⊕ denotes the Whitney sum). This is exactly the point-wise definition of a Lagrangian subspace asking for the coisotropic complement to vary smoothly; These definitions extend naturally to distributions.
Just like in the linear case, a submanifold N ֒→ M is Lagrangian if and only if it is isotropic (or coisotropic) and has maximal (or minimal) dimension. This is a useful characterization that we will use several times in the rest of the paper.
and thus, X defines a Lagrangian submanifold if and only if Taking This implies G i = ∂H ∂x i , for some local function H. It is clear that locally, we have X = X H .
Since the distribution is involutive and regular, the Fröbenius Theorem guarantees the existence of a maximal regular foliation F of N, that is, a decomposition of N in maximal submanifolds tangent to the distribution. In what follows, we suppose that N/F (the space of all leaves) admits a manifold structure so that the projection is a submersion. The main result is the Weinstein reduction theorem [41]: Theorem 3.2 (Coisotropic reduction in the symplectic setting). Let (M, ω) be a symplectic manifold and N ֒→ N be a coisotropic submanifold. If N/F (the spaces of all leaves under the distribution q → (T q N) ⊥ω ) admits a manifold structure such that N π − → N/F is a submersion, there exist an unique 2-form ω N in N/F that defines a symplectic manifold structure such that, if N i − → M is the natural inclusion, then i * ω = π * ω N . The following diagram summarizes the situation: Proof. Uniqueness is guaranteed from the imposed relation since it forces us to define where [u] := T π(q) · u. We only need to check that this is a well-defined closed form and that it is non-degenerate.
We begin showing that our definition does not depend on the representative of the vector [u]. For this, it is sufficient to observe that (ω N ) [q] ([u], [v]) = 0 whenever u is a vector in the distribution. Furthermore, for every vector field X in N with values in (T N ⊥ω ), and this implies the independece of the point (for every two points in the same leave of the foliation can be joined by a finite union of flows of such fields).
It is clearly non-degenerate and it is closed, since d π * ω N = i * d ω = 0 and π is a submersion.

Projection of Lagrangian submanifolds
Definition 3.4 (Clean intersection). We say that two submanifolds L, Proof. It is sufficient to see that is isotropic and that it has maximal dimension in N/F .
. Now, since Ker d q π = (T q N) ⊥ω , the kernel-range formula yields beacause L is Lagrangian and N coisotropic. Substituting (2) in (1) we obtain which is exactly 1 2 dim N/F , as a direct calculation shows.

Poisson structures
A symplectic structure (M, ω) induces a Lie algebra structure in the ring of funtions C ∞ (M).

POISSON STRUCTURES
Definition 4.1 (Poisson bracket). Let (M, ω) be a symplectic manifold and f, g ∈ C ∞ (M). We define the Poisson bracket of f, g as the function It is easily checked that in Darboux coordinates the Poisson bracket is i) It is bilinear with respect to R; Taking into consideration the previous definition, we can generalize the notion of symplectic manifolds as follows: Notice that Im ♯ Λ = S, the characteristic distribution.
A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems In the case of symplectic manifolds ♯ ω = ♯ Λ , and the characteristic distribution is the whole tangent bundle; however, in the general setting ♯ Λ need not be a bundle isomorphism. Actually, if ♯ Λ is a bundle isomorphism, it arises form a symplectic structure defined as This characteristic distribution is involutive [26] and each leaf of the foliation, S, admits a symplectic structure defining for f, g ∈ C ∞ (S) and q ∈ S, for arbitrary extensions f , g ∈ C ∞ (P ) of f, g respectively. It can be easily checked that this definition does not depend on the chosen functions and that it defines a non-degenerate Poisson structure and thus, S is a symplectic manifold [39].
Remark 4.1. The characteristic distribution of a Poisson manifold is an example of generalized distributions, studied by [34,35], that extended the Frobenius theorem for this kind of involutive distributions.
Just as in the symplectic scenario, we say that a subspace ∆ q ⊆ T q P is iii) Lagrangian if ∆ q = ∆ ⊥ Λ q ∩ S q for every q ∈ P . Notice that this is equivalent to ∆ q ∩ S q being Lagrangian in each symplectic vector space S q .
The Λ-orthogonal complement satisfies the following properties: Remark 4.2. For symplectic manifolds, the above definitions coincide with the ones previously given.

