Regularity of the Geodesic Flow of the Incompressible Euler Equations on a Manifold

Abstract

In this paper we investigate the regularity in time of the volume-preserving geodesic flow, which is associated with the incompressible Euler equations on a compact d-dimensional Riemannian manifold with boundary. Our result, which completes the local-in-time well-posedness theory of Ebin and Marsden (Ann Math 92:102–163, 1970), states roughly that the time smoothness of geodesic curves is only limited by the smoothness of the manifold. Such regularity is measured in a broad class of ultradifferentiable functions, which includes the real analytic and Gevrey classes. A by-product of this simple and constructive proof is new ideas to design high-order semi-Lagrangian methods for integrating the incompressible Euler equations on a manifold.

Appendices

Differential Geometry Notation

In this appendix, we recall differential geometry notation, which follow mainly the standard notation of classical monographs. To keep this presentation as short as possible we will not always give the definition of these differential geometry tools. For a precise definition of tools left undefined, we refer the reader to the following classical monographs of differential geometry on manifolds [1, 15, 18, 22, 50, 54]. We also refer the reader to Appendix “Differential geometry in a nutshell” of [7]. Indeed this very short overview of differential geometry tools, used here, can serve as a reminder for the reader.

Consider a LSL–FdB ultradifferentiable compact Riemannian $\partial$-manifold of dimension $d$ with boundary $\partial M$. The set of tangent vectors to M at $a\in M$ forms a vector space $TM_a$, which is called the tangent space to M at a. The union of the tangent spaces to M at the various points of M, i.e., $TM:= \cup _{a\in M}TM_a$, is called the tangent bundle of M. A vector field on M is a (cross-)section of TM. Let us recall that a (cross-)section of a vector bundle assigns to each base point $a\in M$ a vector in the fiber $\pi ^{-1}(a)$ over a and the addition and scalar multiplication of sections take place within each fiber. Here, the mapping $\pi :TM\rightarrow M$, which takes a tangent vector X to the point $a\in M$ at which the vector is tangent to M (i.e., $X\in TM_a$), is called the natural projection. The inverse image of a point $a\in M$ under the natural projection, i.e., $\pi ^{-1}(a)$, is the tangent space $TM_a$. This space is called the fiber of the tangent bundle over the point a. The space $\Gamma (TM)$ of all smooth sections of TM is noted $\mathfrak {X}(M):=\Gamma (TM)$ and describes all smooth vector fields on M. The dual of the tangent bundle, noted $T^*M$, can be constructed through linear forms, called 1-forms or cotangent vectors, acting on vectors of the tangent bundle TM. The cotangent space to M at a is noted $T^*M_a$, and the cotangent bundle is the union of the cotangent spaces to the manifold M at all its points, that is $T^*M:=\cup _{a\in M} T^*M_a$. The space of all smooth sections $\Omega ^1(M):=\Gamma (T^*M)$ is called the space of differential 1-forms on M. The space of 0-forms is the space of smooth functions on M and is noted $\Omega ^0(M)$.

The Riemannian metric is given by the infinitesimal line element $ds^2$, which is defined by the metric tensor g: $ds^2=g=g_{ij}da^ida^j=g_{ij}(a)da^i\otimes da^j.$ The tensor g endows each tangent vector space $TM_a$ with an inner or scalar product, $(\cdot , \cdot )_{g}$ called also Riemannian metric and defined as: $\forall a \in M$, $\forall X,\, Y \in TM_a$, $(X,Y)_{g}=g_{ij}(a)X^i(a)Y^j(a)$. The components of g are LSL–FdB ultradifferentiable functions. Therefore, using the inner product $(\cdot , \cdot )_{g}$, we get an isomorphism between the tangent bundle TM and the cotangent bundle $T^*M$. In particular, it induces an isomorphism of spaces of sections, which is called the raising operator $(\cdot )^\sharp : \Omega ^1(M) \rightarrow \mathfrak {X}(M)$. It is defined by: $\forall \omega \in \Omega ^1(M)$, $\omega ^\sharp =( \omega _ida^i)^\sharp =(\omega ^\sharp )^i\partial /\partial a^i=\omega ^i \partial /\partial a^i$, where contravariant components are given by $(\omega ^\sharp )^i=\omega ^i=g^{ij}\omega _j$. The inverse of the raising operator, named the lowering operator $(\cdot )^\flat : \mathfrak {X}(M) \rightarrow \Omega ^1(M)$, is defined by: $\forall X\in \mathfrak {X}(M)$, $X^\flat =(X^i \partial /\partial a^i)^\flat = (X^\flat )_i da^i=X_ida^i$, where covariant components are given by $(X^\flat )_i=X_i=g_{ij}X^j$. Of course we have $g_{ik}g^{kj}=\delta _{i}^j$ where $\delta _{i}^j$ is the constant diagonal metric with unity on the diagonal.

