Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature

We prove that the equations describing compatible $(N\times N)$ metrics of constant Riemannian curvature define a special class of integrable $N$-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open-closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Frobenius manifolds. We conjecture that open-closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras.