\(F\)-manifolds and integrable systems of hydrodynamic type. (Q2897388)

The integrability of systems of PDE's of hydrodynamic type is closely related to the geometry of \(F\)-manifolds. Such structure on a manifold \(M\) is given by an associative commutative product on every tangent space which satisfies certain first order differential conditions. Denote this product by a \((1,2)\)-type tensor \(c^i_{jk}\). The aim of this article is to study systems of PDE's of the form \(u^i_t = (V_X)^i_j u^j_x\) where \((V_X)^i_j = c^i_{jk}X^k\) for a vector field \(X\) on \(M\). In this general setting, diagonalizability of \(V_X\) and commutativity of flows \(u^i_t = (V_X)^i_j u^j_x\) and \(u^i_\tau = (V_Y)^i_j u^j_x\) for vector fields \(X\) and \(Y\) are discussed.NEWLINENEWLINEAn additional ingredient is an affine connection \(\nabla \) on \(M\) compatible with the product \(c^i_{jk}\). Assuming \(\nabla \) is flat, authors apply the Dubrovin's construction of principal hierarchy [\textit{B. Dubrovin},``Geometry of 2D topological field theories'' in: Francaviglia, M. (ed.) et al., Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 14-22, 1993. Berlin: Springer-Verlag. Lect. Notes Math. 1620, 120-348 (1996; Zbl 0841.58065)] to the case of \(F\)-manifolds. (Note \(\nabla \) does not need to be metric). Motivated by this, authors drop off the flatness assumption of \(\nabla \) and identify the differential condition \(c^i_{jm} \nabla _k X^m = c^i_{km} \nabla _j X^m\); such vector fields \(X\) then define integrable systems of hydrodynamic type. Further, curvature restrictions imposed by existence of such \(X\) are discussed. Finally, the special case of compatible Levi-Civita connections is studied and a class of corresponding metrics is recognized in the Hamiltonian theory of systems of hydrodynamic type.