The space of all anti-symmetric k-linear maps $\omega _{|_a}:TM_a \times \cdots \times TM_a \rightarrow {\mathbb {R}}$ at the point $a\in M$ is noted $\Lambda ^k(T_aM)$. Then the exterior k-form bundle is defined as $\Lambda ^k(TM)=\cup _{a\in M}\Lambda ^k(T_aM)$. The space all smooth sections $\Omega ^k(M):=\Gamma (\Lambda ^k(TM))$ is called the space of differential k-forms on M. The Riemannian $\partial$-manifold is endowed with a metric volume d-form $\mu =\sqrt{|\mathrm {g}|} da^1\wedge \ldots \wedge da^{d}\equiv \sqrt{|\mathrm {g}|}da$, where $\sqrt{|\mathrm {g}|}=\sqrt{\mathrm{det}(g_{ij})}$. We continue by fixing the notation of standard operators acting on differential forms, the properties of which can be found in [1, 15, 18, 50, 54]. The operator $\mathrm{i}_X:\Omega ^k(M) \rightarrow \Omega ^{k-1}(M)$ denotes the interior product of a k-form with the vector field $X\in \mathfrak {X}(M)$. The Riemann-Levi-Civita connection or the covariant derivative on M is noted $\nabla$ and is an operator from $\Omega ^k(M)$ to $\Omega ^{k+1}(M)$. The exterior derivative (resp. coderivative), denoted d (resp. $d^*$), is an operator from $\Omega ^k(M)$ to $\Omega ^{k+1}(M)$ (resp. $\Omega ^{k-1}(M)$). In some textbooks the coderivative $d^*$ is denoted by $\delta$, but we prefer to avoid this notation for not be confused with Kronecker symbol or diagonal metric. The Laplace–De-Rham operator $\Delta$ is defined by $\Delta := dd^* + d^*d$. The Hodge dual operator is defined as the unique isomorphism $\star :\Omega ^k(M)\rightarrow \Omega ^{d-k}(M)$, which satisfies with and where the wedge symbol $\wedge$ denotes the exterior product. Using the metric for k-form on M, we can equipped the space of k-form field $\Omega ^k(M)$ with the $L^2$ scalar product and the induced norm . Let $\varphi : M\rightarrow M$ be a diffeomorphism between compact $\partial$-manifolds. Then, the operator $\varphi ^*: \Omega ^{k}(M) \rightarrow \Omega ^{k}(M)$ denotes the pullback transformation associated with $\varphi$.

To study differential forms on the boundary of M, we consider the inclusion $\jmath : \partial M \rightarrow M$ and its pullback $\jmath ^*: \Omega ^k(M) \rightarrow \Omega ^k(\partial M)$. The boundary manifold $\partial M$ carries a metric $\jmath ^*g$, which is canonically induced from the metric g on M. The corresponding Riemannian volume form $\mu _\partial$ is computed as $\mu _\partial =\mathrm{i}_{\nu }\mu _{|_{\partial M}}$, where the vector field $\nu$ is the outward pointing unit normal to the boundary $\partial M$. The restriction $\omega _{|_{\partial M}}\in \Omega ^{k}(M)_{|_{\partial M}}:=\Gamma (\Lambda ^k(TM)_{|_{\partial M}})$ is called the boundary value of $\omega \in \Omega ^{k}(M)$, and in particular one has $(\omega \wedge \gamma )_{|_{\partial M}}=\omega _{|_{\partial M}} \wedge \gamma _{|_{\partial M}}$ and $\star (\omega _{|_{\partial M}})=(\star \omega )_{|_{\partial M}}$. Every vector field $X\in \Gamma (TM_{|_{\partial M}})$ can be decomposed into its tangential and normal parts as $X=X^\parallel + X^\perp$, where $X^\perp =(X,\nu )_g\,\nu$ and $(X^\parallel ,\nu )_g=0$. The tangent bundle $T\partial M$ should not be confused with the bundle $TM_{|_{\partial M}}$. The latter is the restriction of the full tangent bundle TM to $\partial M$, and contains $T\partial M$ as a sub-bundle of co-dimension 1. Nevertheless there exists a smooth map, the tangent map $T\jmath :T\partial M\rightarrow TM_{|_{\partial M}}$, which induces a natural inclusion from $\Gamma (T\partial M)$ of the vector field on the boundary manifold into the space $\Gamma (TM_{|_{\partial M}})$ of vectors field on M sitting over the boundary. By means of the decomposition of a vector field $X\in \Gamma (TM_{|_{\partial M}})$ into its tangential and normal parts $X=X^\parallel + X^\perp$, we denote the tangential and normal trace operators respectively by $\mathbf{t}$ and $\mathbf{n}$, which are defined as follows: for $k\ge 1$,