Coisotropic reduction in cosymplectic geometry
Cosymplectic structures are just relevant because they are the natural arena to develop time-dependent Lagrangian and Hamiltonian mechanics [9].
Similar to the symplectic setting, there exists canonical coordinates, which will be called Darboux coordinates (q i , p i , t) such that Ω = d q i ∧ d p i and θ = d t. The existence of such coordinate charts is proven in [20].
There are two natural distributions defined on M: i) The horizontal distribution H := Ker θ; ii) The vertical distribution V := Ker Ω.
These distributions induce the following types of tangent vectors in each tangent space. A vector v ∈ T q M will be called: In Darboux coordinates, these distributions are locally generated as follows: Just as before, we can define a bundle isomorphism between the tangent and cotangent bundles: The vector field defined as R := ♯ θ,Ω (θ) is called the Reeb vector field. The Reeb vector field is locally given by Let H be a differentiable function on M. We define the following vector fields: i) The gradient vector field grad H := ♯ θ,Ω (dH); ii) The Hamiltonian vector field X H := grad H − R(H)R; iii) The evolution vector field E H := X H + R.
These vector fields have the local expressions: A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems Notice that the horizontal distribution H is the distribution generated by all Hamiltonian vector fields. Just as in the symplectic case, we can define a Poisson bracket: We can easily check that this is indeed a Poisson structure observing that in coordinates is given by

And, thus, the coordinate expression of the Poisson tensor is
This induces all the definitions from Poisson manifolds given in Section 4. In particular, Note that Ker ♯ Λ = θ and that Im ♯ Λ = H, that is, H is the characteristic distribution of the Poisson structure induced by (θ, Ω). This implies the following result: Proof. It follows from the definition of Lagrangian submanifold (Section 4) and the fact that H is the characteristic distribution on M.
It is also easy to see that observing that we have Ω(X f , X g ) = Ω(grad f, grad g).

Gradient, Hamiltonian and evolution vector fields as Lagrangian submanifolds
Definition 5.3. Given a cosymplectic manifold (M, Ω, θ), we define the symplectic structure on T M as There is another expression of Ω 0 , namely as one can verify [5]. Here, α v , α c denote the complete and vertical lifts of a form α on M to its tangent bundle T M [9]. This implies that in the induced coordinates in T M, Proof. It is easily checked that X * λ 0 = ♭ θ,Ω (X) (just like in Proposition 3.1) and then, that is, X is locally a gradient vector field.
We can also check this in coordinates. Indeed, let be a vector field on M. X : M ֒→ T M defines a Lagrangian submanifold if and only if X * Ω 0 = 0. An easy calculation gives Therefore, X defines a Lagrangian submanifold of (T M, Ω 0 ) if and only if A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems The equations above can be summarized taking We conclude that G i = ∂H ∂x i , for some local function H, that is, locally, X = grad H.
In general, the Hamiltonian and evolution vector field do not define a Lagrangian submanifold in (T M, Ω 0 ). However, modifying the form we can achieve this. First, let us study the Hamiltonian vector field X H . We have is a symplectic form and has the local expression Also, it follows that the evolution vector field E H also defines a Lagrangian submanifold of (T M, Ω H ).
This also gives a way of interpreting both vector fields as Lagrangian submanifolds of a the cosymplectic submanifold (T M × R, Ω H , d s), taking the coordinate in R to be constant.

Coisotropic reduction
We can interpret the orthogonal complement defined by the Poisson structure using the cosymplectic structure. We note that Ω| H defined as Ω restricted to the distribution H induces a symplectic vector space in each H q and thus we have a symplectic vector bunde This last proposition clarifies the situation. The Λ-orthogonal of a subspace ∆ is just the symplectic orthogonal of the intersection with the symplectic leaf. This means that coisotropic reduction in cosymplectic geometry will be performed in each leaf of the characteristic distribution H. Also, because the Λ-orthogonal complement is just the symplectic complement of the intersection with H, we have the following properties: It will also be important to distinguish submanifolds N ֒→ M acording to the position relative to the distributions H, V. ii) Non-horizontal submanifold if T q N ⊆ H q for every q ∈ N;

