Keywords: transitive Lie algebroids, exterior differential systems, exterior differential calculus, Cartan–Kähler theorem, invariant inverse problem of the calculus of variations.

Mathematics Subject Classification: 53D17, 58A15, 58A17, 70F17.

1 Introduction↩︎

Since the early twentieth century, there has been an ever increasing trend to study systems of partial differential equations by means of a geometric approach. One such approach, advocated by e.g., É.Cartan [1], [2], is that of exterior differential systems (EDS). Here, the PDEs are represented by a collection \(\mathcal{I}\) of differential forms on a manifold \(\overline{M}\), satisfying the condition that \(\mathcal{I}\) is closed under exterior differentiation. The main goal is to compute integral manifolds for a given EDS, which are submanifolds \(M\subset \overline{M}\) satisfying the property that all forms in \(\mathcal{I}\) vanish on \(M\). Since any system of partial differential equations can be written as an EDS (with an independence condition), finding solutions to a system of PDEs can be given the desired geometric interpretation, namely, finding integral manifolds of its associated EDS.

Since its conception, EDS has made many valuable contributions in the contexts of e.g.,the isometric embedding problem [3], Cartan’s equivalence method [4] and the inverse problem of the calculus of variations [5]–[7]. Besides, one may also think of the Frobenius theorem as an application of EDS theory, in the special case that the EDS only consists of one-forms [3]: The differential form version of the Frobenius theorem states that the algebraic ideal generated by linearly independent 1-forms \(\theta^1,\dots,\theta^s\in \Omega^1(\overline{M})\), which we denote by \(\mathcal{I}=\langle \theta^1,\dots,\theta^s\rangle_{\mathrm{alg}}\), corresponds to an integrable distribution \(\mathcal{D}=\bigcap_{i=1}^s \ker(\theta^i)\subset \mathrm{T}\overline{M}\) if and only if \(\mathcal{I}\) is a differential ideal, i.e., that \(\mathrm{d}\mathcal{I}\subset \mathcal{I}\).

The theory of EDS aims to generalize the consideration of \(1\)-forms to forms of arbitrary degree. In that case, the Frobenius theorem no longer applies and we need a more general theorem. The analogue of distributions in the Frobenius theorem are the so-called integral elements of the EDS. While the condition that \(\mathcal{I}=\langle \omega^1,\dots,\omega^s\rangle\subset \Omega^{\ast}(\overline{M})\) is a differential ideal is automatically assumed in the theory of EDS, there is also an additional compatibility condition that needs to hold for the integral elements. That additional condition is what is called ordinary. The theorem which generalizes the Frobenius theorem in this setting is called the Cartan–Kähler theorem, which for EDS on manifolds says the following: Let \(\mathcal{I}\subset \Omega^{\ast}(\overline{M})\) be a (real) analytic differential ideal on a manifold \(\overline{M}\). Let \(E_{z}\subset \mathrm{T}_{z} \overline{M}\) be an ordinary integral element of \(\mathcal{I}\). Then there exists an integral manifold \(j\colon X\to \overline{M}\) which passes through \(z\) whose tangent space at \(z\) is \(E_z\) (cf.Chap., § 2, Cor. 2.3 of [3]).

This paper serves as an extension to our previous paper [8], where we have extended the theory of exterior differential systems from manifolds to Lie algebroids. This extension was particularly useful in solving the invariant inverse problem of the calculus of variations (also studied in [9]–[11]), namely: Does a \(G\)-invariant Lagrangian exist for a given \(G\)-invariant second-order system on a Lie group \(G\)? We showed that the problem of finding a reduced multiplier matrix for the second-order system can be formulated as an EDS problem on a very specific (transitive) Lie algebroid. In the current paper, we will outline techniques for determining the existence of integral manifolds for an EDS on a transitive Lie algebroid. In particular, we will prove two extensions of the previously mentioned Cartan–Kähler theorem.

In 2, we present the basic preliminaries, in an attempt to keep the paper self-contained. In particular, we give an introduction to Lie algebroids and define an exterior derivative operator on sections of the dual of the Lie algebroid. We will also recall the statement of the main existence and uniqueness theorem for analytic partial differential equations, called the Cauchy–Kowalevski theorem. The proof of the Cartan–Kähler theorem depends on this existence theorem. In 3, we briefly revisit the ideas in [8], where we introduce prolongation Lie algebroids to define integral manifolds of a set of forms on a transitive Lie algebroid. Since our current approach deviates a bit from the one we used in [8], we have outlined these subtle differences in the Appendix.

In 4, we extend the notion of integral elements. Similarly to the setting of EDS on manifolds, we may also view integral elements as infinitesimal versions of integral manifolds. In particular, we consider Kähler-regular and (a generalization of) ordinary integral elements, which are fundamental in the formulation of the Cartan–Kähler theorem.

In 5, we give the statement of the Cartan–Kähler theorem for real analytic transitive Lie algebroids, which tells us when we can extend a \(p\)-dimensional integral manifold to a \((p+1)\)-dimensional one. In this generalized setting, the fact that the coefficients of the anchor map and the structure functions of such Lie algebroids are real analytic is crucial. We also present an extension of a corollary of the Cartan–Kähler theorem to such Lie algebroids, which, roughly speaking, shows that a so-called “\(l\)-ordinary integral element" is the initial data required for an exterior differential system to be integrable. Arguably, this corollary is more useful than the Cartan–Kähler theorem, and we will also sometimes refer to it by that name.

For instructive purposes, in 6 we demonstrate how the Cartan–Kähler theorem for Lie algebroids can be used to determine existence of integral manifolds in a simple setting. Specifically, we consider an EDS generated algebraically by \(1\)-forms on the Lie algebroid \(\mathrm{T}\mathbb{R}^3\times \mathbb{R}^1\to \mathbb{R}^3\). We then investigate a more complicated application in 7, inspired by the invariant inverse problem, whose EDS approach was discussed extensively in [8]. In this case, the differential ideal includes a \(2\)-form as an algebraic generator, which makes it more difficult to determine the regularity properties of integral elements.

Finally, we conclude the paper with some suggestions for future work.

2 Preliminaries↩︎

We start with the basics on the theory of Lie algebroids. One can refer to e.g., [12]–[14] for a more in-depth treatment. From now on, we denote the space of sections of a vector bundle \(\tau\colon A\to M\) by \(\Gamma(A)\). In particular, when \(A\) is the tangent bundle of a manifold \(M\), the collection of smooth vector fields is written as \(\mathcal{X}(M)\mathrel{\vcenter{:}}= \Gamma(\mathrm{T}M)\).

The tangent bundle \(\mathrm{T}M\) of a manifold \(M\) is a fundamental example of a Lie algebroid (with the Lie bracket of vector fields and \(\rho=\mathrm{id}\)). Another example is a Lie algebra (taking \(M=\{\ast\}\) to be a point, and \(\rho=0\)).

If \(A\) is transitive, the fibers of the anchor map form a Lie algebra bundle \(\ker(\rho)\) over \(M\). One obtains a short exact sequence of vector bundles \[0\longrightarrow \ker(\rho)\longrightarrow A\stackrel{\rho}{\longrightarrow} \mathrm{T}M\longrightarrow 0.\]

We will mainly be interested in transitive Lie algebroids in the real analytic category.

Henceforth, we will simply write ‘analytic’ for ‘real analytic’ for all objects of interest (manifolds, bundles, Lie algebroids, functions, sections and forms, etc.).

Locally, if \(\{e_{\alpha}\}\) is a basis of sections of \(A\), one can write \[\rho(e_{\alpha})=\rho_{\alpha}^i(x)\frac{\partial}{\partial x^i}\in \mathcal{X}(M),\] \[\llbracket e_{\alpha},e_{\beta}\rrbracket=L_{\alpha \beta}^{\gamma}(x)e_{\gamma}.\] It is easy to see that \(L_{\alpha \beta}^{\gamma}=-L_{\beta \alpha}^{\gamma}\). The local functions \(\{L_{\alpha \beta}^\gamma\}\) are called the structure functions. On an analytic Lie algebroid, if \(\{e_{\alpha}\}\) is a basis of analytic sections, then the functions \(\rho_{\alpha}^i\) and \(L_{\alpha \beta}^\gamma\) are analytic.

