In differential geometry a Rizza manifold, named after Giovanni Battista Rizza, [1] is an almost complex manifold also supporting a Finsler structure: this kind of manifold is also referred as almost Hermitian Finsler manifold. [2]
The history of Rizza manifolds follows the history of the structure that such objects carry. According to ShoshichiKobayashi ( 1975 ), the geometry of complex Finsler structures was first studied in Rizza's 1964 paper "F-forme quadratiche ed hermitiane", but Rizza announced his results nearly two years before, in the short communications ( Rizza 1962a ) and ( Rizza 1962b ), proving them in the article ( Rizza 1963 ), nearly one year earlier than the one cited by Kobayashi. Rizza called this differential geometric structure, defined on even-dimensional manifolds, "Struttura di Finsler quasi Hermitiana": [3] his motivation for the introduction of the concept seems to be the aim of comparing two different structures existing on the same manifold. [4] Later Ichijyō (1988 , p. 1) started calling this structure "Rizza structure", and manifolds carrying it "Rizza manifolds". [1]
The content of this paragraph closely follows references ( Rizza 1963 ) and ( Ichijyō 1988 ), borrowing the scheme of notation equally from both sources. Precisely, given a differentiable manifold M and one of its points x ∈ M
Definition 1. Let M be a 2n-dimensional Finsler manifold, n ≥ 1, and let F : TM → ℝ its Finsler function. If the condition
holds true, then M is a Rizza Manifold.
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{{citation}}: CS1 maint: location missing publisher (link). A short research announcement describing briefly the results proved in ( Rizza 1963 ).