$$\begin{aligned}&\forall \, X_1,\ldots ,X_k \in \Gamma (TM_{|_{\partial M}}), \ \ \omega \in \Omega ^k(M), \ \ \\&\quad \mathbf{t} \omega (X_1,\ldots ,X_k) =\omega (X_1^\parallel ,\ldots ,X_k^\parallel ) \ \text{ and } \ \mathbf{n}\omega =\omega _{|_{\partial M}} - \mathbf{t} \omega . \end{aligned}$$

For $k=0$, $\mathbf{t} \omega =\omega$ and $\mathbf{n}\omega =0$. The component $\mathbf{t} \omega$ (resp. $\mathbf{n} \omega$) is called the tangential (resp. normal) component of $\omega \in \Omega ^k( M)$. The tangential component $\mathbf{t} \omega$ is uniquely determinded by $\jmath ^*\omega$, i.e., one has $\mathbf{t} \omega = \jmath ^*\mathbf{t} \omega = \jmath ^*\omega$. We have the commutation relations $\star (\mathbf{n}\omega )=\mathbf{t}(\star \omega )$, $\star (\mathbf{t}\omega )=\mathbf{n}(\star \omega )$, $\mathbf{t}(d\omega )=d(\mathbf{t}\omega )$ and $\mathbf{n}(d^*\omega )=d^*(\mathbf{n}\omega )$. For more details on how to handle differential forms on the boundary of M, we refer the reader to Section 1 of Chapter 1 of [50].

The regularity in space of initial data, i.e., the regularity with respect to the Lagrangian spatial variables, is for convenience measured in Sobolev spaces. Of course the proof can be extended, for instance, in Hölder spaces or ultradifferentiable spaces (e.g., analytic functions). We refer the reader to [6, 32, 50] for an introduction to Sobolev spaces $W^{s,p}\Gamma (\mathbb {F})$ of sections of a general Riemannian vector bundle $\mathbb {F}$ over a $\partial$-manifold M, and especially for the particular case $W^{s,p}\Omega ^k(M)$ of differential forms. We use the notation $H^s\Omega ^k(M):=W^{s,2}\Omega ^k(M)$. The classical Sobolev spaces theory in domains of ${\mathbb {R}}^d$ (see, e.g., [2]) is covered by the general definition of spaces $W^{s,p}\Gamma (\mathbb {F})$. To make the bridge between the theory on section spaces and the classical one in ${\mathbb {R}}^d$ the main ingredient is the invariance of $W^{s,p}$-Sobolev topology under diffeomorphism between compact manifolds. Therefore, a number of central results (continuous and compact Sobolev embeddings, Sobolev inequalities,...) from theory in ${\mathbb {R}}^d$ can be generalized to section spaces $\Gamma (\mathbb {F})$. In particular $\omega \in W^{s,p}\Omega ^k(M)$ implies that coefficients of the k-form $\omega$ are in $W^{s,p}\Omega ^0(M)$. Since $H^s\Omega ^0(M)$ is an algebra with respect to the pointwise multiplication provided that $s> d/2$, there exists a constant $C_a:=C_a(s)$, which depends on s such that, for $s > d/2$, and $k,\,l \ge 0$,

$$\begin{aligned} \Vert \omega \wedge \gamma \Vert _{H^{s}\Omega ^{k+l}(M)} \le C_a \Vert \omega \Vert _{H^{s}\Omega ^k(M)} \Vert \gamma \Vert _{H^{s}\Omega ^l(M)}, \quad \omega \in H^{s}\Omega ^k(M), \ \ \gamma \in H^{s}\Omega ^l(M).\nonumber \\ \end{aligned}$$