We call sections of exterior powers of the dual of the Lie algebroid forms on a Lie algebroid. We denote the set of \(k\)-forms on \(\tau\colon A\to M\) by \(\Lambda^k(A)\).

It is not difficult to verify that \(\delta^2=0\) and that if \(\omega\in \Lambda^k(A)\) and \(\theta\in \Lambda^l(A)\), then \[\delta(\omega\wedge \theta)=\delta \omega\wedge \theta+(-1)^k \omega\wedge \delta \theta.\] If \(\{e^{\alpha}\}\) denotes the dual basis of \(\{e_{\alpha}\}\), then it can be seen that \[\delta f=\frac{\partial f}{\partial x^i}\rho_{\alpha}^i e^{\alpha},\quad \delta e^{\alpha}=-\frac{1}{2}L_{\beta \gamma}^{\alpha}\, e^{\beta}\wedge e^{\gamma}.\]

We now recall the main local existence and uniqueness theorem for analytic partial differential equations, namely, the Cauchy–Kowalevski theorem. Suppose that \(F\) is a vector-valued function in \(n+1\) independent variables \(x^1,\dots,x^n,y\) and let \(\phi\) be a vector-valued function in \(x^1,\dots,x^n\). Let \(x=(x^1,\dots,x^n)\) and consider the Cauchy problem \[\frac{\partial F}{\partial y}=G\left(x,y,F,\frac{\partial F}{\partial x}\right),\] subject to the boundary conditions \[F(x,0)=\phi(x).\] Then if both \(G\) and \(\phi\) are analytic in a neighborhood of \(0\), the Cauchy problem has a unique analytic solution near \(0\).

Finally, we will need the following lemma in the proof of the Cartan–Kähler theorem.

3 Integral manifolds↩︎

In this section, we discuss prolongations of Lie algebroids. From this, we will be able to define integral manifolds of a set of forms on a transitive Lie algebroid. The current setup differs from the one we had in [8]. For readers already familiar with that paper, we have compared the two approaches in the Appendix at the end of the paper.

Let \(B\to N\) be a Lie algebroid with anchor \(\rho_B\) and bracket \(\llbracket \cdot,\cdot\rrbracket_B\), and let \(f\colon N'\to N\) be a smooth map. Consider the manifold \[\mathcal{L}^f B\mathrel{\vcenter{:}}= \rho_B^{\ast}(\mathrm{T}N')=\{(b_{n},v'_{n'})\in B\times \mathrm{T}N'\mid \rho_B(b_{n})=\mathrm{T}f(v'_{n'})\},\] together with the projection \(\tau^f\colon \mathcal{L}^f B\to N'\), \((b_n,v'_{n'})\mapsto n'\), where by construction \(f(n')=n\). Note that \[(\tau^f)^{-1}(n')=\mathop{\mathrm{rank}}(B)+\dim(N')-\dim(\rho_B(B_{f(n')})+(\mathrm{T}_{n'}f)(\mathrm{T}_{n'}N')),\] hence \(\tau^f\) takes the structure of a vector bundle only if \(\dim(\rho_B(B_{f(n')})+(\mathrm{T}_{n'}f)(\mathrm{T}_{n'}N'))\) is constant. In particular, this holds if \(B\) is a transitive Lie algebroid (i.e., \(\rho_B\) is surjective), or if \(f\) is a submersion. In either of these cases, \(\tau^f\) is in fact, a Lie algebroid (see [19], [20] for details). The second projection serves as the anchor map: \[\label{eqn:anchorprolongation} \rho^f\colon \mathcal{L}^f B\longrightarrow \mathrm{T}N',\quad (b_n,v'_{n'})\longmapsto v'_{n'}.\tag{2}\] We now construct its bracket. To this end, we explain what projectable sections of \(\tau^f\) are. A section of \(\tau^f\) is of the form \((\sigma,X)\), where \(\sigma\) is a section of the pullback bundle \(f^{\ast}B\to N'\), and \(X\) is a vector field on \(N'\). We say that \((\sigma,X)\) is projectable if there exists a section \(s\in \Gamma(B)\) such that \(\mu^f\circ (\sigma,X)=s\circ f\), where \(\mu^f\) is the first projection, i.e., \[\mu^f\colon \mathcal{L}^f B\longrightarrow B,\quad (b_n,v'_{n'})\longmapsto b_n.\] For two projectable sections, we define the bracket on \(\mathcal{L}^f B\) by \[\label{eqn:bracketprolongation} \llbracket (\sigma_1,X_1),(\sigma_2,X_2)\rrbracket (n')\mathrel{\vcenter{:}}= \left(\llbracket \sigma_1,\sigma_2\rrbracket_{B}(n),[X_1,X_2](n')\right).\tag{3}\] We can extend the definition of the bracket to all sections of \(\tau^f\) by imposing the Leibniz identity.

We now consider a special case. Let \(\bar{\tau}\colon \overline{A}\to \overline{M}\) be a transitive Lie algebroid of rank \(N\) with anchor \(\bar{\rho}\) and bracket \(\llbracket \cdot,\cdot \rrbracket_{\overline{A}}\), and let \(i\colon M\to \overline{M}\) be an immersion. We consider its \(i\)-prolongation Lie algebroid, which we denote by \(A\mathrel{\vcenter{:}}= \mathcal{L}^i \overline{A}\), i.e., \[\tau\colon A\longrightarrow M,\quad (\bar{a}_{\bar{m}},v_m)\longmapsto m,\] where \[A=\bar{\rho}^*(\mathrm{T}M) = \{ (\bar{a}_{\bar{m}},v_m)\in \overline{A}\times \mathrm{T}M \mid \bar{\rho}(\bar{a}_{\bar{m}}) = \mathrm{T}i(v_m)\}.\] By construction, \(i(m)=\bar{m}\). We denote the anchor map and bracket of \(A\) by \(\rho\colon A\to \mathrm{T}M\) and \(\llbracket \cdot,\cdot \rrbracket\) respectively. Note that \(A\) has rank \(L\mathrel{\vcenter{:}}= N+\dim(M)-\dim(\overline{M})\).

In what follows, it will be useful to construct a convenient basis of \(\overline{A}\) and its \(i\)-prolongation (i.e., \(A\)). Following our discussion in 2, note that the following sequence of vector bundles is short exact: \[0\longrightarrow \ker(\bar{\rho})\longrightarrow \overline{A}\stackrel{\bar{\rho}}{\longrightarrow} \mathrm{T}\overline{M}\longrightarrow 0.\] Since the anchor map \(\bar{\rho}\colon \overline{A}\to \mathrm{T}\overline{M}\) is surjective, this short exact sequence splits, and we will assume throughout that the Lie algebroid is such that an analytic splitting exists. Suppose we denote a splitting of \(\bar{\rho}\) by \(\bar{\gamma}\colon \mathrm{T}\overline{M}\to \overline{A}\). Then \(\overline{A}\simeq \ker(\bar{\rho})\oplus \bar{\gamma}(\mathrm{T}\overline{M})\). We may now choose an adapted basis of \(\overline{A}\) as follows. Choose a basis \(\{\bar{k}_a\}\) of \(\ker(\bar{\rho})\), and let \((z^j)\) be coordinates on \(\overline{M}\). By injectivity of \(\bar{\gamma}\), it follows that \(\left\{\bar{\gamma}_j\mathrel{\vcenter{:}}=\bar{\gamma}\left(\frac{\partial}{\partial z^j}\right)\right\}\) is a basis of \(\bar{\gamma}(\mathrm{T}\overline{M})\). This extends to a basis \(\{\bar{k}_a,\bar{\gamma}_j\}\) of \(\overline{A}\). We denote its dual basis by \(\{\bar{k}^a,\bar{\gamma}^j\}\).

From this, we will construct a basis of \(A\) as follows.