COISOTROPIC REDUCTION IN COSYMPLECTIC GEOMETRY
iii) Vertical submanifold if the Reeb vector field is tangent to N, that is, R(q) ∈ T q N for every q ∈ N.
Lagrangian submanifolds are characterized as follows: ii) It follows from the previous calculation using that θ ∈ T q L 0 because The proof of the equality dim T q L ⊥ Λ = n is straightforward using that T q L ∩ H q is a Lagrangian subspace of (H q , Ω| H ).
Lemma 5.1 guarantees that either dim L = n, in which case L is horizontal, or dim L = n + 1, in which case L is non-horizontal. We have the following useful characterization of Larangian submanifolds: Proof. Both assertions are proved by a comparison of dimensions.
Proof. We start proving that H is an involutive distribution. Let X, Y be vector fields tangent to H. Since θ is closed we have that is, [X, Y ] is tangent to H.

COISOTROPIC REDUCTION IN COSYMPLECTIC GEOMETRY
Denote In order to see this, we take an arbitrary vector field Z on N tangent to H and check that

Vertical coisotropic reduction
We shall now study coisotropic reduction of a vertical submanifold N ֒→ M. Let q ∈ N.
In particular, (T N) ⊥ Λ is a regular distribution.
and they define a cosymplectic structre on N/F . The following diagram summarizes the situation: We only need to verify that the following forms are closed, well defined and define a cosymplectic structure: where [u] := T π(q)·u ∈ T [q] N/F . If they were well defined, it is clear that they are smooth and closed since π * dθ N = dθ 0 = 0, π * Ω N = dΩ 0 = 0 and π is a submersion.

COISOTROPIC REDUCTION IN COSYMPLECTIC GEOMETRY
Let us first check that these definitions do not depend on the chosen representatives of the vectors. It suffices to observe that for vectors in the distribution, say v ∈ (T q N) ⊥ Λ , we have i v Ω 0 = 0 and i v θ 0 = 0. This easily follows from Proposition 5.3 using that the horizontal proyection of every vector u ∈ T q N is tangent to N (here we use the condition R(q) ∈ T q N). To see the independece of the point in the leave chosen, it is enough to observe that L X Ω 0 = 0; L X θ 0 = 0 for every vector field on N tangent to the distribution (T q N) ⊥ Λ (since every two points in the same leave of the foliation can be joined by a finite union of flows of such fields). Indeed, we have Now we check that they define a cosymplectic structure. Assuming k = dim N and 2n + 1 = dim M, from the remark above we have which is equivalent to θ 0 ∧ Ω k−n−1 0 = 0, because π is a submersion. For every point q ∈ N, T qN can be decomposed in It is easy to see that (T q N) H is a coisotropic subspace of (H q , Ω| H ). This implies (using symplectic reduction) that there are dim( Taking the last vector to be R(q), it is clear that

Projection of Lagrangian submanifolds
Now we will proof that Lagrangian submanifolds L ֒→ M project to Lagrangian submanifolds in N/F .
We will check that and prove that dim L N = dim N − n − 1, which with Lemma 5.2 together with the calculation of the dimension of N/F done in Theorem 5.1, yields the result. Then Furthermore, since L is Lagrangian and horizontal, ( Substituting (4) in (3) and using dim L = n, we conclude Proof. The proof follows the same lines as that of Proposition 5.5. That L N is isotropic follows easily from Proposition 5.3. However, in order to calculate dim L N , we need to distinguish whether L ∩ N is horizontal or not.
It is easy to check that dim( We conclude that ii) If L ∩ N is not horizontal, we need to check that dim L N = dim N − n. This follows from the same calculation done in i), using that

Horizontal coisotropic reduction
We will restrict the study to horizontal coisotropic submanifolds N ֒→ M, that is, manifolds satisfying T q N ⊆ H q for every q ∈ N. Note that in this case the distribution (T N) ⊥ Λ is also regular, since Proof. Since N is horizontal and the horizontal distribution is integrable, N will be contained in an unique symplectic leaf and thus, we are performing symplectic reduction. The proof is just repeating what has been done in Theorem 3.2.

JACOBI STRUCTURES
We can generalize this process to arbitrary submanifolds. Let N ֒→ M be a coisotropic submanifold. Since in general we cannot guarantee the well-definedness of the 2-form in the quotient, we will reduce the intersection of N with each one of the symplectic leaves. It is clear that T N ∩ H is an involutive distribution, since T N and H are. If this distribution was regular, for every q ∈ N there would exist an unique maximal leaf of the distribution, say S q . Notice that S q ֒→ M is an horizontal submanifold. We can perform coisotropic reduction in each of this submanifolds. Proof. It follows from Proposition 3.4.