(90)

Let $s, \ p, \ k$ be integers such that $s\in {\mathbb {N}}$, $1<p<\infty$ and $k\ge 1$. Since the differential operators considered below can be described in terms of the induced covariant derivative $\nabla$ on $\Omega ^k(M)$, we then have the following continuous mappings,

$$\begin{aligned} \begin{array}{rll} d&{} : &{} W^{s+1,p}\Omega ^k(M) \longrightarrow W^{s,p}\Omega ^{k+1}(M),\\ d^*&{}:&{} W^{s+1,p}\Omega ^k(M) \longrightarrow W^{s,p}\Omega ^{k-1}(M),\\ \Delta &{}:&{} W^{s+2,p}\Omega ^k(M) \longrightarrow W^{s,p}\Omega ^{k}(M),\\ \star &{}:&{} W^{s,p}\Omega ^k(M) \longrightarrow W^{s,p}\Omega ^{d-k}(M), \\ \varphi ^*&{}:&{} W^{s,p}\Omega ^k(M) \longrightarrow W^{s,p}\Omega ^{k}(M), \end{array} \end{aligned}$$

(91)

where $\varphi : M\rightarrow M$ is a diffeomorphism between compact $\partial$-manifolds and X is a smooth vector field on M (e.g., $X\in \mathfrak {X}(M)$ and is bounded or X is an element of $W^{s+1,p}$ sections of TM with $s>d/p$). More details can be found in the monograph [50].

Hodge Decomposition of Differential Forms on a Manifold

In this appendix, we present the Hodge decomposition theorem for k-forms on compact $\partial$-manifolds. We refer the reader to the monograph [50] for a proof of the following Theorem 3, and especially to Sect. 2.4 of [50]. For a compact $\partial$-manifolds, Hodge decomposition of differential forms in the spaces $L^2$ and $H^1$, was first given by Friedrichs [23] and Morrey [45, 46]. Its generalisation to differential forms of Sobolev class $W^{s,p}$, with $s\in {\mathbb {N}}$ and $1<p<\infty$ was proven by Schwarz [50]. Before stating the Hodge decomposition theorem, we need to introduce some spaces described by

Definition 2

Let M be a compact $\partial$-manifold. Let $s, \ p, \ k$ be integers such that $s\in {\mathbb {N}}$, $1<p<\infty$, and $k\ge 1$. We set the following spaces,

$$\begin{aligned} H^1\Omega _D^k(M)= & {} \left\{ \omega \in H^1\Omega ^k(M) \ |\ \mathbf{t}\omega =0 \right\} , \\ H^1\Omega _N^k(M)= & {} \left\{ \omega \in H^1\Omega ^k(M) \ |\ \mathbf{n}\omega =0 \right\} , \\ \mathcal {H}^k(M)= & {} \left\{ \omega \in H^1\Omega ^k(M) \ |\ d\omega =0 \text{ and } d^*\omega =0\right\} , \ \text{(space } \text{ of } \text{ harmonic } \text{ fields) }, \\ \mathcal {H}_N^k(M)= & {} H^1\Omega _N^k(M) \cap \mathcal {H}^k(M),\ \text{(space } \text{ of } \text{ Neumann } \text{ fields) },\\ \mathcal {H}_D^k(M)= & {} H^1\Omega _D^k(M) \cap \mathcal {H}^k(M),\ \text{(space } \text{ of } \text{ Dirichlet } \text{ fields) },\\ \mathcal {H}_\mathrm{ex}^k(M)= & {} \left\{ \omega \in \mathcal {H}^k(M) \ |\ \omega =d\gamma \right\} , \\ \mathcal {E}^k(M)= & {} \left\{ d\omega \ |\ \omega \in H^1\Omega _D^{k-1}(M)\right\} \subset L^2\Omega ^k(M),\\ \mathcal {C}^k(M)= & {} \left\{ d^*\gamma \ |\ \gamma \in H^1\Omega _N^{k+1}(M)\right\} \subset L^2\Omega ^k(M),\\ W^{s,p}\mathcal {E}^k(M)= & {} W^{s,p}\Omega ^k(M) \cap \mathcal {E}^k(M),\\ W^{s,p}\mathcal {C}^k(M)= & {} W^{s,p}\Omega ^k(M) \cap \mathcal {C}^k(M). \end{aligned}$$