Note that \(\gamma(v_m)\in A\), because \[\bar{\rho}(\bar{\gamma}(\mathrm{T}i(v_m)))=\mathrm{T}i(v_m),\] and that \(\gamma\) is indeed a splitting because \[\rho(\gamma(v_m))=\rho(\bar{\gamma}(\mathrm{T}i(v_m)),v_m)=v_m.\]

Now suppose that \((x^i)\) are coordinates on \(M\). A basis of sections of \(A\) is then given by \(\{k_a,\gamma_i\}\), where \[k_a\mathrel{\vcenter{:}}= (\bar{k}_a,0),\quad \gamma_i\mathrel{\vcenter{:}}= \gamma\left(\frac{\partial}{\partial x^i}\right)=\left(\bar{\gamma}\left(\mathrm{T}i\left(\frac{\partial}{\partial x^i}\right)\right),\frac{\partial}{\partial x^i}\right),\] where \(\gamma\) is the \((i,\bar{\gamma})\)-induced splitting. Analogously, we denote its dual basis by \(\{k^a,\gamma^i\}\).

Let \(I\mathrel{\vcenter{:}}= \mu^i\), which we will call the \(i\)-induced map be the first projection, i.e., \[I\colon A \longrightarrow \overline{A},\quad (\bar{a}_{\bar{m}},v_m) \longmapsto \bar{a}_{\bar{m}}.\] Then, the following diagram commutes: \[\begin{gather} \xymatrix{ A \ar[d]_{\rho} \ar@<3pt>[r]^{I} \ar@/^-2pc/[dd]_{\tau} & \overline{A} \ar[d]_{\bar{\rho}} \ar@/^2pc/[dd]^{\bar{\tau}} \\ \mathrm{T}M \ar@<3pt>[r]^{\mathrm{T}i} \ar[d] & \mathrm{T}\overline{M} \ar[d] \\ M \ar@<3pt>[r]^{i} & \overline{M} } \end{gather}\]

From \(I\), we may also define a pullback operator \(I^{\ast}\colon \Lambda^k(\overline{A})\to \Lambda^k(A)\), as follows. On functions \(\bar f\in C^{\infty}(\overline{M})\) and one-forms \(\bar\theta\) on \(\overline{A}\), we set \[(I^{\ast}{\bar f})(m) = \bar f(i(m)),\qquad (I^{\ast}{\bar \theta})_m(\bar{a}_{\bar{m}},v_m)= \bar\theta_{i(m)}(I(\bar{a}_{\bar{m}},v_m))=\bar{\theta}_{i(m)}(\bar{a}_{\bar{m}}),\] which we extend to \(k\)-forms by imposing linearity of \(I^{\ast}\) and the rule \(I^{\ast}(\omega\wedge \theta)=I^{\ast}\omega\wedge I^{\ast}\theta\) for all \(\omega\in \Lambda^k(\overline{A})\) and \(\theta\in \Lambda^l(\overline{A})\).

Locally, if \(i\) takes the form \[i\colon M\longrightarrow \overline{M},\quad (x^i)\longmapsto (z^j=\bar{z}^j(x^i)),\] then \[I^{\ast}\bar{k}^a=k^a,\quad I^{\ast}\bar{\gamma}^j=\frac{\partial \bar{z}^j}{\partial x^i}\gamma^i.\]

We may now state the definition of an integral manifold of a set of forms on a (transitive) Lie algebroid. A subspace \(\mathcal{I}\subset \Lambda^{\ast}(\overline{A})\) is called an algebraic ideal if it decomposes as a direct sum of subspaces \(\mathcal{I}^k\subset \Lambda^{k}(\overline{A})\), and is closed under the wedge product with arbitrary differential forms (i.e., \(\mathcal{I}\) is a graded ideal of \(\Lambda^{\ast}(\overline{A})\)). If, in addition, it satisfies \(\bar{\delta} \mathcal{I}\subset \mathcal{I}\), we say that \(\mathcal{I}\) is a differential ideal.

Suppose we are given a finite set of forms of arbitrary degree \(\{\omega^1,\dots,\omega^n\}\subset \Lambda^{\ast}(\overline{A})\). Then the smallest algebraic ideal containing these forms is given by \[\langle \omega^1,\dots,\omega^n\rangle_{\mathrm{alg}}\mathrel{\vcenter{:}}= \left\{\alpha^j\wedge \omega^j\mid \alpha^1,\dots,\alpha^n\in \Lambda^{\ast}(\overline{A})\right\},\] and the smallest differential ideal containing these forms is given by \[\langle \omega^1,\dots,\omega^n\rangle\mathrel{\vcenter{:}}= \left\{\alpha^j\wedge \omega^j+\beta^j\wedge \bar{\delta} \omega^j\mid \alpha^1,\dots,\alpha^n,\beta^1,\dots,\beta^n\in \Lambda^{\ast}(\overline{A})\right\}.\] In particular, if the algebraic ideal generated by \(\{\omega^1,\dots,\omega^n\}\subset \Lambda^{\ast}(\overline{A})\) is also a differential ideal then \(\langle \omega^1,\dots,\omega^n\rangle=\langle \omega^1,\dots,\omega^n\rangle_{\mathrm{alg}}\).

We denote throughout this paper the collection of \(k\)-forms in a differential ideal \(\mathcal{I}\) by \(\mathcal{I}^k\mathrel{\vcenter{:}}= \mathcal{I}\cap \Lambda^k(\overline{A})\). For the rest of this paper, we assume that all differential ideals satisfy \(\mathcal{I}^0=\varnothing\).

4 Integral elements↩︎

In this section, we extend the concept of an integral element to transitive Lie algebroids. We refer to e.g.,[3], [21]–[23] for this well-known concept, in the standard theory of EDS on manifolds. Roughly speaking, integral elements can be viewed as infinitesimal versions of integral manifolds, in the sense that the defining property for an integral manifold is only required to hold on a fiber of \(\bar{\tau}\colon \overline{A}\to \overline{M}\). We also consider special classes of integral elements, as they are useful in our extension of the Cartan–Kähler theorem.

4.1 Introduction↩︎

Given a vector bundle \(\pi\colon V\to X\), we denote its fiber at \(x\in X\) by \(V_x\mathrel{\vcenter{:}}= \pi^{-1}(\{x\})\). We define the \(k\)-th Grassmann bundle of \(V\) to be the vector bundle \(G_k(V)\to X\) whose fibers at \(x\in X\) are the \(k\)-th Grassmannians of \(V_x\) (i.e., the set of all \(k\)-dimensional subspaces of \(V_x\)).

Let \(\mathcal{I}\) be a differential ideal of a transitive Lie algebroid \(\bar{\tau}\colon \overline{A}\to \overline{M}\). In this section, we fix a number \(k\leq N= \mathop{\mathrm{rank}}(\overline{A})\).

In the following, most of the time, we will simply write \(E_{\bar{m}}\) for \((\bar{m},E_{\bar{m}})\in G_k(\overline{A})\).

We denote the set of all \(k\)-dimensional integral elements by \(V_k(\mathcal{I})\), i.e., \[V_k(\mathcal{I})\mathrel{\vcenter{:}}= \{E_{\bar m}\in G_k(\overline{A})\mid \theta_{E_{\bar m}}=0~\mathrm{for}~\mathrm{all}~\theta\in \mathcal{I}^k\},\] where \(\theta_{E_{\bar m}}\) is the restriction of \(\theta|_{\bar{m}}\) to \(E_{\bar m}\). One can then see the following results:

The concept of an integral element and integral manifold, are in fact, related. To establish this relationship, we first prove the following lemma.

In the statement of our main theorem (5), we wish to extend a \(p\)-dimensional integral manifold to a \((p+1)\)-dimensional one. The infinitesimal analogue of this is to extend a \(k\)-dimensional integral element \(E_{\bar{m}}\in V_k(\mathcal{I})\) to a \((k+1)\)-dimensional integral element \(E_{\bar{m}}^+\in V_{k+1}(\mathcal{I})\) containing \(E_{\bar{m}}\). To do this, it is useful to introduce the concept of a polar space.

Note that \(E_{\bar{m}}\subset H(E_{\bar{m}})\). With the following result, we can see why this definition is useful.

