Received: 9 Nov 2024; Accepted: 27 Nov 2024
Integrable geodesic flows with simultaneously diagonalisable quadratic integrals
Abstact.
Mathematics Subject Classification:
Abstact.
37J35, 70H06
Mathematics Subject Classification:
Key words and phrases: 
1 Introduction 
We work locally, on a smooth \(n\)-dimensional pseudo-Riemannian manifold \((M,g)\) of any signature. By geodesic flow we understand the Hamiltonian system on \(T^{*}M\) generated by the Hamiltonian
| \(H(x,p)=\tfrac{1}{2}g^{ij}p_{i}p_{j}.\) |
We will study the situation when the geodesic flow admits \(n\), including the Hamiltonian, integrals
| \(\overset{1}{I}(x,p)=2H\,,\ \overset{2}{I}(x,p)\,,\ \dots\,,\ \overset{n}{I}(x,p)\) |
such that the following conditions are fulfilled:
The integrals are quadratic in momenta, that is, \(\overset{\alpha}{I}(x,p)=\overset{\alpha}{K}^{ij}p_{i}p_{j}\). In particular, \(g^{ij}=\overset{1}{K}^{ij}\). We assume without loss of generality that the (2,0)-tensor fields \(\overset{\alpha}{K}^{ij}\) are symmetric in upper indexes.
At almost every point \(x\in M\), there exists a basis in \(T_{x}M\) such that, for every \(\alpha=1,\dots,n\), the matrix \(\left(\overset{\alpha}{K}^{ij}(x)\right)\) is diagonal.
The differentials of the integrals are linearly independent at least at one111Using ideas of [11], is is easy to show that linear independence of the differentials of polynomial in momenta integrals at one point implies their linear independence at almost every point, provided the manifold is connected point of \(T^{*}M\).
In many publications on this topic, e.g. in [2, 5, 8], it is assumed that for almost every point \(x\in M\) the restrictions of the tensor fields \(\overset{\alpha}{K}^{ij}\), \(\alpha=1,\dots,n\), to \(T_{x}M\) are linearly independent. Our main result, Theorem 1 below, shows that this assumptions follows from conditions (1,2,3):
Under the assumptions above, for almost every point \(x\) the restrictions of the tensor fields \(\overset{\alpha}{K}^{ij}\), \(\alpha=1,\dots,n\), to \(T_{x}M\) are linearly independent. In particular, for a generic linear combination \(I=\sum_{\alpha=2}^{n}\lambda_{\alpha}\overset{\alpha}{I}\) of the integrals, the corresponding (1,1)-tensor field \(K^{i}_{j}:=K^{si}g_{sj}\), where \(K^{ij}=\sum_{\alpha=2}^{n}\lambda_{\alpha}\overset{\alpha}{K}^{ij}\), has \(n\) different eigenvalues.
In dimension \(n=3\), Theorem 1 and Corollary 2 below were proven in [1, Theorem 2] by other methods.
Assume the integrals \(\overset{\alpha}{I}\) satisfy the conditions (1,2,3) above and in addition are in involution with respect to the standard Poisson bracket. Then, near almost every point, the metric \(g\) and the integrals come from the Stäckel construction.
In view of Theorem 1, Corollary 2 follows from [7, Theorem 6], [8, Proposition 1.1.3], [2, Theorem 8.6] or, possibly,222By [9, Note on page 185] the diploma thesis of A. Thimm 1976, which we were not able to find, contains this result from A. Thimm 1976. In these references, it was shown that \(n\) quadratic functionally independent integrals in involution such that the corresponding Killing tensors are simultaneously diagonalisable at every tangent space and such that at least one of the Killing tensors with one index raised by the metric has \(n\) different eigenvalues, come from the Stäckel construction which we recall below.
As mentioned above, the difference between our conditions (1,2,3) and the assumptions used in [7, Theorem 6], [8, Proposition 1.1.3] or [2, Theorem 8.6] is as follows: in [7, Theorem 6], [8, Proposition 1.1.3] or [2, Theorem 8.6] it was assumed that one of the Killing tensors, with one index raised by the metric, has \(n\) different eigenvalues. We do not have this condition as an assumption and prove that it follows from other assumptions.