Using Definition 2 we have

Theorem 3

(Hodge decomposition, Schwarz [50]). On a compact $\partial$-manifold M, the space $W^{s,p}\Omega ^k(M)$, with $k\ge 0$, $s\in {\mathbb {N}}$, and $1<p<\infty$, can be orthogonally decomposed into

$$\begin{aligned} W^{s,p}\Omega ^k(M) = W^{s,p}\mathcal {E}^k(M)\oplus W^{s,p}\mathcal {C}^k(M)\oplus \mathcal {H}_\mathrm{ex}^k(M) \oplus \mathcal {H}_N^k(M). \end{aligned}$$

(92)

Proof

See Corollary 2.4.9 of [50]. $\quad \square$

Remark 9

The space $W^{s,p}\Omega ^k(M)$ admits other Hodge decompositions, which depend on boundary conditions, among other criteria (see Section 2.4 of [50]). The orthogonal decomposition (92) is the so-called normal Hodge decomposition. There exist other orthogonal decompositions such as the tangential or the mixed Hodge decompositions (see, e.g., [3, 26, 50]). In Theorem 3, we choose the normal Hodge decomposition, because it is well suited for the boundary data available in our problem.

Derivation of the Cauchy Invariants Equation from the Relabelling Symmetry and a Variational Principle

In this appendix, from the relabelling symmetry, i.e., the invariance of the action under relabelling transformations, we obtain the Cauchy invariants equation on a d-dimensional Riemannian manifold. This derivation was originally performed in [7], but with an improper treatment of boundary terms. We give here the corrected version of this derivation, which does not directly make use of Noether’s theorem, but is reminiscent of its proof. Before stating the result, we give the formal definition of a relabelling transformation and we establish a Green formula. Another proof of existence of Cauchy invariants equation, rooted in differential geometry, is given by the particular application of Theorem 1 of [7] to the Lie-advected vorticity 2-form $\omega :=d\mathfrak {u}^\flat$.

Definition 3

(Relabelling transformation) .

Let $(M,g,\nabla ,\mu )$ be a Riemannian $\partial$-manifold. A relabelling transformation is a map $M \ni a\rightarrow \gamma (a)\in M$ such that

$$\begin{aligned} \gamma (a)=a + \delta a(a), \quad \delta a \in \mathfrak {g}, \end{aligned}$$

i.e., with,

$$\begin{aligned} \nabla _i \delta a^i=0 \quad \text{ and } \quad (\delta a, \nu )_g=0. \end{aligned}$$

In other words the vector field $\delta a$ is the infinitesimal generator of a group of volume-preserving diffeomorphisms of M that leave the boundary $\partial M$ invariant.

Lemma 3

(Green formula). Let $\mu _\partial$ be the metric volume form on the boundary manifold $\partial M$. Let $\nu$ be the outward pointing unit normal to the boundary $\partial M$. Then we have, for all $f\in \Omega ^0(M)$ and $X\in \mathfrak {X}(M)$,

$$\begin{aligned} \int _M \mu f \nabla _i X^i= -\int _M \mu \, \mathrm{i}_Xdf + \int _{\partial M} \mu _\partial f\, \mathrm{i}_\nu X^\flat . \end{aligned}$$

(93)

Proof

Equation (93) is the same as the following Green formula,

(94)

The proof of (94), which is based on Stokes’ theorem and the commutation relation $\mathbf{t}(\star \omega ) =\star (\mathbf{n}\omega )$, is given in Proposition 2.1.2 of [50]. Let us now retrieve (93) from (94). Using the definition of the scalar product, and using the relations $\star X^\flat = \mathrm{i}_X \mu$ and $df \wedge \mathrm{i}_X \mu = \mathrm{i}_Xdf \wedge \mu$, we obtain

