On complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral
For a 2-dimensional torus \(T^2\) with a Riemannian metric \(g=\sum_{i,j=1}^2 g_{ij}dx_idx_j\) a function \(H=\sum_{i,j=1}^2g^{ij}p_ip_j\) on the cotangent bundle \(T^*(T^2)\) with local coordinates \((x_i,p_i)\) is a Hamiltonian function on \(T^*(T^2)\), and a symplectic manifold \(T^*(T^2)\) with \(H\) is called a geodesic flow. A function \(F\) on \(T^*(T^2)\) is called an integral of this flow if \(\sum_{i=i}^2{\partial H\over\partial x_i} {\partial F\over\partial p_i}-{\partial H\over \partial p_i} {\partial F\over \partial x_i}=0\). \(F\) is a quadratic integral if it is of the form \(F=\sum_{i+j=2}a_{ij}p_1^ip_2^j\), where \((a_{ij})\) is a symmetric matrix. A quotient set \(G_c\) of Riemannian metrics by the conformal relation on \(T^2\) is the set of complex structures on \(T^2\). In this paper, the authors describe a subset of the set \(G_c\) of complex structures on \(T^2\) admitting Riemannian metrics with quadratic integrals.
scientific article; zbMATH DE number 921617