Geodesic
Canonical forms for the~invariant tensors and $A$-$B$-$C$-cohomologies of integrable Hamiltonian systems
Sbornik. Mathematics, Tome 189 (1998) no. 3, pp. 315-357

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The canonical forms for the $(\ell ,m)$ tensors, $\ell +m\leqslant 3$, that are invariant with respect to a Liouville-integrable non-degenerate Hamiltonian system $V$ on a symplectic manifold $M^{2k}$ are derived. It is proved that the characteristic polynomial of any invariant $(1,1)$ tensor $A^\alpha _\beta$ is a perfect square; therefore its eigenvalues have even multiplicities. Any invariant metric $g_{\alpha \beta }$ is indefinite and has signature $\sigma \leqslant k$. The derived canonical forms are applied to the calculation of the $A$-$B$-$C$-cohomologies of Liouville-integrable Hamiltonian systems. 
@article{SM_1998_189_3_a0,
     author = {O. I. Bogoyavlenskii},
     title = {Canonical forms for the~invariant tensors and $A$-$B$-$C$-cohomologies of integrable {Hamiltonian} systems},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {189},
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     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_3_a0/}
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O. I. Bogoyavlenskii. Canonical forms for the~invariant tensors and $A$-$B$-$C$-cohomologies of integrable Hamiltonian systems. Sbornik. Mathematics, Tome 189 (1998) no. 3, pp. 315-357. http://geodesic.mathdoc.fr/item/SM_1998_189_3_a0/