Using the relations $\star d^*X^\flat = (d^*X^\flat )\star 1 = \mu d^*X^\flat =-\mu \nabla _i X^i$, we obtain

The proof of (93) ends by using the relation $\mathbf{t}f\wedge \star \mathbf{n} X^\flat = \mu _\partial f\mathrm{i}_\nu X^\flat ,$ which is proved in Proposition 1.2.6 of [50]. $\quad \square$

Theorem 4

(Cauchy invariants equation from the relabelling symmetry and variational principle) Let $\eta _t \in \mathrm{SDiff}(M,\mu )$ be the geodesic flow solving the incompressible Euler equations. We set $x:=\eta _t$, and $v:=\dot{\eta }_t :=\partial _t {\eta }_t=\mathfrak {u} \circ \eta _t$, with $v_0=\dot{\eta }_0=\mathfrak {u}_0$. Then the invariance of the action

$$\begin{aligned} \mathcal {A}:=\frac{1}{2}\int _0^Tdt\, \langle \mathfrak {u}(t), \mathfrak {u}(t) \rangle _{L^2(M)}= \frac{1}{2}\int _0^Tdt \int _{M} \mu \,(\mathfrak {u}(t),\mathfrak {u}(t))_g, \end{aligned}$$

(95)

under relabelling transformations of Definition 3 implies the following Cauchy invariants conservation law,

$$\begin{aligned} dv_k\wedge dx^k = \omega _0:= dv_0^\flat . \end{aligned}$$

(96)

Proof

The idea is first to compute the first-order variation of the action integral

$$\begin{aligned} \mathcal {A}(\eta ,M)=\frac{1}{2} \int _0^Tdt \int _M \mu (a) g_{ij}(\eta _t(a)) \partial _t\eta _t^i(a) \partial _t\eta _t^j(a), \end{aligned}$$

induced by the relabelling transformations of Definition 3. The variation of $\mathcal {A}(\eta ,M)$ is given by

$$\begin{aligned} \delta \mathcal {A}(\eta ,M)[\delta \eta ]= & {} \frac{1}{2} \delta \int _0^Tdt\int _{M}\mu (a)\, g_{ij}(\eta _t(a))\partial _t\eta _t^i(a) \partial _t\eta _t^j(a) \nonumber \\= & {} \frac{1}{2} \int _0^Tdt\int _{M}\mu (a)\, \partial _l g_{ij}(\eta _t(a))\delta \eta _t^l(a) \partial _t\eta _t^i(a) \partial _t\eta _t^j(a)\nonumber \\&+\, \int _0^Tdt\int _{M}\mu (a)\, g_{ij}(\eta _t(a)) \partial _t\delta \eta _t^i(a) \partial _t\eta _t^j(a). \end{aligned}$$

(97)

The relabelling transformation of Definition 3 induces a change in the Lagrangian flow $\eta _t$ at time t, given by

$$\begin{aligned} \delta \eta _t = \frac{\partial \eta _t}{\partial a^i} \delta \gamma ^i = \frac{\partial \eta _t}{\partial a^i} \delta a^i. \end{aligned}$$

(98)