The space of all extensions is a (possibly empty) projective space \(\mathbb{P}\left(H(E_{\bar{m}})\big\slash E_{\bar{m}}\right)\). It is useful to determine the dimension of this projective space to determine whether extensions exist and, if they do, for understanding the size of the space of extensions. To this end, we define a function \(r\colon V_k(\mathcal{I})\to \mathbb{Z}_{\geq -1}\) by \[r(E_{\bar{m}})\mathrel{\vcenter{:}}= \dim(H(E_{\bar{m}}))-(k+1).\] Note that \(r(E_{\bar{m}})=-1\) if and only if \(E_{\bar{m}}\) lies in no \((k+1)\)-dimensional integral element of \(\mathcal{I}\). When \(r(E_{\bar{m}})\geq 0\), the space of \((k+1)\)-dimensional integral elements of \(\mathcal{I}\) which contain \(E_{\bar{m}}\) is a real projective space of dimension \(r(E_{\bar{m}})\).

4.2 Kähler-regular integral elements and manifolds↩︎

In order to define the notion of Kähler-regularity, we need a few preliminary definitions. If \(\mathcal{F}\subset C^{\infty}(X)\) is a set of smooth functions on a smooth manifold \(X\), we denote by \(Z(\mathcal{F})\subset X\) the set of common zeros of the functions in \(\mathcal{F}\).

Let \(\Omega\) be any \(k\)-form on \(\overline{A}\). Then we denote by \(G_k(\overline{A},\Omega)\) the set of all \(E_{\bar{m}}\in G_k(\overline{A})\) such that \(\Omega_{E_{\bar{m}}}\neq 0\). If \(\theta\) is any other \(k\)-form on \(\overline{A}\), we define a function \(\theta_{\Omega}\) on \(G_k(\overline{A},\Omega)\) by \[\label{eqn:formfunction} \theta_{E_{\bar{m}}}=\theta_{\Omega}(E_{\bar{m}})\Omega_{E_{\bar{m}}},\tag{5}\] where we consider \(\theta_{E_{\bar{m}}}\) and \(\Omega_{E_{\bar{m}}}\) as \(k\)-forms on \(\Lambda^k(E_{\bar{m}})\). This function is well-defined since \(\Lambda^k(E_{\bar{m}})\) is \(1\)-dimensional with basis \(\Omega_{E_{\bar{m}}}\).

Now consider the set of functions \[\mathcal{F}_{\Omega}(\mathcal{I})\mathrel{\vcenter{:}}= \{\theta_{\Omega}\mid \theta\in \mathcal{I}^k\}\] on the manifold \(X=G_k(\overline{A},\Omega)\). Since \(\theta_{\Omega}(E_{\bar{m}})=0\) if and only if \(\theta_{E_{\bar{m}}}=0\), the set of common zeros of \(\mathcal{F}_{\Omega}(\mathcal{I})\) is given by \(V_k(\mathcal{I},\Omega)\mathrel{\vcenter{:}}= V_k(\mathcal{I})\cap G_k(\overline{A},\Omega)\).

We are now in position to define Kähler-ordinary and Kähler-regular integral manifolds. We say that a submanifold \(i\colon M\to \overline{M}\) is a Kähler-ordinary integral manifold if \(I(A_m)\subset \overline{A}_{i(m)}\) is a Kähler-ordinary integral element for all \(m\in M\). Similarly, we say that \(i\colon M\to \overline{M}\) is a Kähler-regular integral manifold if \(I(A_m)\subset \overline{A}_{i(m)}\) is a Kähler-regular integral element for all \(m\in M\). If \(i\colon M\to \overline{M}\) is connected and Kähler-regular, we set \[r(M)\mathrel{\vcenter{:}}= r(I(A_m)),\] which is well-defined since \(r\) is locally constant.

Later, it will be useful in the formulation of the corollary of the Cartan–Kähler theorem (1) to assume the existence of an increasing sequence of Kähler-regular integral elements. To this end, we define an integral flag.

5 The Cartan–Kähler theorem↩︎

In this section, we prove two versions of the Cartan–Kähler theorem for transitive Lie algebroids, stated in 5 and 1. From now on, we will no longer make any notational distinction between the exterior derivative \(\delta\) on \(A\) and \(\bar{\delta}\) on \(\overline{A}\) (we will denote both by \(\delta\)).

The core ideas of the proof below stem from [3], [21], [24]. Note that in the case that \(\overline{A}=\mathrm{T}\overline{M}\), the condition ?? simplifies to the usual transversality condition \[\mathrm{T}_x R+H(\mathrm{T}_x M)=\mathrm{T}_x\overline{M},\quad \mathrm{for}~x\in M.\] For a better comparison with the classical case, we have also retained some of the notation from [3].

Applying this theorem inductively gives the following useful corollary (we will also call this the Cartan–Kähler theorem).

Note that when \(\overline{A}=\mathrm{T}\overline{M}\), 1 reduces to the classical statement of the Cartan–Kähler theorem which we have also mentioned in the introduction (Chap., § 2, Cor. 2.3 of [3]): If \(E_z\subset \mathrm{T}_z\overline{M}\) is an ordinary integral element, there exists an integral manifold through \(z\) whose tangent space at \(z\) is precisely \(E_z\).

6 A simple example↩︎

In this section, we provide an example where we apply the Cartan–Kähler theorem. In particular, for a specific EDS on a Lie algebroid, we will construct a family of integral elements which we will denote by \(\{E_z\mid z\in \overline{M}\}\). We will show that for each element in this family, there exists an integral manifold with the necessary properties in 1.

Let \(\overline{A}=\mathrm{T}\mathbb{R}^3\times \mathbb{R}^1\) be a vector bundle over \(\overline{M}=\mathbb{R}^3\). If \((x^1,x^2,x^3)\) are coordinates on \(\overline{M}\), then a basis of sections of \(\overline{A}\) is given by \(\{\frac{\partial}{\partial x^1},\frac{\partial}{\partial x^2},\frac{\partial}{\partial x^3},\bar{k}_1\}\), where \(\bar{k}_1\colon \overline{M}\to \overline{A}\) is given by \(z\mapsto (0,0,0,1)\). We denote its dual basis by \(\{\mathrm{d}x^1,\mathrm{d}x^2,\mathrm{d}x^3,\bar{k}^1\}\). We define a Lie algebroid structure on \(\overline{A}\) by defining the anchor \(\bar{\rho}\colon \overline{A}\to \mathrm{T}\mathbb{R}^3\) on the given basis of sections such that \[\bar{\rho}\left(\frac{\partial}{\partial x^i}\right)=\frac{\partial}{\partial x^i},\quad \bar{\rho}(\bar{k}_1)=0,\] and its Lie bracket such that \[\left\llbracket\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right\rrbracket=\left\llbracket\frac{\partial}{\partial x^i},\bar{k}_1\right\rrbracket=0.\] Let \(\mathcal{I}=\langle \theta^1=\mathrm{d}x^3-x^2\, \mathrm{d}x^1,\theta^2=\mathrm{d}x^2\rangle_{\mathrm{alg}}\). Since \[\delta(\mathrm{d}x^i)=0,\quad \delta x^2=\mathrm{d}x^2,\] it follows that \[\delta \theta^1=-\mathrm{d}x^2\wedge \mathrm{d}x^1=\mathrm{d}x^1\wedge \mathrm{d}x^2=\mathrm{d}x\wedge \theta^2\in \mathcal{I},\] \[\delta \theta^2=0\in \mathcal{I}.\] Hence, \(\mathcal{I}\) is a differential ideal. At each \(z\in \overline{M}\), there is a unique integral element of dimension \(2\), namely \[E_z\mathrel{\vcenter{:}}= \{v\in \overline{A}_z\mid \theta^a(v)=0~\mathrm{for}~1\leq a\leq 2\}=\mathop{\mathrm{span}}\left\{\left(\frac{\partial}{\partial x^1}+x^2 \frac{\partial}{\partial x^3}\right)\Big|_z,\bar{k}_1(z)\right\}.\] In fact, every integral element of \(\mathcal{I}\) based at \(z\) must be a subspace of \(E_z\), since \(H((0)_z)=E_z\). Thus, if \[(E_1)_z\subset (E_2)_z=E_z\] is any \(1\)-integral flag, then \(H((E_1)_z)=E_z\). We must now show that \((E_2)_z=E_z\) is a \(1\)-ordinary integral element. We therefore only need to find a \(1\)-dimensional Kähler-regular subspace \((E_1)_z\) of \(E_z\).

