Geodesic Equations on Diffeomorphism Groups

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.


Introduction
Some conservative systems of hydrodynamical type can be written as geodesic equations on the group of diffeomorphisms or the group of volume preserving diffeomorphisms of a Riemannian manifold, as well as on extensions of these groups. Considering right invariant L 2 or H 1 metrics on these infinite dimensional Lie groups, the following geodesic equations can be obtained: the Euler equation of motion of a perfect fluid [2,10], the averaged Euler equation [31,50], the equations of ideal magneto-hydrodynamics [54,32], the Burgers inviscid equation [7], the template matching equation [18,55], the Korteweg-de Vries equation [44], the Camassa-Holm shallow water equation [8,38,29], the higher dimensional Camassa-Holm equation (also called EPDiff or averaged template matching equation) [20], the superconductivity equation [49], the equations of motion of a charged ideal fluid [57], of an ideal fluid in Yang-Mills field [14] and of a stratified fluid in Boussinesq approximation [61,58].
For a Lie group G with right invariant metric, the geodesic equation written for the right logarithmic derivative u of the geodesic is a first order equation on the Lie algebra g, called the Euler equation. Denoting by ad(u) ⊤ the adjoint of ad(u) with respect to the scalar product on g given by the metric, Euler's equation can be written as d dt u = − ad(u) ⊤ u. In this survey type article we do the formal derivation of all the equations of hydrodynamical type mentioned above, starting from this equation.
By writing such partial differential equations as geodesic equations on diffeomorphism groups, there are various properties one can obtain using the Riemannian geometry of right invariant metrics on these diffeomorphism groups. We will not focus on them in this paper, but we list some of them below, with some of the references.
For some of these equations smoothness of the geodesic spray on the group implies local wellposedness of the Cauchy problem as well as smooth dependence on the initial data. This applies for the following right invariant Riemannian metrics: L 2 metric on the group of volume preserving diffeomorphisms [10], H 1 metric on the group of volume preserving diffeomorphisms on a boundary free manifold [50], on a manifold with Dirichlet boundary conditions [31,51] and with Neumann or mixt boundary conditions [51,13], H 1 metric on the group of diffeomorphisms of the circle [50,29] and on the Bott-Virasoro group [9], and H 1 metric on the group of diffeomorphisms on a higher dimensional manifold [15].

Euler's equation
Given a regular Fréchet-Lie group in the sense of Kriegl-Michor [28], and a (positive definite) scalar product , : g × g → R on the Lie algebra g, we can define a right invariant metric on G by g x (ξ, η) = ξx −1 , ηx −1 for ξ, η ∈ T x G. The energy functional of a smooth curve c : I = [a, b] → G is defined by where δ r denotes the right logarithmic derivative (angular velocity) on the Lie group G, i.e. δ r c(t) = c ′ (t)c(t) −1 ∈ g. We assume the adjoint of ad(X) with respect to , exists for all X ∈ g and we denote it by ad(X) ⊤ , i.e.
The corresponding notation in [3] is B(X, Y ) = ad(Y ) ⊤ X for the bilinear map B : g × g → g. Proof . We denote the given curve by c 0 and its logarithmic derivative by u 0 . For any variation with fixed endpoints c(t, s) ∈ G, t ∈ [a, b], s ∈ (−ε, ε) of the given curve c 0 , we define u = (∂ t c)c −1 and v = (∂ s c)c −1 . In particular u(·, 0) = u 0 , and we denote v(·, 0) by v 0 . Following [35] we show first that For each h ∈ G we consider the map F h (t, s) = (t, s, c(t, s)h) for t ∈ [a, b] and s ∈ (−ε, ε). The bracket of the following two vector fields on [a, b] × (−ε, ε) × G vanishes: The reason is they correspond under the mappings F h , h ∈ G, to the vector fields ∂ t and ∂ s on because the bracket of right invariant vector fields corresponds to the opposite bracket on the Lie algebra g, so the claim (2.2) follows. As in [34] we compute the derivative of E(c) = 1 2 b a u, u dt with respect to s, using the fact that v(a, s) = v(b, s) = 0.
The curve c 0 in G is a geodesic if and only if this derivative vanishes at s = 0 for all variations c The Euler equation for a left invariant metric on a Lie group is d dt u = ad(u) ⊤ u. In the case G = SO(3) one obtains the equations of the rigid body.
Denoting by ( , ) the pairing between g * and g, the inertia operator [3] is defined by It is injective, but not necessarily surjective for infinite dimensional g. The image of A is called the regular part of the dual and is denoted by g * reg . Let ad * be the coadjoint action of g on g * given by (ad * (X)m, Y ) = (m, − ad(X)Y ), for m ∈ g * . The inertia operator relates ad(X) ⊤ to the opposite of the coadjoint action of X, i.e.
Hence the inertia operator transforms the Euler equation (2.1) into an equation for m = A(u): result known also as the second Euler theorem. First Euler theorem states that the solution of (2.4) with m(a) = m 0 is where u = δ r c and c(a) = e. Indeed, d dt m = ad * (δ r c) Ad * (c)m 0 = ad * (u)m.
Remark 2. The Euler-Lagrange equation for a right invariant Lagrangian L : T G → R with value l : g → R at the identity is: also called the right Euler-Poincaré equation [47,30]. The Hamiltonian form (2.4) of Euler's equation is obtained for l(u) = 1 2 u, u since the functional derivative δl δu is A(u) in this case.

