Quantum geodesic flows and curvature

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 $\ast$ 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.

arXiv comment32 pages amslatex, 3 pdf figures; minor corrections and clarifications, more background in the preliminaries

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Cited by in corpus (4)

• Qubit Geodesics on the Bloch Sphere from Optimal-Speed Hamiltonian Evolutions
• Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model
• Differential geometry and general relativity with algebraifolds