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Through the eigenvalue problem we associate to the classical orthogonal polynomials two classes of weighted Riemannian $$1$$ 1 -manifolds having the coordinate $$x$$ x . For the first class the eigenvalues contains $$x$$ x and the metric is fixed as being the Euclidean one while for the second class the eigenvalues are independent of this variable and the metric and weight function are founded. The Hermite polynomials is the only case which generates the same manifold. The geometry of second class of weighted manifolds is studied from several points of view: geodesics, distance and exponential map, harmonic functions and their energy density, volume, zeta function, heat kernel. A partial heat equation is studied for these metrics and for the Poincaré ball model of hyperbolic geometry.
Analysis and Mathematical Physics – Springer Journals
Published: Apr 18, 2015
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