We claim that \((E_1)_z\mathrel{\vcenter{:}}= \mathop{\mathrm{span}}\{\bar{k}_1(z)\}\) is a Kähler-regular integral element. Since \((E_1)_z\subset E_z\), it follows from 2 that \((E_1)_z\) is an integral element. Now, consider \(\Omega=\bar{k}^1\in \Lambda^1(\overline{A})\). Then \(\Omega_{(E_1)_z}\neq 0\), and we have that \[\mathcal{F}_{\Omega}(\mathcal{I})=\{\theta_{\Omega}\mid \theta\in \mathcal{I}^1\}=\mathop{\mathrm{span}}\{\theta_{\Omega}^1,\theta_{\Omega}^2\}.\] Every \((F_1)_{\bar{m}}\in G_1(\overline{A},\Omega)\) has a unique basis of the form \[X_1((F_1)_{\bar{m}})=\bar{k}_1(\bar{m})+p^1((F_1)_{\bar{m}}) \frac{\partial}{\partial x^1}\Big|_{\bar{m}}+p^2((F_1)_{\bar{m}}) \frac{\partial}{\partial x^2}\Big|_{\bar{m}}+p^3((F_1)_{\bar{m}}) \frac{\partial}{\partial x^3}\Big|_{\bar{m}}.\] Therefore, the functions \(x^1,x^2,x^3,p^1,p^2,p^3\) form a coordinate system on \(G_1(\overline{A},\Omega)\). Note that \[\theta_{\Omega}^1((F_1)_{\bar{m}})=\frac{\theta^1(X_1)}{\Omega(X_1)}=-x^2 p^1+p^3,\quad \theta_{\Omega}^2((F_1)_{\bar{m}})=p^2.\] Note that the common zeros of \(\theta_{\Omega}^1\) and \(\theta_{\Omega}^2\) are precisely \(V_1(\mathcal{I},\Omega)\). We also have that \[Z(\mathcal{F}_{\Omega}(\mathcal{I}))=\{(F_1)_{\bar{m}}\in G_1(\overline{A},\Omega)\mid \theta_{\Omega}^1((F_1)_{\bar{m}})=\theta_{\Omega}^2((F_1)_{\bar{m}})=0\},\] and moreover, \(\theta_{\Omega}^1, \theta_{\Omega}^2\) have linearly independent differentials, since \[\mathrm{d}\theta_{\Omega}^1=-p^1\, \mathrm{d}x^2-x^2\, \mathrm{d}p^1+\mathrm{d}p^3,\quad \mathrm{d}\theta_{\Omega}^2=\mathrm{d}p^2.\] It follows that \((E_1)_z\) is Kähler-ordinary (in fact, our reasoning shows that every point in \(V_1(\mathcal{I},\Omega)\) is Kähler-ordinary). It is also Kähler-regular, since \(H((F_1)_{\bar{m}})=E_{\bar{m}}\). Since \(\theta^1,\theta^2\) are pointwise linearly independent, it follows that \(\dim(H((F_1)_{\bar{m}}))=2\) for all \(\bar{m}\in \overline{M}\), and therefore \(r\) is a locally constant function. We may therefore conclude using the second version of the Cartan–Kähler theorem that for every \(z\in \overline{M}\), there exists an integral manifold \(j\colon X\to \overline{M}\) of \(\mathcal{I}\) which passes through \(z\) and satisfies \(J((\mathcal{L}^j \overline{A})_x)=E_z\), where \(x\in X\) is a point such that \(z=j(x)\).

Although the Cartan–Kähler theorem only guarantees existence, in this case it is possible to give an explicit example of such an integral manifold for each \(z=(x_0^1,x_0^2,x_0^3)\in \overline{M}\). We claim that \[j\colon \mathbb{R}\longrightarrow \overline{M},\quad x^1\longmapsto \left(x^1,x_0^2,x_0^2(x^1-x_0^1)+x_0^3\right),\] defines such an integral manifold. Note that if \(\bar{a}_{\bar{m}}=a_1\frac{\partial}{\partial x^1}+a_2 \frac{\partial}{\partial x^2}+a_3 \frac{\partial}{\partial x^3}+a_4 \bar{k}_1\) and \(v_m=b_1 \frac{\partial}{\partial x^1}\), then \(\bar{\rho}(\bar{a}_{\bar{m}})=\mathrm{T}j(v_m)\) requires \(a_1=b_1\) and \(a_3=x_0^2 b_1\). Hence, one has \[\mathcal{L}^j \overline{A}=\mathop{\mathrm{span}}\left\{\left(\frac{\partial}{\partial x^1}+x_0^2 \frac{\partial}{\partial x^3},\frac{\partial}{\partial x^1}\right),\left(\bar{k}_1,0\right) \right\}.\] Applying \(J\) gives \[J((\mathcal{L}^j \overline{A})_{x_0^1})=\mathop{\mathrm{span}}\left\{\frac{\partial}{\partial x^1}\Big|_{j(x_0^1)}+x_0^2 \frac{\partial}{\partial x^3}\Big|_{j(x_0^1)},\bar{k}_1(j(x_0^1)) \right\}=E_z,\] as required.

7 The invariant inverse problem↩︎

This section is based on the discussion of the EDS approach to the time-dependent invariant inverse problem in [8]. In that article, we showed that we may solve the following problem by finding integral manifolds of an appropriate EDS on a Lie algebroid.

Given a (possibly time-dependent) \(G\)-invariant second-order system on a Lie group \(G\), when does a (possibly time-dependent) regular \(G\)-invariant Lagrangian \(L\) whose Euler–Lagrange equations are equivalent to this system exist?

The appropriate Lie algebroid is constructed as follows. Let \(\mathfrak{g}\) be the Lie algebra of \(G\). Consider the vector bundle \[A=\mathbb{R}\times \mathfrak{g}\times \mathbb{R}\times \mathfrak{g}\times \mathfrak{g}\longrightarrow M=\mathbb{R}\times \mathfrak{g}.\] We put coordinates \((w^i)\) on \(\mathfrak{g}\). A basis of sections of this bundle is given by \(\{T_0,e_i,W_i\}\), where \[T_0\colon (t,w)\longmapsto (t,w,1,0,0),\] \[e_i\colon (t,w)\longmapsto (t,w,0,E_i,0),\quad W_i\colon (t,w)\longmapsto (t,w,0,0,E_i),\] where \(E_i\) is a basis of \(\mathfrak{g}\). If \(C_{ij}^k\) are the structure constants of the Lie algebra, the anchor map satisfies \[\rho(T_0)=\frac{\partial}{\partial t},\quad \rho(e_i)=w^k C_{ki}^j \frac{\partial}{\partial w^j},\quad \rho(W_i)=\frac{\partial}{\partial w^i},\] and its bracket is given by \[\llbracket T_0,e_i\rrbracket=0,\quad \llbracket T_0,W_i\rrbracket=0,\] \[\llbracket e_i,e_j\rrbracket=C_{ij}^k e_k,\quad \llbracket e_i,W_j\rrbracket=C_{ij}^k W_k,\quad \llbracket W_i,W_j\rrbracket=0.\] We call \(A\) the IP Lie algebroid. We then carried out a convenient change of basis, yielding the basis \(\{\Gamma_0,W_i,H_i\}\), whose dual we denoted by \(\{\Gamma^0,\Psi^i,\Theta^i\}\). We then showed (cf. Thm. 2 of [8]) that a reduced multiplier matrix exists for a given \(G\)-invariant second order system if and only if integral manifolds of a special type exist for an EDS generated by a set of \(1\)-forms on a specific \(p\)-prolongation Lie algebroid of \(A\). The “special type” in the previous sentence refers to a discussion on an independence condition, that we will come back to at the end of this section.