Ideal hydrodynamics
Let G = Diff µ (M ) be the regular Fréchet Lie group of volume preserving diffeomorphisms of a compact Riemannian manifold (M, g) with induced volume form µ. Its Lie algebra is g = X µ (M ), the Lie algebra of divergence free vector fields, with Lie bracket the opposite of the usual bracket of vector fields ad(X)Y = −[X, Y ]. We consider the right invariant metric on G given by the L 2 scalar product on vector fields with (∇X) ⊤ denoting the adjoint of the (1,1)-tensor ∇X relative to the metric: g(∇ Z X, Y ) = g(Z, (∇X) ⊤ Y ). In particular ad(X) ⊤ X = P (∇ X X) = ∇ X X + grad p, with p the smooth function uniquely defined up to a constant by ∆p = div(∇ X X). Now Theorem 1 assures that the geodesic equation in Diff µ (M ), in terms of the right logarithmic derivative u of the geodesic, is Euler's equation for ideal f low with velocity u and pressure p [41,2,10]: The geodesic equation (3.2) written for the vorticity 2-form ω = du ♭ , ♭ denoting the inverse of the Riemannian lift ♯ and L the Lie derivative, is because (∇ u u) ♭ = L u u ♭ − 1 2 d(g(u, u)) and (grad p) ♭ = dp.

Burgers equation
Let G = Diff(S 1 ) be the group of orientation preserving diffeomorphisms of the circle and g = X(S 1 ) the Lie algebra of vector fields. The Lie bracket is [X, Y ] = X ′ Y − XY ′ , the negative of the usual bracket on vector fields (vector fields on the circle are identified here with their coefficient functions in C ∞ (S 1 )). We consider the right invariant metric on G given by the L 2 scalar product X, Y = S 1 XY dx on g. The adjoint of ad(X) is ad(X) ⊤ Y = 2X ′ Y + XY ′ , because: It follows from Theorem 1 that the geodesic equation on Diff(S 1 ) in terms of the right logarithmic derivative u : I → C ∞ (S 1 ) is Burgers inviscid equation [7]: The higher dimensional Burgers equation is the template matching equation, used for comparing images via a deformation induced distance. It is the geodesic equation on Diff(M ), the diffeomorphism group of a compact Riemannian manifold (M, g), for the right invariant L 2 metric [18,55]: Indeed, because as in Section 3 we compute ad(X) In particular for M = S 1 and u a curve in X(S 1 ), identified with C ∞ (S 1 ), div u = u ′ and g(u, u) = u 2 , so each of the three terms in the right hand side of (4.2) is −uu ′ and we recover Burgers equation (4.1).

Abelian extensions
A bilinear skew-symmetric map ω : g × g → V is a 2-cocycle on the Lie algebra g with values in the g-module V if it satisfies the condition where b : g → L(V ) denotes the Lie algebra action on V . It determines an Abelian Lie algebra extensionĝ := V ⋊ ω g of g by the g-module V with Lie bracket There is a 1-1 correspondence between the second Lie algebra cohomology group H 2 (g, V ) and equivalence classes of Abelian Lie algebra extensions 0 → V →ĝ → g → 0.
When G is infinite dimensional, the two obstructions for the integrability of such an Abelian Lie algebra extension to a Lie group extension of the connected Lie group G involve π 1 (G) and π 2 (G) [43]. The Lie algebra 2-cocycle ω is integrable if • the period group Π ω ⊂ V (the group of spherical periods of the equivariant V -valued 2-form on G defined by ω) is discrete and • the flux homomorphism F ω : π 1 (G) → H 1 (g, V ) vanishes.
Then for any discrete subgroup Γ of the subspace of g-invariant elements of V with Γ ⊇ Π ω , there is an Abelian Lie group extension 1 → T →Ĝ → G → 1 of G by T = V /Γ. There are two special cases: 1. Semidirect product:ĝ = V ⋊ g, obtained when ω = 0. An example is the semidirect product g * ⋊ G for the coadjoint G-action on g * , called the magnetic extension in [3]. It has the Lie algebra g * ⋊ g, a semidirect product for the coadjoint g-action b = ad * on g * .

