Applications of Nijenhuis Geometry V: Geodesic Equivalence and Finite-Dimensional Reductions of Integrable Quasilinear Systems
Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev
Abstract
Abstract We describe all metrics geodesically compatible with a $$\textrm{gl}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>gl</mml:mtext></mml:math> -regular Nijenhuis operator L . The set of such metrics is large enough so that a generic local curve $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> to be a g -geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g -geodesics is naturally equivalent to the Poisson action of $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> on the cotangent bundle generated by the integrals coming from geodesic compatibility.