Jacobi structures
Contact and cocontact manifolds are not Poisson manifolds. However, there is still a Lie bracket defined in the algebra of functions, as we will see. This bracket induces what is called a Jacobi manifold. In this section we define and study such structures (see [23,26] for more details). Every Jacobi bracket can be uniquely expressed as where Λ is a bivector field (called the Jacobi tensor) and E is a vector field. Λ and E satisfy the equalities where [·, ·] is the Schouten-Nijenhuis bracket. Conversely, given a bivector field Λ and a vector field E, defines a Jacobi bracket if and only if both equalities above hold.
Remark 6.1. It is clear that Poisson manifolds are Jacobi manifolds, taking E = 0.
A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems The Jacobi tensor allows us to define the morphism Define the Λ-orthogonal of distributions ∆ as We can define the Hamiltonian vector field defined by a function H as Just like in the Poisson case, we say that a distribution ∆ is: These definitions extend naturally to submanifolds.

Coisotropic reduction in contact geometry
Contact manifolds are the natural settings for Hamiltonian systems with dissipation, instead of symplectic Hamiltonian systems where the antisymmetry of the symplectic form provides conservative properties. In the Lagrangian picture, contact Lagrangian systems correspond to the so-called Lagrangians depending on the action, and instead of Hamilton principle, one has to use the so-called Herglotz principle to obtain the dynamics [13].
In this case we also have Darboux coordinates (q i , p i , z) in M [20] such that We have also have a bundle isomorphism defined as in the cosymplectic case its inverse ♯ η = ♭ −1 η , and a couple of natural distributions:

COISOTROPIC REDUCTION IN CONTACT GEOMETRY
i) The horizontal distribution H := Ker η; ii) The vertical distribution V := Ker d η.
We can find different types of tangent vectors at a point q ∈ M. Indeed, a tangent vector v ∈ T q M will be called This time, however, we cannot define a canonical Poisson structure since the bivector field is not a Poisson tensor. In fact, where R is the Reeb vector field defined as R := ♯ η (η) (locally R = ∂ ∂z ). This is easily seen performing a direct calculation in Darboux coordinates using the local expresion This defines a Jacobi structure in M taking Λ as above and E = −R (see Section 6). The Jacobi bracket is locally expressed as

The morphism induced by the Jacobi tensor Λ satisfies
Ker ♯ Λ = η , Im ♯ Λ = H. Locally, it has the local expression

Hamiltonian and evolution vector fields as Lagrangian and Legendrian submanifolds
We can define a symplectic structure in T M taking Ω 0 := ♭ * η Ω M , where Ω M is the canonical symplectic structure in T * M. In local coordinates (q i , p i , z,q i ,ṗ i ,ż), it has the expression Locally, it is given by We have the following relation between both vector fields Just like in the previous sections, a vector field X : M → T M is locally a gradient vector field if and only if it defines a Lagrangian submanifold in (T M, Ω 0 ). The proof is straight-forward, checking that We can also interpret the Hamiltonian vector field X H as a Lagrangian submanifold of T M, but we need to modify slightly the symplectic form. It is easy to verify that Now we study evolution vector fields, an important vector field in the application of contact geometry to thermodynamics. For more details, see [32,33]. A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems Locally, the evolution vector field is written Let us see how we can modify the sympelctic form Ω 0 in such a way that E H defines a Lagrangian submanifold. We have where We can also interpret Hamiltonian and evolution vector fields as Legendrian submanifolds of a certain contact structure defined on T M × R.
where η c , η v are the complete and vertical lifts [9]. It is easily checked thatη defines a contact structure [7].
In local coordinates it has the expression: We have the following [7]: is a Legendrian submanifold of (T M × R,η).