Substituting (98) in (97), and using the product rule, we obtain

$$\begin{aligned}&\delta \mathcal {A}(\eta ,M)[\delta a]\nonumber \\&\quad =\int _0^T\int _{M}\mu (a)\,\left\{ \frac{1}{2} \partial _l g_{ij}(\eta _t(a)) \frac{\partial \eta _t^l(a)}{\partial a^m} \partial _t\eta _t^i(a) \partial _t\eta _t^j(a) \delta a^m \right. \nonumber \\&\qquad \left. +\, g_{ij}(\eta _t(a)) \partial _t\left( \frac{\partial \eta _t^i(a)}{\partial a^n}\right) \partial _t\eta _t^j(a) \delta a^n\right\} \nonumber \\&\quad =\int _0^T\int _{M}\mu (a)\,\left\{ \frac{1}{2} \partial _l g_{ij}(\eta _t(a)) \frac{\partial \eta _t^l(a)}{\partial a^m} \partial _t\eta _t^i(a) \partial _t\eta _t^j(a) \delta a^m \right. \nonumber \\&\qquad \left. +\, \partial _t\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \delta a^n - \partial _t\left( g_{ij}(\eta _t(a)) \partial _t\eta _t^j(a)\right) \frac{\partial \eta _t^i(a)}{\partial a^n}\delta a^n \right\} \nonumber \\&\quad =\int _0^T\int _{M}\mu (a)\,\left\{ \frac{1}{2} \partial _l g_{ij}(\eta _t(a)) \frac{\partial \eta _t^l(a)}{\partial a^m} \partial _t\eta _t^i(a) \partial _t\eta _t^j(a) \delta a^m\right. \nonumber \\&\qquad \left. - \partial _kg_{ij}(\eta _t(a)) \partial _t\eta _t^k(a)\partial _t\eta _t^j(a) \frac{\partial \eta _t^i(a)}{\partial a^m}\delta a^m -g_{ij}(\eta _t(a)) \partial _t^2\eta _t^j(a) \frac{\partial \eta _t^i(a)}{\partial a^m}\delta a^m \right\} \nonumber \\&\qquad +\,\int _0^T\int _{M}\mu (a)\, \partial _t\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \delta a^n\nonumber \\&\quad = I_1 + I_2. \end{aligned}$$

(99)

First, we show that $I_1=0$. From (99) and using the definition of the covariant derivative, we obtain

$$\begin{aligned} I_1= & {} \int _0^Tdt\int _{M}\mu (a) \,\frac{\partial \eta _t^j}{\partial a^m} \delta a^m\left\{ - g_{ij}(\eta _t)\left[ \partial _t\mathfrak {u}^i(t,\eta _t) + \mathfrak {u}^k(t,\eta _t)\partial _k\mathfrak {u}^i(t,\eta _t) \right] \right. \nonumber \\&\left. +\, \frac{1}{2}\partial _j g_{ik}(\eta _t)\mathfrak {u}^i(t,\eta _t)\mathfrak {u}^k(t,\eta _t) - \partial _k g_{ij}(\eta _t)\mathfrak {u}^i(t,\eta _t)\mathfrak {u}^k(t,\eta _t) \right\} \nonumber \\= & {} -\int _0^Tdt\int _{M}\mu (a) \,\frac{\partial \eta _t^j}{\partial a^m} \delta a^m g_{ij}(\eta _t)\left\{ \partial _t\mathfrak {u}^i(t,\eta _t) + \mathfrak {u}^k(t,\eta _t)\partial _k\mathfrak {u}^i(t,\eta _t)\right. \nonumber \\&\left. +\, \frac{1}{2} g^{im}(t,\eta _t)(\partial _kg_{lm}(t,\eta _t)+\partial _lg_{km}(t,\eta _t)-\partial _mg_{lk}(t,\eta _t)) \mathfrak {u}^k(t,\eta _t)\mathfrak {u}^l(t,\eta _t) \right\} \nonumber \\= & {} -\int _0^Tdt\int _{M}\mu (a) \, \frac{\partial \eta _t^j}{\partial a^m} \delta a^m g_{ij}(\eta _t)\left\{ \partial _t\mathfrak {u}^i(t,\eta _t)+ \mathfrak {u}^k(t,\eta _t)\nabla _k\mathfrak {u}^i(t,\eta _t) \right\} . \end{aligned}$$

Using the Euler equations, $\partial _t \mathfrak {u}^i+\mathfrak {u}^k\nabla _k \mathfrak {u}^i=-g^{ik}\partial _kp$, the term $I_1$ becomes

$$\begin{aligned} I_1= & {} \int _0^Tdt\int _{M}\mu (a) \,\delta a^m \frac{\partial \eta _t^j}{\partial a^m} g_{ij}(\eta _t) g^{ik}(\eta _t) \partial _k p (t,\eta _t) \\= & {} \int _0^Tdt\int _{M}\mu (a) \, \delta a^m \frac{\partial \eta _t^j}{\partial a^m} \delta _j^k\partial _k p (t,\eta _t) \nonumber \\= & {} \int _0^Tdt\int _{M}\mu (a) \,\delta a^m \frac{\partial \eta _t^k}{\partial a^m} \partial _k p (t,\eta _t) =\int _0^Tdt\int _{M}\mu (a) \, \delta a^m \frac{\partial p}{\partial a^m}. \end{aligned}$$