Let us recall the Stäckel333The construction appeared already in [12, §§13-14], see also discussion in [13, pp. 703–705] construction following [5, 14]. Take a non-degenerate \(n\times n\) matrix \(S=(S_{ij})\) with \(S_{ij}\) being a function of the \(i\)-th variable \(x^{i}\) only . Next, consider the functions \(\overset{\alpha}{I}\), \(\alpha=1,\dots,n\), given by the following system of linear equations
| \(S\mathbb{I}=\mathbb{P},\) | (1) |
where \(\mathbb{I}=\left(\overset{1}{I},\overset{2}{I},\dots,\overset{n}{I}\right)^{\top}\) and \(\mathbb{P}=\left(p_{1}^{2},p_{2}^{2},\dots,p_{n}^{2}\right)^{\top}\). It is known that the functions \(\overset{\alpha}{I}\) are in involution. Taking one of them (say, the first one, provided the inverse matrix to \(S\) has no zeros in the first raw) as twice the Hamiltonian of the metric, one obtains an integrable geodesic flow whose integrals satisfy the conditions (1,2,3). Corollary 2 says that locally, near almost every points, there exist no other examples of geodesic flows admitting \(n\) independend quadratic in momenta integrals in involution, such that the corresponding Killing tensors are simultaneously diagonalisable at almost every tangent space.
It is known that metrics coming from the Stäckel construction admit orthogonal separation of variables in the Hamilton-Jacobi equations, so the equation for their geodesics can be locally solved in quadratures [4, 6]. Namely, J. Liouville [12] and, independently, P. Stäckel [14] has shown that the metrics are precisely those admitting othogonal separation of variables. L. P. Eisenhart, in his widely cited and very influential paper [5], has shown that locally the metrics coming from the Stäckel construction are precisely those whose geodesic flows admit \(n\) functionally independent integrals in involution satisfying the following conditions: the integrals are quadratic in momenta, the corresponding matrices are simultaneously diagonalisable in a coordinate system, and at every point the corresponding matrices are linearly independent. In [2, 7, 8] it was shown that the assumption that the integrals are simultaneously diagonalisable in a coordinate system may be replaced by a weaker assumption that the matrices of the integrals are diagonalisable in a frame. Our result further improves the result of Eisenhart and shows that the condition that the matrices of the integrals are linearly independent at each point is not necessary as this assumption follows from other conditions.
2 Proof of Theorem 1
Under the assumptions (1,2,3) from Section 1, near almost every point, there exist smooth vector fields \(v_{1}(x),...,v_{n}(x)\in T_{x}M\) such that they are linearly independent at every tangent space and such that the metric \(g\) and the matrices \(\overset{\alpha}{K}^{ij}\) are diagonal in the basis \((v_{1},...,v_{n})\). After re-arranging and re-scaling the vectors \(v_{i}\), there exists \(m\in\mathbb{N}\), \(m\leq n\), \(k_{1},...,k_{m}\in\mathbb{N}\) with \(k_{1}+\cdots+k_{m}=n\) and smooth local functions \(g_{1},..,g_{m},\overset{\alpha}{\rho}_{1},\dots,\overset{\alpha}{\rho}_{m},\) \(\alpha\in\{2,\dots,n\}\), on \(M\) such that the Hamiltonian and the integrals \(\overset{\alpha}{I}\) with \(\alpha=2,\dots,n\) are given by the formulas
| \(\begin{array}[]{rcl}2H&=&V_{1}+V_{2}+\cdots+V_{m}\\ \overset{\alpha}{I}&=&\overset{\alpha}{\rho}_{1}V_{1}+\overset{\alpha}{\rho}_{% 2}V_{2}+\cdots+\overset{\alpha}{\rho}_{m}V_{m}\end{array}\) | (2) |
In the formulas above, \(V_{i}\) are the functions on the cotangent bundle given by
| \(\begin{array}[]{lcl}V_{1}&=&(v_{1})^{2}\varepsilon_{1}+(v_{2})^{2}\varepsilon_% {2}+\cdots+(v_{k_{1}})^{2}\varepsilon_{k_{1}},\\ V_{2}&=&(v_{k_{1}+1})^{2}\varepsilon_{k_{1}+1}+(v_{k_{1}+2})^{2}\varepsilon_{k% _{1}+2}+\cdots+(v_{k_{1}+k_{2}})^{2}\varepsilon_{k_{1}+k_{2}},\\ &\vdots&\\ V_{m}&=&(v_{k_{1}+\cdots+k_{m-1}+1})^{2}\varepsilon_{k_{1}+\cdots+k_{m-1}+1}+(% v_{k_{1}+\cdots+k_{m-1}+2})^{2}\varepsilon_{k_{1}+\cdots+k_{m-1}+2}+\cdots+(v_% {n})^{2}\varepsilon_{n},\end{array}\) |
where \(v_{i}\) is the linear function on the \(T^{*}M\) generated by the vector field \(v_{i}\) via the canonical identification444In naive terms, we consider the vector field \(v=\sum_{i}v^{i}\partial_{i}\) as the linear function \(p\mapsto\sum_{i}v^{i}p_{i}\) on \(T^{*}M\). This identification of vector fields on \(M\) and linear in momenta functions on the cotangent bundle is independent of a coordinate system \(TM\equiv T^{**}M\), and \(\varepsilon_{i}\in\{-1,1\}\).