Proof. Using the properties of complete and vertical lifts we have
Using Lemma 7.2 it will be sufficient to see that L X H η = −R(H). This is a straight-forward verification using A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems
The following definition will result useful. Given a subspace ∆ q ⊆ T q M, we define the dη-orthogonal complement as Furthermore, if R(q) ∈ ∆ q or ∆ q ⊆ H q , the equality holds.
Now, if R(q) ∈ ∆ q , we compare dimensions. Since Ker ♯ Λ = η and η ∈ ∆ 0 q , we have Furthermore, ∆ ⊥ dη ∩ H q has the same dimension, since This latter is just the symplectic complement in (H q , dη| H ) and hence, A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems

COISOTROPIC REDUCTION IN CONTACT GEOMETRY
This proposition allows us to characterize Legendrian submanifolds: Lemma 7.1. If L ֒→ M is a Legendrian submanifold, then L is horizontal and dim L = n (where dim M = 2n + 1). Furthermore, if L is horizontal and isotropic (or coisotropic) with dim L = n, L is Legendrian.
Proof. Since ♯ Λ takes values in H, it is clear that every Legendrian submanifold is horizontal. Since L is horizontal, From the previous equation and using that T q L ⊥ Λ = T q L, we deduce that dim L = dim M − dim L−1. This implies dim L = n. The last property is easily seen via a direct comparison of dimensions.
We will also need characterization of isotropic submanifolds in contact geometry: Proof. Necessity is clear, since ♯ Λ takes values in H. Now suppose that N is horizontal. We have i * η = 0 and thus, Proposition 7.4. Let i : N ֒→ M be a coisotropic submanifold such that R(q) ∈ T q N for every q ∈ N or T q N ⊆ H q for every q ∈ N. Define η 0 := i * η. Then Proof. Let q ∈ N. Proposition 7.3 implies that But, since T q N is coisotropic, then it is just Ker d η 0 ∩ Ker η 0 .
We then have the following result: Proposition 7.5. Let i : N ֒→ M be a coisotropic subamnifold such that R(q) ∈ T q N for every q ∈ N or T q N ⊆ H q for every q ∈ N. Then, the distribution T N ⊥ Λ defined by Proof. Denote η 0 := i * η and let X, Y be vector fields along N taking values in T N ⊥ Λ . Proposition 7.4 implies that It suffices to check that

COISOTROPIC REDUCTION IN CONTACT GEOMETRY
Indeed, taking Z an arbitrary vector field in N, we have where we have used that X, Y ∈ Ker dη 0 . In a similar way we obtain

Vertical coisotropic reduction
We will restrict the study to vertical submanifolds, that is, submanifolds satisfying R(q) ∈ T q N, for every q ∈ N. Notice that if N is a coisotropic vertical submanifold, the distribu- It only remains to check well-definedness and that it defines a contact manifold. That this definition does not depend on the chosen representative vector is clear since a vector tangent to the distribution is necessarily in the kernel of η. Furthermore, if X is a vector field tangent to the distribution T N ⊥ Λ , Proposition 7.4 implies To check that it is a contact manifold, we calculate the dimension of N/F . We have that and therefore, (N/F , η N ) is a contact manifold if and only if Since π is a submersion, this is equivalent to η 0 ∧ (d η 0 ) k−n−1 = 0. This is straightforward using Proposition 7.4. Proof. It suffices to check that L N is horizontal, isotropic and dim L N = dim N − n − 1 using Lemma 7.2. Since L is horizontal, L N is horizontal and thus, L N is isotropic.
For the comparison of dimensions, we have Using dim(T q L ∩ T q N ⊥ Λ ) = dim(T q L + T q N) − 1 and substituting (6) in (5), we obtain

Horizontal coisotropic reduction
We will retsrict the study to horizontal coisotropic submanifolds N → M, that is, manifolds satisfying T q N ⊆ H q for every q ∈ N.
Remark 7.1. Notice that in this case reduction is trivial, since the only coisotropic horizontal submanifolds of a contact manifold are those that are Legendrian. This would imply dim N/F = 0, making the resulting manifold trivial.
Given an arbitrary coisotropic submanifold N ֒→ M, we cannot guarantee the welldefinedness of the 2-form in the quotient N/F (actually, in the contact setting, we cannot even guarantee the integrability of T N ⊥ Λ ) so this time (referring to horizontal reduction in cosymplectic geometry) we cannot obtain a foliation of N in symplectic leaves, since T N ∩ H|N is not integrable in the general setting.

Projection of Legendrian submanifolds
The triviality of this case makes the projection of Lagrangian submanifolds trivial.

Coisotropic reduction in cocontact geometry
Cocontact manifolds have been introduced in [6] just to provide a setting for dissipative systems which also depend on time. In geometric terms, we are combining cosymplectic and contact structures. (M, θ, η), where M is a (2n+2)-dimensional manifold, θ is a closed 1-form, η is a 1-form and, θ∧η∧(d η) n = 0 is a volume form.