2.
Central extension:ĝ = V × ω g, obtained when b = 0. An example is the Virasoro algebra R × ω X(S 1 ), a central extension of the Lie algebra of vector fields on the circle given by the Virasoro cocycle ω(X, It has a corresponding Lie group extension of the group Diff(S 1 ) of orientation preserving diffeomorphisms of the circle, defined by the Bott group cocycle: An example of a general Abelian Lie algebra extension is C ∞ (M ) ⋊ ω X(M ), the Abelian extension of the Lie algebra of vector fields on the manifold M with the opposite bracket by the natural module of smooth functions on M , the Lie algebra action being b( is given by a closed differential 2-form η on M . If η is an integral form, then there is a principal circle bundle P over M with curvature η. In this case the group of equivariant automorphisms of P is a Lie group extension integrating the Lie algebra cocycle ω:

Geodesic equations on Abelian extensions
Following [57] we write down the geodesic equations on an Abelian Lie group extensionĜ of G with respect to the right invariant metric defined with the scalar product on its Lie algebraĝ = V ⋊ ω g. Here , g and , V are scalar products on g and V . We have to assume the existence of the following maps: the adjoint ad(X) ⊤ : g → g and the adjoint b(X) ⊤ : V → V for any X ∈ g, the linear map h : V → L skew (g) taking values in the space of skew-adjoint operators on g, defined by and the bilinear map l : The diamond operation ⋄ : V × V * → g in [19] corresponds to our map l via , V .
Proposition 1. The geodesic equation on the Abelian extensionĜ for the right invariant metric defined by the scalar product (6.1) onĝ, written for the right logarithmic derivative (f, u), i.e. for curves u in g and f in V , is Proof . We compute the adjoint of ad(v, X) in V ⋊ ω g w.r.t. the scalar product (6.1) The result follows now from Euler's equation (2.1).

Remark 3. When the scalar product on
then l is skew-symmetric and the geodesic equation becomes

Geodesic equations on semidirect products
A special case of Proposition 1, obtained for ω = 0, is: The geodesic equation on the semidirect product Lie group V ⋊ G for the right invariant metric defined by the scalar product (6.1), written for the curve

It reduces to
when the scalar product on V is g-invariant.

Passive scalar motion
The geodesic equation on the semidirect product C ∞ (M ) ⋊ Diff µ (M ) with L 2 right invariant metric, written for the right logarithmic derivative (f, u) : I → C ∞ (M )⋊X µ (M ) models passive scalar motion [17]: In this case the L 2 scalar product on C ∞ (M ) is X µ (M )-invariant and we apply Corollary 1 to get this geodesic equation.

Magnetohydrodynamics
Let A : g → g * be the inertia operator defined by a fixed scalar product , on g. The scalar product on the regular dual g * reg = A(g) induced via A by this scalar product in g is again denoted by , . Next we consider the subgroup g * reg ⋊ G of the magnetic extension g * ⋊ G, with right invariant metric of type (6.1) [56]. Proposition 2. If the adjoint of ad(X) exists for any X ∈ g, then the geodesic equation on the magnetic extension g * reg ⋊ G with right invariant metric, written for the curve Proof . We have to compute the map l : g * reg × g * reg → g and the adjoint b(X) ⊤ : g * reg → g * reg for b = ad * . We use the fact (2.3) that the coadjoint action on the image of A comes from the opposite of ad(·) ⊤ . Then The result follows now from Corollary 1.
For G = SO (3) and left invariant metric on its magnetic extension g * ⋊ G one obtains Kirchhoff equations for a rigid body moving in a fluid.
Let G = Diff µ (M ) be the group of volume preserving diffeomorphisms on a compact manifold M and g = X µ (M ). The regular part g * reg of g * is naturally isomorphic to the quotient space Ω 1 (M )/dΩ 0 (M ) of differential 1-forms modulo exact 1-forms, the pairing being ( Considering the right invariant L 2 metric on the magnetic extension g * reg ⋊ G determined by the L 2 scalar product (3.1) on vector fields, the geodesic equations for the time dependent divergence free vector fields u and B are (by Proposition 2) We specialize to a three dimensional manifold M . The curl of a vector field X is the vector field defined by the relation i curl X µ = dX ♭ and the cross product of two vector fields X and Y is the vector field defined by the relation ( . The geodesic equations above are in this case the equations of ideal magnetohydrodynamics with velocity u, magnetic field B and pressure p [54,32]:

Magnetic hydrodynamics with asymmetric stress tensor
Let M be a 3-dimensional compact parallelizable Riemannian manifold with induced volume form µ and let G = Diff µ (M ) with g = X µ (M ). Each vector field X on M can be identified with a smooth function in C ∞ (M, R 3 ), and j(X) ∈ C ∞ (M, gl(3, R)) denotes its Jacobian. Then ω(X, Y ) = [tr(j(X)dj(Y ))] ∈ Ω 1 (M )/dΩ 0 (M ) is a Lie algebra 2-cocycle on g with values in the regular dual g * reg . Considering the L 2 scalar product on the Abelian extension g * reg ⋊ ω g, we get the following Euler equation [5] for time dependent divergence free vector fields u and B: modeling magnetic hydrodynamics with asymmetric stress tensor T = j(B) • j(u).

Geodesic equations on central extensions
When V = R is the trivial g-module, then the Lie algebra action b vanishes and we get a central extension R × ω g defined by the cocycle ω : g × g → R. A consequence of Proposition 1 is: where u is a curve in g and k ∈ L skew (g) is defined by the Lie algebra cocycle ω via Proof . The central extension is a particular case of an Abelian extension, so Proposition 1 can be applied. The linear map h :

KdV equation
The geodesic equation on the Bott-Virasoro group (5.2) for the right invariant L 2 metric is the Korteweg-de Vries equation [44]. In this case the Lie algebra is the central extension of g = X(S 1 ) (identified with C ∞ (S 1 )) given by the Virasoro cocycle ω(X, 10 Superconductivity equation Given a compact manifold M with volume form µ, each closed 2-form η on M defines a Lichnerowicz 2-cocycle ω η on the Lie algebra of divergence free vector fields, The kernel of the flux homomorphism is the Lie algebra X ex µ (M ) of exact divergence free vector fields. On a 2-dimensional manifold it consists of vector fields X possessing stream functions f ∈ C ∞ (M ), i.e. i X µ = df (X is the Hamiltonian vector field with Hamiltonian function f ). On a 3-dimensional manifold it consists of vector fields X possessing vector potentials A ∈ X(M ), i.e. i X µ = dA ♭ (X is the curl of A).
The Lie algebra homomorphism flux µ integrates to the flux homomorphism (due to Thurston) Flux µ on the identity component of the group of volume preserving diffeomorphisms: For η integral, the Lichnerowicz cocycle is integrable to Diff ex µ (M ) [23]. When M is 3dimensional, there exists a vector field B on M defined with η = −i B µ. The 2-form η is closed if and only if B is divergence free. The integrality condition of η expresses as S (B · n)dσ ∈ Z on every closed surface S ⊂ M .
The superconductivity equation models the motion of a high density electronic gas in a magnetic field B with velocity u: It is the geodesic equation on a central extension of the group of volume preserving diffeomorphisms for the right invariant L 2 metric [61,57], when M is simply connected. Indeed, hence the map k ∈ L skew (g) determined by the Lichnerowicz cocycle ω η is k(X) = P (X × B), with P denoting the orthogonal projection on the space of divergence free vector fields. Now we apply Corollary 2.