For the sake of simplicity, we will consider the invariant inverse problem for the canonical connection on the \(1\)-dimensional Lie group \(G=\mathbb{R}^1\) of the real line (as considered in Example 4 of [8]). In this case, the \(p\)-prolongation Lie algebroid, which we will denote by \(\overline{A}\mathrel{\vcenter{:}}= \mathcal{L}^p A\) is given by \[\overline{A}=A\times \mathrm{T}\mathbb{R}^1\times \mathrm{T}\mathbb{R}^2\longrightarrow \overline{M}=M\times \mathbb{R}^1\times \mathbb{R}^2,\] where \(\tau\colon A\to M\) has a basis of sections \(\{\Gamma_0,W_1,H_1\}\). If \((t,w,s,P,Q)\) are the coordinates of \(\overline{M}\) (note that in [8], we used the notation \((t,w^1,s_{11},P_{111},Q_{111})\) instead), the given basis of \(A\) extends to a basis of \(\overline{A}\), given by \(\{\Gamma_0,W_1,H_1,\frac{\partial}{\partial s},\frac{\partial}{\partial P},\frac{\partial}{\partial Q}\}\). Its dual basis is given by \(\{\Gamma^0,\Psi^1,\Theta^1,\delta s,\delta P,\delta Q\}\). The suitable differential ideal for this problem is given by \(\mathcal{I}=\langle \sigma_{11}\rangle=\langle \sigma_{11},\delta \sigma_{11} \rangle_{\mathrm{alg}}\), where \[\sigma_{11}=\delta s+P \Psi^1+Q \Theta^1,\] \[\delta \sigma_{11}=(\delta P+Q \Gamma^0)\wedge \Psi^1+\delta Q\wedge \Theta^1.\] There, it was shown that integral manifolds of \(\mathcal{I}\) are generally given by \[i\colon M\longrightarrow \overline{M},\quad (t,w)\longmapsto \left(t,w,s=g(w),P=-\frac{\partial g(w)}{\partial w},Q=0\right),\] where \(g\) is any nowhere zero smooth function.

We now use the Cartan–Kähler theorem to construct integral manifolds of this type. Let \(z\mathrel{\vcenter{:}}= (0,0,c_1,c_2,0)\), where \(c_1\neq 0\) and \(c_2\in \mathbb{R}\). Let \(d_1,d_2,d_3\in \mathbb{R}\) and consider \[E_z\mathrel{\vcenter{:}}= \mathop{\mathrm{span}}\left\{\Gamma_0(z),W_1(z)+d_1 \frac{\partial}{\partial s}\Big|_z+d_2 \frac{\partial}{\partial P}\Big|_z+d_3\frac{\partial}{\partial Q}\Big|_z,H_1(z)\right\}.\] Note that \(E_z\) is an integral element based at \(z\) if and only if \(d_1=-c_2\) and \(d_3=0\), because \[\begin{gather} (\beta\wedge \sigma_{11})_z\left(\Gamma_0(z),W_1(z)+d_1 \frac{\partial}{\partial s}\Big|_z+d_2 \frac{\partial}{\partial P}\Big|_z+d_3\frac{\partial}{\partial Q}\Big|_z,H_1(z)\right)\\=\beta_z\left(\Gamma_0(z),W_1(z)+d_1 \frac{\partial}{\partial s}\Big|_z+d_2 \frac{\partial}{\partial P}\Big|_z+d_3\frac{\partial}{\partial Q}\Big|_z\right)Q(z)-\beta_z(\Gamma_0(z),H_1(z))(P(z)+d_1)\\=-\beta_z(\Gamma_0(z),H_1(z))(c_2+d_1), \end{gather}\] and \[\begin{gather} (\alpha\wedge \delta \sigma_{11})_z\left(\Gamma_0(z),W_1(z)+d_1 \frac{\partial}{\partial s}\Big|_z+d_2 \frac{\partial}{\partial P}\Big|_z+d_3\frac{\partial}{\partial Q}\Big|_z,H_1(z)\right)\\=\alpha_z(\Gamma_0(z)) d_3+\alpha_z(H_1(z))Q(z)=\alpha_z(\Gamma_0(z)) d_3, \end{gather}\] for every \(\alpha\in \Lambda^1(\overline{A})\) and \(\beta\in \Lambda^2(\overline{A})\). From now on, we assume that \(d_1=-c_2\) and \(d_3=0\), so that \[E_z\mathrel{\vcenter{:}}= \mathop{\mathrm{span}}\left\{\Gamma_0(z),W_1(z)-c_2 \frac{\partial}{\partial s}\Big|_z+d_2 \frac{\partial}{\partial P}\Big|_z,H_1(z)\right\}\] is an integral element. We claim that there exists an integral manifold \(j\colon X\to \overline{M}\) of \(\mathcal{I}\) such that \(J((\mathcal{L}^j \overline{A})_x)=E_z\) for all \(d_2\in \mathbb{R}\), where \(x\in X\) is a point such that \(z=j(x)\).

Let \[(E_1)_z\mathrel{\vcenter{:}}= \mathop{\mathrm{span}}\{H_1(z)\},\quad (E_2)_z\mathrel{\vcenter{:}}= \mathop{\mathrm{span}}\left\{\Gamma_0(z),H_1(z)+W_1(z)-c_2 \frac{\partial}{\partial s}\Big|_z+d_2\frac{\partial}{\partial P}\Big|_z\right\}.\] We will show that \((E_1)_z\subset (E_2)_z\subset E_z\) is a \(1\)-ordinary integral flag. It follows from 2 that \((E_1)_z\) and \((E_2)_z\) are integral elements. We now show that \((E_1)_z\) is Kähler-ordinary. Let \(\Omega=\Theta^1\). Then \(\Omega|_{(E_1)_z}\neq 0\) because \(\Omega(H_1)=1\). We have that \[\mathcal{F}_{\Omega}(\mathcal{I})=\mathop{\mathrm{span}}\{(\sigma_{11})_{\Omega}\}.\] Every \((F_1)_{\bar{m}}\in G_1(\overline{A},\Omega)\) has a unique basis of the form \[\begin{gather} X_1((F_1)_{\bar{m}})=H_1(\bar{m})+p^1((F_1)_{\bar{m}}) \Gamma_0(\bar{m})+p^2((F_1)_{\bar{m}}) W_1(\bar{m})\\+p^4((F_1)_{\bar{m}}) \frac{\partial}{\partial s}\Big|_{\bar{m}}+p^5((F_1)_{\bar{m}}) \frac{\partial}{\partial P}\Big|_{\bar{m}}+p^6((F_1)_{\bar{m}}) \frac{\partial}{\partial Q}\Big|_{\bar{m}}. \end{gather}\] A coordinate system on \(G_1(\overline{A},\Omega)\) is hence given by \(t,w,s,P,Q,p^1,p^2,p^4,p^5,p^6\). Note that \[(\sigma_{11})_{\Omega}((F_1)_{\bar{m}})=\frac{\sigma_{11}(X_1)}{\Omega(X_1)}=p^4+P p^2+Q.\] Note that \[Z(\mathcal{F}_{\Omega}(\mathcal{I}))=V_1(\mathcal{I},\Omega)=\{(F_1)_{\bar{m}}\in G_1(\overline{A},\Omega)\mid (\sigma_{11})_{\Omega}((F_1)_{\bar{m}})=0\},\] and therefore \((E_1)_z\) is Kähler-ordinary.