Definition 8.1 (Cocontact manifold). A cocontact manifold is a triple
The bundle isomorphism in this case is defined as and its inverse is denoted by ♯ θ,η = ♭ −1 θ,η .
In cocontact geometry there exists aswell a set of canonical coordinates (q i , p i , z, t), which will be called Darboux coordinates, such that We can define aswell the Reeb vector fields as which can be expressed locally as We also have vertical and horizontal distributions: i) The z-horizontal distribution, H z := Ker η; ii) The t-horizontal distribution, H t := Ker θ; iii) The tz-horizontal distribution H tz := H t ∩ H z ; iv) The t-vertical distribution, V t := R t ; v) The z-vertical distribution, V z := R z .

Hamiltonian vector fields as Lagrangian and Legendrian submanifolds
Just like in previous sections, define the gradient vector field of certain Hamiltonian Locally, the gradient vector field is expressed: We can define a symplectic structure in T M taking where Ω M is the canonical symplectic form on the cotangent bundle. In the induced coordinates (q i , p i , z,q i ,ṗ i ,ż), Ω 0 takes the form It is easy to verify that a vector field X : M → T M is a locally gradient vector field if and only if it defines a Lagrangian submanifold in (T M, Ω 0 ).

Definition 8.2 (Hamiltonian vector field). Given a Hamiltonian H on M, define its
Hamiltonian vector field as The Hamiltonian vector field has the local expression In general, X H does not define a Lagrangian submanifold of (T M, Ω 0 ); but, just like in the cosymplectic and contact scenario, we can achieve this by modifying the symplectic form. Indeed, since defining we have that X H defines a Lagrangian submanifold of (T M, Ω H ).
A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems

COISOTROPIC REDUCTION IN COCONTACT GEOMETRY
Now we interpret the Hamiltonian vector field X H as a Legendrian submanifold of T M × R × R with the cocontact structure given by the forms where (s, e) are the parameters in R × R. In local coordinates (q i , p i , z, t,q i ,ṗ i ,ż,ṫ, s, e), these forms have the expression: It is easy to see that these forms define a cocontact structure. Now, given a vector field X : M → T M and two functions f, g on M, define Applying the properties of complete and vertical lifts, namely X * α c = L X (α), we have Proof. Using the observation above and Lemma 8.1, it is sufficient to observe that

Coisotropic reduction
A cocontact manifold is also a Jacobi manifold defining and thus, we have the Λ -orthogonal and the corresponding definitions of isotropic, coisotropic or Legendrian submanifolds and distributions.
Notice that H t is an integrable distribution and that each leave of its foliation inherits a contact structure. Indeed, H t is the characteristic distribution defined by the Jacobi structure, S. Now we give a symplectic interpretation of the Λ-orthogonal. Notice that the restriction of d η to H tz defines a symplectic structure on the distribution. Denote by ⊥ dη| its symplectic orthogonal. The Λ-orthogonal is just the symplectic orthogonal of the intersection with H tz .
Proposition 8.2. Given a distribution ∆ on a cocontact manifold (M, η, θ), Proof. We check one inclusion and compare dimensions: Let α ∈ ∆ 0 q and u ∈ ∆ 0 q ∩ (H tz ) q . We will see that d η q (u, ♯ Λ (α)) = 0. Indeed, Now we compare both dimensions. Let k := dim ∆, r q := dim(∆ 0 q ∩ θ q , η q ). Since we have It only remains to observe that and that Proof. Necessity is clear, since L is necessarily horizontal. Sufficiency follows from i * d η = 0 which, together with dim L = n, implies that T q L is a Lagrangian submanifold of (H tz ) q , for every q ∈ L. Using Proposition 8.2, we have the equivalence.
This allows us to express the Λ-orthogonal complement of a coisotropic distribution in a more convenient way: where η 0 and θ 0 are the restrictions of η and θ to ∆, respectively. Now, given a coisotropic submanifold N ֒→ M, since T N ⊥ Λ is involutive, it provides a maximal foliation, F . We assume that N/F inherits a manifold structure such that the canonical projection π : N → N/F is a submersion.
Just like in the previous cases, for the well-definedness and non-degeneracy of the forms in the quotient, we need to restrict the coisotropic submanifolds we are studying. Consequently, we will say that a submanifold N ֒→ M is: ii)) tz-vertical, if it is both t-vertical and z-vertical.
iv) tz-horizontal if it is both t-horizontal and z-horizontal, that is, if T N ⊆ H tz .