Charged ideal fluid
Let M be an n-dimensional Riemannian manifold with Levi-Civita connection ∇ and volume form µ, and η a closed integral differential two-form. Let B be an (n − 2) vector field on M (i.e. B ∈ C ∞ (∧ n−2 T M )) such that η = (−1) n−2 i B µ is a closed two-form. The cross product of a vector field X with B is the vector field X×B = (i X∧B µ) ♯ = (i X η) ♯ , ♯ denoting the Riemannian lift. When M is 3-dimensional, then B is a divergence free vector field with η = −i B µ and × is the cross product of vector fields. From the integrality of η follows the existence of a principal T-bundle π : P → M with a principal connection 1-form α on P having curvature η. The associated Kaluza-Klein metric κ on P , defined at a point x ∈ P by κ x (X,Ỹ ) = g π(x) (T x π.X, T x π.Ỹ ) + α x (X)α x (Ỹ ),X,Ỹ ∈ T x P determines the volume formμ = π * µ ∧ α on P .
The group Diffμ(P ) T of volume preserving automorphisms of the principal bundle P is an Abelian Lie group extension of Diff µ (M ) [P ] , the group of volume preserving diffeomorphisms preserving the bundle class [P ], by the gauge group C ∞ (M, T) (an extension contained in (5.3)). The corresponding Abelian Lie algebra extension is described again by the Lie algebra cocycle ω : X µ (M ) × X µ (M ) → C ∞ (M ) given by η.
The Kaluza-Klein metric on P determines a right invariant L 2 metric on the group of volume preserving automorphisms of the principal T-bundle P . The geodesic equation written in terms of the right logarithmic derivative (ρ, u), with ρ a time dependent function and u a time dependent divergence free vector field on M , is: It models the motion of a charged ideal f luid with velocity u, pressure p and charge density ρ in a fixed magnetic field B [57].
Indeed, the connection α defines a horizontal lift and identifying the pair (f, X), f ∈ C ∞ (M ), X ∈ X µ (M ) with the sum of the horizontal lift of X and the vertical vector field given by f , we get an isomorphism between the Abelian Lie algebra extension C ∞ (M ) ⋊ ω X µ (M ) and the Lie algebra Xμ(P ) T of invariant divergence free vector fields on P . Under this isomorphism the L 2 metric defined by the Kaluza-Klein metric κ is ( where P denotes the orthogonal projection on the space of divergence free vector fields on M . The result follows from Remark 3, knowing that ad(X) ⊤ X = P (∇ X X).

Geodesics on general extensions
A general extension of Lie algebras is an exact sequence of Lie algebras 0 → h →ĝ → g → 0. (12.1) A section s : g →ĝ (i.e. a right inverse to the projectionĝ → g) induces the following mappings [1]: with properties: The Lie algebra structure on the extensionĝ, identified as a vector space with h ⊕ g via the section s, can be expressed in terms of b and ω: In particular for h an Abelian Lie algebra this is the Lie bracket (5.1) on an Abelian Lie algebra extension.
We consider scalar products , g on g and , h on h and, as in Section 6, we impose the existence of several maps: ad(X) ⊤ : g → g for any X ∈ g, ad(f ) ⊤ : h → h for any f ∈ h, b(X) ⊤ : h → h for any X ∈ g, as well as the linear map h : h → L skew (g) defined by h(f )X 1 , X 2 g = ω(X 1 , X 2 ), f h , and the bilinear map l : h × h → g, defined by A result similar to Proposition 1 is: Proposition 3. The geodesic equation on the Lie group extensionĜ of G by H integrating (12.1), with right invariant metric determined by the scalar product written in terms of the right logarithmic derivative (ρ, u) is:

Ideal fluid in a fixed Yang-Mills field
Let π : P → M be a principal G-bundle with principal action σ : P × G → P and let Ad P = P × G g be its adjoint bundle. The space Ω k (M, Ad P ) of differential forms with values in Ad P is identified with the space Ω k hor (P, g) G of G-equivariant horizontal forms on P . In particular C ∞ (M, Ad P ) = C ∞ (P, g) G .
Let g be a Riemannian metric on M and γ a G-invariant scalar product on g. These data, together with the connection α, define a Kaluza-Klein metric on P : κ x (X,Ỹ ) = g π(x) (T x π.X, T x π.Ỹ ) + γ(α x (X), α x (Ỹ )),X,Ỹ ∈ T x P.
The canonically induced volume form on P isμ = π * µ∧α * det γ , where µ is the canonical volume form on M induced by the Riemannian metric g and α * det γ is the pullback by α : T P → g of the determinant det γ ∈ ∧ dim g g * induced by the scalar product γ on g.
The gauge group of the principal bundle is identified with C ∞ (P, G) G , the group of Gequivariant functions from P to G, with G acting on itself by conjugation. The group of automorphisms of P , i.e. the group of G-equivariant diffeomorphisms of P , is an extension of Diff(M ) [P ] , the group of diffeomorphisms of M preserving the bundle class [P ], by the gauge group. This is the analogue of (5.3) for non-commutative structure group. Restricting to volume preserving diffeomorphisms, we get the exact sequence: On the Lie algebra level the exact sequence is The horizontal lift provides a linear section : X µ (M ) → Xμ(P ) G , thus identifying the pair (f, X) ∈ C ∞ (P, g) G ⊕ X µ (M ) withX =σ(f ) + X hor ∈ Xμ(P ) G . With this identification, the L 2 metric on Xμ(P ) G given by the Kaluza-Klein metric can be written as A particular case of a result in [14] is the fact that the geodesic equation on the group Diffμ(P ) G of volume preserving automorphisms of P with right invariant L 2 metric gives the equations of motion of an ideal f luid moving in a f ixed Yang-Mills f ield. Written for the right logarithmic derivative (ρ, u) : I → C ∞ (P, g) G ⊕ X µ (M ), these are: Here u denotes the Eulerian velocity, ρ, viewed as a time dependent section of Ad P , denotes the magnetic charge, η, viewed as a 2-form on M with values in Ad P , denotes the fixed Yang-Mills field. The scalar product γ being G-invariant, can be viewed as a bundle metric on Ad P . This result follows from Proposition 3. Indeed, in this particular case the cocycle is ω = η and the Lie algebra action is b( . Moreover ad(X) ⊤ X = P ∇ X X with P the projection on divergence free vector fields and ad(f ) ⊤ f = [f, f ] = 0, so (13.1) follows.
The equations of a charged ideal fluid from Section 11 are obtained for the structure group G equal to the torus T.
14 Totally geodesic subgroups From the Euler equation (2.1) we see that this is the case if ad(X) ⊤ X ∈ h for all X ∈ h. If there is a geodesic in G in any direction of h, then this condition is necessary and sufficient, so we give the following definition: the Lie subalgebra h is called totally geodesic in g if ad(X) ⊤ X ∈ h for all X ∈ h. Remark 4. Given two totally geodesic Lie subalgebras h and k of the Lie algebra g, the intersection h ∩ k is totally geodesic in g, but also in h and in k. In particular the ideal fluid flow on the 2-torus preserves the property of having a stream function [3] and the ideal fluid flow on the 3-torus preserves the property of having a vector potential.

Superconductivity
Given a compact Riemannian manifold M , from the Hodge decomposition follows that X µ (M ) = X ex µ (M ) ⊕ X harm (M ). On a flat torus the harmonic vector fields are those with all components constant.
In the setting of Section 10, the next proposition determines when is R ⋊ ωη X ex µ (M ) totally geodesic in R ⋊ ωη X µ (M ), for M = T 3 and η = −i B µ. Proof . Any exact divergence free vector field X on the 3-torus admits a potential 1-form α Then the totally geodesicity condition which, in this case, says that P (X × B) is exact divergence free for all X exact divergence free, is equivalent to [Y, B] = 0 for all harmonic vector fields Y . This is further equivalent to the fact that the three components of the magnetic field B are constant.

Passive scalar motion
On the trivial principal T bundle P = M × T we consider the volume formμ = µ ∧ dθ. Noticing that i (f,X)μ = i X µ∧dθ+f µ, we get the Lie algebra isomorphisms Xμ(M ×T) is the subspace of functions with vanishing integral.
From [55] we know that the group of equivariant volume preserving diffeomorphisms is totally geodesic in the group of volume preserving diffeomorphisms and from Theorem 2 we know that the group of exact volume preserving diffeomorphisms of a torus is totally geodesic in the group of volume preserving diffeomorphisms, hence by Remark 4 we obtain that for M = T 2 the subgroup Diff ex In other words equation (7.1), describing passive scalar motion, preserves the property of having a stream function if f has zero integral at the initial moment. Moreover, f will have zero integral at any moment.

Quasigeostrophic motion
Given a closed 1-form α on the compact symplectic manifold (M, σ), the Roger cocycle on the Lie algebra X ex σ (M ) of Hamiltonian vector fields on M is [49] ω α (H f , H g ) = M f α(H g )σ n .
Here f and g are Hamiltonian functions with zero integral for the Hamiltonian vector fields H f and H g . The integrability of the 2-cocycle ω α to a central extension of the group of Hamiltonian diffeomorphisms is an open problem. Partial results are given in [24]. For M = T 2 the cocycle ω α can be extended to a cocycle on the Lie algebra of symplectic vector fields X σ (T 2 ) by ω α (∂ x , ∂ y ) = ω α (∂ x , H f ) = ω α (∂ y , H f ) = 0 [27]. The extendability of ω α to X σ (M ) for M an arbitrary symplectic manifold is studied in [59]. To a divergence free vector field X on the 2-torus one can assign a smooth function ψ X on the 2-torus uniquely determined by X through dψ X = i X σ − i X σ and T 2 ψ X σ = 0. Here denotes the average of a 1-form on the torus: adx + bdy = ( T 2 aσ)dx + ( T 2 bσ)dy. In particular ψ H f = f whenever f has zero integral.