Now, we recall that $\nabla _i\delta a^i=|\mathrm {g}|^{-1/2}\partial _i(\sqrt{|\mathrm {g}|}\delta a^i)=0$, and $(\delta a, \nu )_g=0$. Therefore, using Green formula (93), the term $I_1$ becomes

$$\begin{aligned} I_1= & {} \int _0^Tdt\int _{M}\mu (a) \, \delta a^i \frac{\partial p}{\partial a^i} =- \int _0^Tdt\int _{M}\mu (a) \, \nabla _i \delta a^i p \\&+ \int _0^Tdt\int _{\partial M} \mu _\partial \, p\, (\delta a, \nu )_g =0. \end{aligned}$$

Finally, we deal with the term $I_2$ defined in (99). For this, we use the property that $\delta a\in \mathfrak {g}$, i.e., $\nabla _i \delta a^i=0$, and $(\delta a,\nu )_g=0$. Such a vector $\delta a$ can be constructed from a skew-symmetric 2-contravariant tensor $\pi$ satisfying the following constraints,

$$\begin{aligned} \pi ^{ij}+\pi ^{ji}=0\ \ \text{ on }\ \ M, \quad g_{ij}\pi ^{ik}\nu ^j=0\ \ \forall k \ \ \text{ on }\ \ \partial M, \quad \text{ and } \quad g_{ij}\pi ^{ik}\nabla _k\nu ^j=0 \ \ \text{ on }\ \ \partial M.\nonumber \\ \end{aligned}$$

(100)

Indeed, if we set

$$\begin{aligned} \delta a^i := \nabla _j \pi ^{ij}=\frac{1}{\sqrt{|\mathrm {g}|}} \partial _j(\sqrt{|\mathrm {g}|} \pi ^{ij}), \end{aligned}$$

(101)

then, using (100), we find that $\nabla _i \delta a^i=0$, and $(\delta a,\nu )_g=0$ on $\partial M$. We observe that a smooth skew-symmetric 2-contravariant tensor $\pi$ with compact support in M and vanishing smoothly at the boundary $\partial M$ satisfies (100). Using (100)–(101) and Green formula (93), the term $I_2$ becomes

$$\begin{aligned} I_2= & {} \int _0^T\int _{M}\mu (a)\, \partial _t\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \delta a^n\nonumber \\= & {} -\int _0^T\int _{M}\mu (a)\, \partial _k \partial _t\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \pi ^{nk} \nonumber \\&+\, \int _0^Tdt\int _{\partial M} \mu _\partial (a)\, \partial _t\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \pi ^{nk}\nu ^mg_{km} \nonumber \\= & {} -\int _0^T\int _{M}\mu (a)\, \partial _t\partial _k\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \pi ^{nk}. \end{aligned}$$

The action $\mathcal {A}(\eta ,M)$ should be invariant under relabelling transformations. Thus the variation of the action integral, i.e., $\delta \mathcal {A}$, must vanish. Therefore we have $I_2=0$, i.e.,

$$\begin{aligned} -\int _0^T\int _{M}\mu (a)\, \partial _t\partial _k\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) \pi ^{nk} = 0. \end{aligned}$$

Since the $\pi ^{nk}$’s are arbitrary, we obtain

$$\begin{aligned} \frac{d}{dt}\partial _k\left( g_{ij}(\eta _t(a)) \frac{\partial \eta _t^i(a)}{\partial a^n} \partial _t\eta _t^j(a)\right) =0, \quad \forall k,\, n =1,\ldots , d. \end{aligned}$$

Integration in time of these equations leads to

$$\begin{aligned} \partial _k\left( {\mathfrak {u}}_i(t,\eta _t(a))\frac{\partial \eta _t^i(a)}{\partial a^n}\right) =\partial _{k}v_{0n}, \quad \forall k,\, n =1,\ldots , d. \end{aligned}$$

(102)

Using the identity $v_i(t,a) = {\mathfrak {u}}_i(t,\eta _t(a))$ and multiplying (102) by $da^k\wedge da^n$, followed by a summation over the indices k and n, we obtain

$$\begin{aligned} d(v_i dx^i)=dv_0^\flat \quad \text{ i.e., } \quad dv_i\wedge dx^i=\omega _0:=dv_0^\flat , \end{aligned}$$

which ends the proof. $\quad \square$