tzvertical reduction
Let i : N ֒→ M be a tz-vertical submanifold. It is easy to check that under these conditions Before proving the theorem, let us calculate the dimension of the quotient. Let k + 2 := dim N. We have dim T N ⊥ Λ = 2n + 2 − (k + 2) = 2n − k and, therefore, Proof. Uniqueness is clear, since π is a submersion. We only need to check the well- Independence of the vector is clear, using Proposition 8.2. For independence on the point, let X be a vector field on N tangent to the distribution. It is easy to check that L X η 0 = 0, L X θ 0 = 0; and thus, well-definedness follows. For non-degeneracy, it is enough to proof that This follows easily from T N ⊥ Λ = Ker d η 0 ∩ Ker η 0 ∩ Ker θ 0 . Proof. Using Lemma 8.1, L N is clearly isotropic. Now we need to check that dim L N = k − n, given that dim N/F = 2(k − n) + 2. We have Now, using the Grassman formula: From (13) and (16) we obtain Substituting in (12) yields dim π(L ∩ N) = k − n. This means that reduction is trivial, leaving the trivial cosymplectic submanifold of dimension 1:

tvertical, zhorizontal reduction
Theorem 8.2. Let i : N ֒→ M be a t-vertical, z-horizontal coisotropic submanifold of a coconatct manifold (M, θ, η). Denote by F the maximal foliation defined by the distribution (T N) ⊥ Λ . If N/F has a manifold structure such that the canonical projection π : N → N/F defines a submersion, then N/F is one-dimensional and there exists and unique volume form θ N on N/F such that i * θ = π * θ N .

Projection of Legendrian submanifolds
Given the triviality of reduction in the t-vertical and z-horizontal case, projection of Legendrian submanifolds in M will always result in 0-dimensional Lagrangian submanifolds in N/F .

zvertical, thorizontal reduction
Let i : N → M be a z-vertical and t-horizontal coisotropic submanifold. It is easy to check that this time we have the equality Since H t is integrable, we have that coisotropic reduction of N is actually happening in one of the leaves of the foliation that inhertits a contact structure from the cocontact structure. We conclude, from Theorem 7.1: Proof. It is clearly horizontal and, therefore, using Lemma 7.2, it is isotropic. Now, supposing k = dim N, we only need to check that since dim N/F = 2(k − n) + 1. This is straight-forward, following the same steps given in Proposition 8.3.

tzhorizontal reduction
Let i : N ֒→ M be a tz-horizontal coisotropic submanifold. Since η 0 = 0, d η 0 = 0 and θ 0 = 0, we have leaving a trivial symplectic manifold, having as many points as path components of N. This means that if N/F admits a manifold structure, it will be a symplectic manifold.

Projection of Legendrian submanifolds
Again, the projection of Legendrian submanifolds is trivial.