Proposition 5 ([60]). The Euler equation for the
where the function ψ u is uniquely determined by u through dψ u = i u σ − i u σ and T 2 ψ u σ = 0.

Proposition 6 ([60]
). If the two components of the 1-form α on T 2 are constant, then equation (15.1) preserves the property of having a stream function, i.e. R × ωα X ex σ (T 2 ) is totally geodesic in R × ωα X σ (T 2 ). In this case the restriction of (15.1) to Hamiltonian vector fields is Proof . By Theorem 2 on the 2-torus P (∇ X X) is Hamiltonian for X Hamiltonian, hence the totally geodesicity condition in this case is equivalent to the fact that P (ψ X α ♯ ) is Hamiltonian for X Hamiltonian. By Hodge decomposition this means ψ X α ♯ is orthogonal to the space of harmonic vector fields, so On the torus the harmonic vector fields Y are the vector fields with constant components and the functions ψ X have vanishing integral by definition, so the expression above vanishes for all constant vector fields Y if the 1-form α has constant coefficients.

Central extensions of semidirect products
Let g be a Lie algebra with scalar product , g and V a g-module with g-action b and g-invariant scalar product , V . Each Lie algebra 1-cocycle α ∈ Z 1 (g, V ) (i.e. a linear map α : g → V which satisfies α([X 1 , X 2 ]) = b(X 1 )α(X 2 ) − b(X 2 )α(X 1 )) defines a 2-cocycle ω on the semidirect product V ⋊ g [45]: Proposition 7. The Euler equation on the central extension (g ⋉ V ) × ω R with respect to the scalar product , g + , V , written for curves u in g and f in V , is Proof . The map k ∈ L skew (V ⋊ g) defined by ω is k(v, X) = (α(X), −α ⊤ (v)) because The result follows from Corollaries 1 and 2.
Remark 5. More generally, a 1-cocycle α on g with values in the dual g-module V * defines a 2-cocycle on V ⋊ g by where ( , ) denotes the pairing between V * and V . :

2)
with ∇ the Levi-Civita covariant derivative and ♯ the Riemannian lift.
Proof . We apply Proposition 7 for g = X µ (M ) and V = C ∞ (M ). In this case b(X)f = −L X f and ad(X) ⊤ X = P ∇ X X. We compute α ⊤ (f ), Because d(f α ♯ ) ♭ = df ∧ α, the equation (17.2) written for vorticity 2-form ω = du ♭ becomes Proof . We know from Section 14 that for M = T 2 , X ex µ (M ) ⋉ C ∞ 0 (M ) is totally geodesic in X µ (M ) ⋉ C ∞ (M ). But α(u) has zero integral for u exact divergence free, α being closed. We have to make sure that f α ♯ is orthogonal to the space of harmonic vector fields for all f with zero integral, i.e. for all functions f such that f µ is exact (f µ = dν). But M g(f α ♯ , Y )µ = M α(Y )dν = − M L Y α ∧ ν = 0 because L Y α = 0 for all harmonic vector fields Y on the 2-torus, α being a constant 1-form.
Hence on the 2-torus, for constant α and initial conditions u 0 Hamiltonian vector field and f 0 function with zero integral, u will be Hamiltonian and f will have zero integral at every time t. The Hamiltonian vector field is H ψ = ∂ y ψ∂ x − ∂ x ψ∂ y and the Poisson bracket is the Jacobian of f and ψ. If α = −βdy and a = −1 we get the equation for stream function ψ and vorticity function ω = ∆ψ: Let ξ = g ρ−ρ 0 ρ 0 be a buoyancy variable measuring the deviation of a density ρ from a background value ρ 0 , with g the gravity acceleration. The background stratification ρ 0 is assumed to be exponential, characterized by the constant Brunt-Väisälä frequency N = (−g d log ρ 0 dy ) 1 2 . The equation for a stratif ied f luid in Boussinesq approximation [61] is the geodesic equation (17.3) for β = N constant and f = N −1 ξ: When the Brunt-Väisälä frequency N is an integer and ξ has zero integral (at time zero), then the stratified fluid equation is a geodesic equation on a Lie group [58].