The Poisson bracket of \(H\) and \(I=\overset{\alpha}{I}\) reads (we omit the index \(\alpha\) since the equations hold for any \(\overset{\alpha}{I}\)):
| \(0=\{2H,I\}=\sum_{i,j=1}^{m}\left(\{V_{i},\rho_{j}\}V_{j}+\rho_{j}\{V_{i},V_{j}% \}\right).\) | (3) |
The right hand side of (3) is a cubic polynomial in momenta so all its coefficients are zero. For every point \(x\in M\), this gives us a system of linear equations on the directional derivatives \(v_{s}(\rho_{j})\) with \(s\in\{1,\dots,n\}\) and \(j\in\{1,\dots,m\}\). The coefficients and free terms of this system depend on \(\rho_{j}(x)\), on the entries of the vector fields \(v_{s}\) at \(x\), and on the derivatives of the entries of the vector fields \(v_{s}\) at \(x\). Let us show that all directional derivatives \(v_{s}(\rho_{i})\) can be reconstructed from this system. We will show this for the directional derivatives \(v_{1}(\rho_{2})\) and \(v_{1}(\rho_{1})\), since this will cover two principle cases \(i=j\) and \(i\neq j\); the proof for all other \(v_{i}(\rho_{j})\) is completely analogous.
In order to extract \(v_{1}(\rho_{2})\), note that the cubic in momenta component \((v_{k_{1}+1})^{2}v_{1}\) shows up only in the addends \(\{V_{1},\rho_{2}\}V_{2}\), \(\rho_{1}\{V_{2},V_{1}\}\) and \(\rho_{1}\{V_{1},V_{2}\}\). In these addends, the coefficient containing a derivative of one of the functions \(\rho_{s}\) is \(v_{1}(\rho_{2})\). Thus, equating the coefficient of \((v_{k_{1}+1})^{2}v_{1}\) to zero gives us \(v_{1}(\rho_{2})\) as a function of \(\rho_{1},\rho_{2}\) and the entries of \(\{V_{1},V_{2}\}\).
Similarly, in order to extract \(v_{1}(\rho_{1})\), we note that the cubic in momenta component \((v_{1})^{3}\) shows up only in the addends \(\{V_{1},\rho_{1}\}V_{1}\) and \(\rho_{1}\{V_{1},V_{2}\}\). Its coefficient containing the derivatives of \(\rho\)’s is \(v_{1}(\rho_{1})\). Thus, equating the coefficient of \((v_{1})^{3}\) to zero gives us \(v_{1}(\rho_{1})\) as a function of \(\rho_{1}\).
Thus, all directional derivatives \(v_{s}(\rho_{j})\) can be obtained from the system (3). Let us now view the system (3) as a linear PDE-system on unknown functions \(\rho_{i}\). The coefficients of this system come from the vector fields \(v_{s}\) and are given by certain nonlinear expressions in the components of \(v_{s}\) and their derivatives. Since the directional derivatives of all functions \(\rho_{i}\) are expressed in the terms of the functions \(\rho_{i}\), the system can be solved with respect to all derivatives of the functions \(\rho_{i}\). Therefore, the initial values of the functions \(\rho_{i}\) at one point \(x_{0}\) determine the local solution of the system. This implies that the space of solutions is at most \(m\)-dimensional. Finally, the linear vector space of the integrals \(\overset{\alpha}{I}\) is at most \(m\)-dimensional. Since \(n\) of them are functionally independent by our assumptions, \(n=m\) and Theorem 1 is proved.
The proof of Theorem 1 is motivated by [3, proof of Lemma 1.2], [8, §1.1] and [10, §2].
Acknowledgments V. M. thanks the DFG (projects 455806247 and 529233771) and ARC Discovery Programme (DP210100951) for the support. This research was partially supported by FAPESP (research fellowship 2018/20009-6), in particular the visit of V. M. to São José do Rio Preto, was supported by this grant. We thank C. Chanu and G. Rastelli for pointing out the references [2, 7] and for useful discussions. We are grateful to the anonymous referee for useful suggestions.
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