We now show that \((E_2)_z\) is Kähler-ordinary. Then \(\Omega=\Gamma^0\wedge \Theta^1\in \Lambda^2(\overline{A})\) is such that \(\Omega|_{(E_2)_z}\neq 0\). We have that \[\mathcal{F}_{\Omega}(\mathcal{I})=\{(\alpha\wedge \sigma_{11})_{\Omega}+(b\, \delta \sigma_{11})_{\Omega}\mid \alpha\in \Lambda^1(\overline{A}),b\in C^{\infty}(\overline{M})\}.\] Every \((F_2)_{\bar{m}}\in G_2(\overline{A},\Omega)\) has a unique basis of the form \[\begin{gather} Y_1((F_2)_{\bar{m}})=\Gamma_0(\bar{m})+p_1^2((F_2)_{\bar{m}}) W_1(\bar{m})+p_1^4((F_2)_{\bar{m}}) \frac{\partial}{\partial s}\Big|_{\bar{m}}\\+p_1^5((F_2)_{\bar{m}}) \frac{\partial}{\partial P}\Big|_{\bar{m}}+p_1^6((F_2)_{\bar{m}}) \frac{\partial}{\partial Q}\Big|_{\bar{m}}, \end{gather}\] \[\begin{gather} Y_2((F_2)_{\bar{m}})=H_1(\bar{m})+p_2^2((F_2)_{\bar{m}}) W_1(\bar{m})+p_2^4((F_2)_{\bar{m}}) \frac{\partial}{\partial s}\Big|_{\bar{m}}\\+p_2^5((F_2)_{\bar{m}}) \frac{\partial}{\partial P}\Big|_{\bar{m}}+p_2^6((F_2)_{\bar{m}}) \frac{\partial}{\partial Q}\Big|_{\bar{m}}. \end{gather}\] The functions \(t,w,s,P,Q,p_i^j\) form a coordinate system on \(G_2(\overline{A},\Omega)\). Note that \[\begin{align} (\alpha\wedge \sigma_{11})_{\Omega}((F_2)_{\bar{m}})=\frac{(\alpha\wedge \sigma_{11})(Y_1,Y_2)}{\Omega(Y_1,Y_2)}&=\alpha(Y_1)\sigma_{11}(Y_2)-\alpha(Y_2)\sigma_{11}(Y_1)\\&=\alpha(Y_1)(p_2^4+P p_2^2)-\alpha(Y_2)(p_1^4+P p_1^2+Q). \end{align}\] and \[(\delta \sigma_{11})_{\Omega}((F_2)_{\bar{m}})=p_2^2(p_1^5+Q)-p_2^5 p_1^2+p_1^6.\] It follows that \[Z(\mathcal{F}_{\Omega}(\mathcal{I}))=\{(F_2)_{\bar{m}}\in G_2(\overline{A},\Omega)\mid p_2^4+P p_2^2=p_1^4+P p_1^2+Q=p_2^2(p_1^5+Q)-p_2^5 p_1^2+p_1^6=0\}.\] The differentials of \(p_2^4+P p_2^2\), \(p_1^4+P p_1^2+Q\) and \(p_2^2(p_1^5+Q)-p_2^5 p_1^2+p_1^6\) are linearly independent, hence \((E_2)_z\) is Kähler-ordinary.

We now show that both \((E_1)_z\) and \((E_2)_z\) are Kähler-regular. Let \((F_1)_{\bar{m}}=\langle X_1\rangle\) be any \(1\)-dimensional integral element. Then \[H(\langle X_1\rangle)=\left\{v\in \overline{A}_{\bar{m}}\mid \alpha \wedge \sigma_{11}(v,X_1)=0, \delta \sigma_{11}(v,X_1)=0~\mathrm{for}~\alpha\in \Lambda^1(\overline{A})\right\}.\] Note that \[(\alpha\wedge \sigma_{11})(v,X_1)=\alpha(v)\sigma_{11}(X_1)-\alpha(X_1)\sigma_{11}(v)=-\alpha(X_1)\sigma_{11}(v),\] which is zero for all \(\alpha\in \Lambda^1(\overline{A})\) if and only if \(\sigma_{11}(v)=0\). If \[v=\left(a_1 \Gamma_0+a_2 W_1+a_3 H_1+a_4 \frac{\partial}{\partial s}+a_5 \frac{\partial}{\partial P}+a_6 \frac{\partial}{\partial Q}\right)\Big|_{\bar{m}},\] then this equation is given by \[a_4+P a_2+Q a_3=0.\] Note that \[\begin{align} \delta \sigma_{11}(v,X_1)&=\delta \sigma_{11}\left(v,H_1+p^1 \Gamma_0+p^2 W_1+p^4 \frac{\partial}{\partial s}+p^5 \frac{\partial}{\partial P}+p^6 \frac{\partial}{\partial Q}\right)\\&=p^2(a_5+Q a_1)-a_2(p^5+Q p^1)+a_6-a_3 p^6=0 \end{align}\] We thus have that \[H(\langle X_1\rangle)=\{(a_1,\dots,a_6)\in \mathbb{R}^6\mid P a_2+Q a_3+a_4=Q p^2 a_1-(p^5+Q p^1)a_2-p^6 a_3+p^2 a_5+a_6=0\},\] which gives that \(\dim H((F_1)_{\bar{m}})=4\) for all \((F_1)_{\bar{m}}\in G_1(\overline{A},\Omega)\). Hence, \((E_1)_{z}\) is Kähler-regular.

We now check that \((E_2)_{z}\) is Kähler-regular. If \((F_2)_{\bar{m}}=\langle Y_1,Y_2\rangle\) is any \(2\)-dimensional integral element, then note that \[\beta\wedge \sigma_{11}(v,Y_1,Y_2)=\beta(v,Y_1)\sigma_{11}(Y_2)-\beta(v,Y_2)\sigma_{11}(Y_1)+\beta(Y_1,Y_2)\sigma_{11}(v)=\beta(Y_1,Y_2)\sigma_{11}(v),\] for every \(\beta\in \Lambda^2(\overline{A})\), hence \(\beta\wedge \sigma_{11}(v,Y_1,Y_2)=0\) for all \(\beta\) if and only if \(\sigma_{11}(v)=0\). Also, note that \[\begin{align} \alpha\wedge \delta \sigma_{11}(v,Y_1,Y_2)&=\alpha(v)\delta \sigma_{11}(Y_1,Y_2)-\alpha(Y_1)\delta \sigma_{11}(v,Y_2)+\alpha(Y_2)\delta \sigma_{11}(v,Y_1)\\&=-\alpha(Y_1)\delta \sigma_{11}(v,Y_2)+\alpha(Y_2)\delta \sigma_{11}(v,Y_1), \end{align}\] which is zero for all \(\alpha\in \Lambda^1(\overline{A})\) if and only if \(\delta \sigma_{11}(v,Y_1)=\delta \sigma_{11}(v,Y_2)=0\). This gives that \[\begin{gather} H(\langle Y_1,Y_2\rangle)=\Bigg\{(a_1,\dots,a_6)\in \mathbb{R}^6\mid P a_2+Q a_3+a_4\\=Q p_1^2 a_1-(p_1^5+Q)a_2-p_1^6 a_3+p_1^2 a_5+a_6=Q p_2^2 a_1-p_2^5 a_2-p_2^6 a_3+p_2^2 a_5+a_6=0\Bigg\} \end{gather}\] Note that the last two equations are linearly dependent if and only if \(p_1^2=p_2^2\), \(p_1^6=p_2^6\) and \(p_1^5+Q=p_2^5\). We therefore have that \[\dim H((F_2)_{\bar{m}})=\begin{cases} 4,\quad \mathrm{if}~p_1^2=p_2^2,p_1^6=p_2^6,p_1^5+Q=p_2^5, \\ 3,\quad \mathrm{otherwise}. \end{cases}\]

Note that \(\dim(H((E_2)_z))=3\), since \(p_1^2((E_2)_z)=0\) and \(p_2^2((E_2)_z)=1\). It follows that \(r\) is a constant function on a neighborhood of \((E_2)_z\), hence \((E_2)_z\) is Kähler-regular.

Therefore, by the Cartan–Kähler theorem, there exists an integral manifold \(j\colon X\to \overline{M}\) such that \(J((\mathcal{L}^j\overline{A})_x)=E_z\) where \(x\in X\) is a point such that \(z=j(x)\) for all points of the form \(z=(0,0,c_1,c_2,0)\), where \(c_1\neq 0\) and \(c_2\in \mathbb{R}\).