Camassa-Holm equation
The Camassa-Holm equation is the geodesic equation for the right invariant metric on Diff(S 1 ) given by the H 1 scalar product [29]. Indeed, one gets from Indeed, the identity ω(X, Y ) = k(X), Y for the Virasoro cocycle ω(X, Y ) = 2 S 1 X ′′′ Y dx and the H 1 scalar product implies k(X) = 2(1−∂ 2 x ) −1 X ′′′ . Now by Corollary 2 the geodesic equation is the extended Camassa-Holm equation above.

Higher dimensional Camassa-Holm equation
The higher dimensional Camassa-Holm equation (also called EPDiff or averaged template matching equation) [19,20] is the geodesic equation for the right invariant H 1 metric X, Y = M (g(X, Y ) + α 2 g(∇X, ∇Y ))µ, (18.2) on Diff(M ) for compact M . Because ∇ * ∇ = ∆ + Ric, this scalar product can be rewritten with the help of the rough Laplacian ∆ R = ∆ + Ric as X, Y = M (g(X − α 2 ∆ R X, Y )µ, so the momentum density of the fluid m = A(u) is m = (1 − α 2 ∆ R )u. It follows that the adjoint of ad(X) with respect to (18.2) is conjugate by 1 − α 2 ∆ R to the adjoint of ad(X) with respect to the L 2 metric (3.1) computed to be (4.3).
. We get as geodesic equation the higher dimensional Camassa-Holm equation

In Hamiltonian form this equation is
In particular for M = S 1 we get the Camassa-Holm equation (18.1). When M is a manifold with boundary and we put Neumann or mixed conditions on the boundary, then the H 1 scalar product (18.2) has to be replaced by X, Y = M (g(X, Y ) + 2α 2 g(Def X, Def Y ))µ, (18.3) where Def X = 1 2 (∇X + (∇X) ⊤ ) denotes the deformation (1, 1)-tensor of X [15].

Averaged Euler equation
For a compact Riemannian manifold M we consider the right invariant metric on the group Diff µ (M ) of volume preserving diffeomorphisms given by the H 1 scalar product (18.2) on vector fields. The geodesic equation is the (Lagrangian) averaged Euler equation [31,50], also called LAE-α equation: and we use Euler's equation (

Systems of two evolutionary equations
From [12] we know that a basis for H 2 (X(S 1 ), C ∞ (S 1 )) is represented by σ(X, Y ) = X ′ Y − XY ′ and the Virasoro cocycle ω(X, Y ) = S 1 (X ′ Y ′′ − X ′′ Y ′ )dx ∈ R ⊂ C ∞ (S 1 ); a basis for H 2 (X(S 1 ), Ω 1 (S 1 )) is represented by the cocycles a basis for H 2 (X(S 1 ), Ω 2 (S 1 )) is represented by the cocycles Only the cocycles ω, ω 1 and ω 2 (whose expressions involve only derivatives of X and Y ) integrate to group cocycles [45]. The Euler equations for the L 2 or H 1 scalar product on the corresponding Abelian extensions provide systems of two equations, generalizing Burgers (4.1) or Camassa-Holm (18.1) equation. We exemplify with the 2-cocycle σ taking values in the module of functions on the circle. The Euler equations for the L 2 scalar product on C ∞ (S 1 ) ⋊ X(S 1 ) and on C ∞ (S 1 ) ⋊ σ X(S 1 ) are The Euler equations for the H 1 scalar product on C ∞ (S 1 ) ⋊ X(S 1 ) and on C ∞ (S 1 ) ⋊ σ X(S 1 ) are One can consider central extensions of semidirect products of X(S 1 ) with modules of densities as in Remark 5 [45]. For instance the 1-cocycle α(X) = X ′′ on X(S 1 ) with values in Ω 1 (S 1 ), the module dual to C ∞ (S 1 ), gives the 2-cocycle ω((f 1 , X 1 ), (f 2 , X 2 )) = S 1 (X ′′ 1 f 2 − X ′′ 2 f 1 )dx on the semidirect product C ∞ (S 1 ) ⋊ X(S 1 ). The geodesic equation for the L 2 scalar product on the central extension (C ∞ (S 1 ) ⋊ X(S 1 )) × ω R is

Conclusions
This survey article presents the formal deduction as geodesic equations on diffeomorphism groups with right invariant metrics of several PDE's of hydrodynamical type. Sometimes extensions of diffeomorphism groups by central or Abelian sugroups come into play and the corresponding Lie algebra 2-cocycles introduce additional terms to the geodesic equations.
These equations are Hamiltonian equations too, possessing rich geometric structures. Some of them are completely integrable. But presenting these results is beyond the scope of this article.