2011/03/30 by Joakim Arnlind, Gerhard Huisken, Arnlind, Joakim +1 · 1 citation
Mathematics · Medicine · #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.1103.5862
openalex publication_date 2011/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, Kähler-Poisson algebras are introduced, and it is shown that a corresponding purely algebraic theory of geometry and curvature can be developed. As an illustration of the new concepts we give an algebraic proof of the statement that a bound on the (algebraic) Ricci curvature induces a bound on the eigenvalues of the (algebraic) Laplace operator, in analogy with the well-known theorem in Riemannian geometry. As the correspondence between Poisson brackets of smooth functions and commutators of operators lies at the heart of quantization, a purely Poisson algebraic proof of, for instance, such a "Gap Theorem", might lead to an understanding of spectral properties in a corresponding quantum mechanical system.