As before, the Cartan–Kähler theorem only guarantees the existence of an integral manifold, but does not give one explicitly. We now claim that \[\label{eqn:specialj} j\colon M\longrightarrow \overline{M},\quad (t,w)\longmapsto \left(t,w,s=c_1 e^{-h(w)},P=c_1 e^{-h(w)} h'(w),Q=0\right),\tag{15}\] where \(h(w)\mathrel{\vcenter{:}}= \frac{c_2}{c_1}w+\frac{d_2 c_1+c_2^2}{2c_1^2} w^2\) is such an integral manifold. For a generic element \(\bar{a}_{\bar{m}}=a_1\Gamma_0+a_2 W_1+a_3 H_1+a_4 \frac{\partial}{\partial s}+a_5 \frac{\partial}{\partial P}+a_6 \frac{\partial}{\partial Q}\in \overline{A}\), one has that \[\bar{\rho}(\bar{a}_{\bar{m}})=a_1\frac{\partial}{\partial t}+a_2 \frac{\partial}{\partial w}+a_4 \frac{\partial}{\partial s}+a_5 \frac{\partial}{\partial P}+a_6 \frac{\partial}{\partial Q}.\] A generic element \(v_m=b_1 \frac{\partial}{\partial t}+b_2\frac{\partial}{\partial w}\in \mathrm{T}M\) transforms under (the tangent map of) \(j\) via \[\mathrm{T}j(v_m)=b_1\frac{\partial}{\partial t}+b_2 \frac{\partial}{\partial w}-b_2 c_1 e^{-h(w)}h'(w) \frac{\partial}{\partial s}+b_2 c_1 e^{-h(w)}\left(h''(w)-(h'(w))^2\right)\frac{\partial}{\partial P}.\] Points in \(\mathcal{L}^j \overline{A}\) must satisfy the equations \(a_1=b_1\), \(a_2=b_2\), \(a_4=-b_2 c_2 e^{-h(w)}h'(w)\), \(a_5=b_2 c_1 e^{-h(w)}\left(h''(w)-(h'(w))^2\right)\) and \(a_6=0\). Hence, a basis of \(\mathcal{L}^j \overline{A}\) is given by \[\left\{\left(\Gamma_0,\frac{\partial}{\partial t}\right),\left(W_1-c_1 e^{-h(w)}h'(w)\frac{\partial}{\partial s}+c_1 e^{-h(w)}\left(h''(w)-(h'(w))^2\right)\frac{\partial}{\partial P},\frac{\partial}{\partial w}\right),(H_1,0)\right\}.\] Applying \(J\) and using the values of \(h\), \(h'\) and \(h''\) at \(0\) gives \[J((\mathcal{L}^j \overline{A})_{(0,0)})=\mathop{\mathrm{span}}\left\{\Gamma_0(j(0,0)),W_1(j(0,0))-c_2 \frac{\partial}{\partial s}\Big|_{j(0,0)}+d_2 \frac{\partial}{\partial P}\Big|_{j(0,0)},H_1(j(0,0))\right\}=E_z,\] as required.

The integral manifold \(j\) in 15 has a further property that is not automatically guaranteed by the Cartan–Kähler Theorem. It is, in fact, a section of the fiber bundle \(p\colon \overline{M}\to M\) and satisfies, for this reason, an independence condition, namely \(j^{\ast}(\mathrm{d}t\wedge \mathrm{d}w^1\wedge \dots\wedge \mathrm{d}w^n)\neq 0\), where \((t,w^i)\) are the coordinates of \(M=\mathbb{R}\times \mathfrak{g}\) (see the next section for more details). This means that \(j\) is indeed of the “special type” which induces a multiplier matrix for the invariant inverse problem (see the reference we had made earlier to Thm. 2 of [8]).

8 Outlook↩︎

We end the paper with some different directions for future research.

First, it may be possible to extend Cartan’s test to (transitive) Lie algebroids. For EDS on manifolds, it expresses the condition of integrability in terms of certain integers (called characters). With such an extension in place, one could then attempt to extend the Frobenius theorem to transitive Lie algebroids in the analytic category (in other words, showing that all integral elements \(E_z\subset \overline{A}_z\) of dimension \(N-s\) are \(\dim(\ker(\bar{\rho}_z))\)-ordinary for an EDS on a transitive Lie algebroid of rank \(N\), generated algebraically by \(s\) linearly independent \(1\)-forms).

Second, in the classical theory of EDS on manifolds, one often works with an extra assumption called an independence condition. This means that, given a choice of a differential \(n\)-form \(\Omega\) on \(\overline{M}\), integral manifolds \(i\colon M\to \overline{M}\) must additionally be \(n\)-dimensional, and they must satisfy \(i^{\ast}(\Omega)\neq 0\) at each point of \(M\) (besides the condition that \(i^{\ast}\theta=0\) for any \(\theta\in \mathcal{I}\subset \Omega^{\ast}(\overline{M})\)). We have explained in [8] that if \(p\colon \overline{M} \to M\) is a fiber bundle, any integral manifold \(i\colon M\to \overline{M}\) that is also a section implicitly satisfies a certain independence condition. For the specific case of an EDS with an independence condition, finding the characters in Cartan’s test is easy using tableaux. Roughly speaking, a tableau is a matrix encoding such an EDS. It also serves as a computational tool, to compute the characters using only linear algebra [23], [24]. We believe that extending tableaux to transitive Lie algebroids will be very useful in determining existence of integral manifolds. A natural starting point would be to study extensions of so-called linear Pfaffian systems (a linear EDS generated by only one-forms together with an independence condition) in the Lie algebroid setting, since the theory of tableaux is simplest in this situation (in the classical theory of EDS).

A third possible extension of this work is motivated by our discussion on the invariant inverse problem of the calculus of variations. While in 7 we have shown that our version of the Cartan–Kähler theorem is consistent with the fact that a \(G\)-invariant Lagrangian exists for the geodesic spray of the canonical connection on the Lie group \(G=\mathbb{R}\), we were not able to conclude using the Cartan–Kähler theorem that a reduced multiplier matrix exists. This is because the existence of a reduced multiplier requires the additional condition that there exists an integral manifold \(j\colon M\to \overline{M}\) which is a section of \(p\colon \overline{M}\to M\) (and therefore satisfies an independence condition), which is not guaranteed by the Cartan–Kähler theorem. We would therefore like to devise techniques to solve this problem using our version of the Cartan–Kähler theorem. Since this is an EDS with an independence condition, we suspect that an extension of the theory of tableaux will be very helpful for this.

If such an approach is successful, we may be able to study more nontrivial cases of this problem. Following the notation in [8], the natural starting point for study is the case where \(\langle \Sigma^0\rangle_{\mathrm{alg}}\) is a differential ideal. In the future, a classification structure (analogous to Douglas’ classification [25], or the classification in [7]) for this problem may be identified.

8.0.0.1 Acknowledgements.

The authors thank the Research Fund of the University of Antwerp (BOF) for its support through the DOCPRO4 project 46954 “Symmetry reduction and unreduction in mechanics and geometry".

Appendix: Comparison with the approach in [8]↩︎

The current framework defines integral manifolds by starting from a given transitive Lie algebroid \(\bar{\tau}\colon \overline{A}\to \overline{M}\). We then constructed a new Lie algebroid \(\tau\colon A\to M\) by setting \(A\mathrel{\vcenter{:}}= \mathcal{L}^i \overline{A}\) to be the \(i\)-prolongation Lie algebroid, where \(i\colon M\to \overline{M}\) is an immersion. We then said that \(i\colon M\to \overline{M}\) is an integral manifold if \(I^{\ast}\theta=0\) for all \(\theta\in \mathcal{I}\), where \(I\) is the \(i\)-induced map. Integral manifolds in this framework are not required to satisfy any independence condition, i.e., if \(n\) is the dimension of the integral manifolds one must find, there is no requirement that \(i^{\ast}(\Omega)\neq 0\) for some \(\Omega\in \Omega^n(\overline{M})\).

Howewer, in a previous paper [8], we defined integral manifolds by instead beginning with the Lie algebroid \(\tau\colon A\to M\). We then constructed \(\bar{\tau}\colon \overline{A}\to \overline{M}\) by defining \(\overline{A}\mathrel{\vcenter{:}}= \mathcal{L}^p A\), where \(p\colon \overline{M}\to M\) is a fiber bundle. In this framework, \(i\) is no longer an arbitrary immersion, but a section of \(p\). In order to distinguish the two definitions, we use \(s\) instead of \(i\). The definition given there is as follows:

This approach generalizes the definition of an integral manifold of an EDS with an independence condition. Indeed, if \(p\colon \overline{M}\to M\), \((x^i,y^{\mu})\mapsto (x^i)\) is a fiber bundle, then any section \(s\) of \(p\) satisfies \(s^{\ast}(\mathrm{d}x^1\wedge \dots \wedge \mathrm{d}x^n)\neq 0\).

We now claim that in fact, \(I=S\) in the case where \(i\) is a section of a fiber bundle. Recall that \(I\colon A=\mathcal{L}^i \overline{A}\to \overline{A}\) was given by \((\bar{a}_{\bar{m}},w_m)\mapsto \bar{a}_{\bar{m}}\). To this end, we first prove the following lemma:

We now prove the equivalence of the two frameworks in the case when one imposes that \(i\) is a section.

As a result, also \(I^{\ast}=S^{\ast}\).

References↩︎