Coisotropic reduction in stable Hamiltonian structures
There were several attempts to combine cosymplectic and contact structures. The first one is due to Albert [3], using a combination of a 1-form and a 2-form; however, the setting is not useful for us since the lack of integrability. The second attempt in this direction is due to Acakpo [2], which is studied in this section. There exists, just like in the previous cases, a natural isomorphism and its inverse ♯ λ,ω := ♭ −1 λ,ω . Let us perform some calculations in coordinates. Since d ω is closed and of constant range 2n, around any point, there exists coordinates (q i , p i , z) such that ω = d q i ∧ d p i (see [20]). In this coordinate chart λ will have an expression of the form Since we conclude that c = 0. Let ϕ t (q i , p i , z) be the (local) flow of the vector field 1 c ∂ ∂z . Fix some value z 0 , and define the map It is clear that this defines a local diffeomorphism. Take the new set of coordinates to be Therefore, in the new coordinate chart, We conclude: A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems We call these coordinates Darboux coordinates.
In Darboux coordinates, the condition Ker d ω ⊆ d λ translates to Also, the musical isomorphisms take the expression: Imitating the definitions in the contact case, we can define a bivector field on M as Λ q (α q , β q ) := ω q (♯ λ,ω (α q ), ♯ λ,ω (β q )), and the morphism with the induced Λ-orthogonal complement for distributions In coordinates (q i , p i , z) the bivector field Λ takes the local form: We also have the distributions i) H q := Ker λ q , ii) V q := Ker ω q , and the Reeb vector field R q := ♯ λ,ω (λ q ). Locally, A natural question to ask is wether the bivector field Λ arises from a Jacobi bracket. We have Taking an arbitrary vector field It is easily checked that the first equality holds when This is easily seen to be equivalent to We have concluded the following: Proposition 9.2. The bivector field Λ arises from a Jacobi structure if and only if there exists some f ∈ C ∞ (M) such that And, in that case, the Jacobi structure is defined by the pair (Λ, f R).
Remark 9.1. Notice that we recover the cosymplectic scenario when f = 0 and the contact scenario when f = 1 (because the definition of Λ in contact geometry is the opposite of the definition we gave in SHS).
Let us return to the study of coisotropic reduction. It is easy to see that ω induces a symplectic form in H, ω| H . This induces the symplectic orthogonal for ∆ q ⊂ H q : ∆ ⊥ ω| H q . = {v ∈ H q | ω(v, w) = 0 ∀w ∈ ∆ q }.
A distribution ∆ in M will be called: We have the following equality: Proposition 9.3. Let ∆ be a distribution on M. Then Proof. The proof follows the same lines as that of Proposition 5.3 Just like in previous sections, we say that a Lagrangian submanifold L ֒→ M is horizontal if T q L ⊆ H q ∀q ∈ L and say that it is non-horizontal if T q L ⊆ H q ∀q ∈ L. We have the following characterization: Lemma 9.1. Let L ֒→ M be an isotropic (or coisotropic) submanifold. We have i) If L is horizontal and dim L = n, then L is Lagrangian.
ii) If L is non-horizontal and dim L = n + 1, then L is Lagrangian.
Proof. The proof goes as Lemma 5.2, since we only need to check the condition at each tangent space. Now, given a coisotropic submanifold N ֒→ M (that is, (T q N) ⊥ Λ ⊆ T q N), the distribution (T N) ⊥ Λ is not necessarily integrable and we shall assume it in what follows:

Gradient and Hamiltonian vector fields as Lagrangian submanifolds
We can define a symplectic structure on T M taking where Ω M is the canonical symplectic form on T * M. In Darboux coordinates it has the expression: In Darboux coordinates, the gradient vector field is written It is easily checked that X : M → T M is locally a gradient vector field if and only if X(M) is a Lagrangian submanifold of (T M, Ω 0 ). Indeed, we have the equality When Λ comes from a Jacobi bracket on M, that is, when In Darboux coordinates it has the expression: Let us interpret the Hamiltonian vector field as a Lagrangian submanifold of T M, with some sympelctic form. First, observe that Therefore, defining the symplectic form we have that X H defines a Lagrangian submanifold of (T M, Ω H ).

Vertical coisotropic reduction
Theorem 9.1 (Vertical coisotropic reduction in stable Hamiltonian structures). Let i : N ֒→ M be a vertical coisotropic submanifold such that (T N) ⊥ Λ defines an integrable distribution. Let F be the set of leaves and suppose that N/F admits a manifold structure such that the canonical projection π : N → N/F defines a submersion. If i * d λ = 0 in T N ∩ H, then N/F admits an unique stable Hamiltonian system structure (ω N , λ N ) such that π * ω N = i * ω and π * λ N = i * λ. The following diagram summarizes the situation: Proof. The proof goes as Theorem 5.1. Asking i * d λ = 0 is necessary to guarantee the well-definedness of λ N in the quotient using where λ 0 = i * λ. It would only remain to check that Ker ω N ⊆ Ker d λ N .
Indeed, since Ker ω N = R N , and R N = π * R, it follows from π * (i R N d λ N ) = i R d λ = 0.
A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact systems 10. CONCLUSIONS

Projection of Lagrangian submanifolds
We have the result: Proof. The proof goes as Proposition 5.5 and Proposition 5.6, since the proof reduces to the study of each tangent space. Proof. The proof goes as Proposition 3.4 since we only need to check it in every tangent space.

Conclusions
The interpretations of the different types of vector fields in the different types of geometry as Lagrangian or Legendrian submanifolds can be summarized in Table 1. Also, the results on coisotropic reduction can be summarized in Table 2