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arXiv · 2201.08244

Quantum geodesic flows and curvature

Edwin Beggs·Shahn Majid
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Abstract

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical $*$ operation on noncommutative vector fields. We show on a classical manifold how the Ricci tensor arises naturally in our approach as a term in the convective derivative of the divergence of the geodesic velocity field, and use this to propose a similar object in the noncommutative case. Examples include quantum geodesic flows on the algebra of 2 x 2 matrices, fuzzy spheres and the $q$-sphere.

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Edwin Beggs, Shahn Majid. 2022-01-17. Quantum geodesic flows and curvature. https://doi.org/10.1007/s11005-023-01687-7

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