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Let us consider a function of n independent variables F (t n) satisfying the following two conditions: 1. The matrix η αβ = ∂ 3 F ∂t 1 ∂t α ∂t β is constant and nondegenerate. Note that the matrix η αβ completely determines dependence of the function F on the fixed variable t 1 .
We solve the problem of description for nonsingular pairs of compatible flat metrics in the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is(More)
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlin-ear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in(More)