The history of Rizza manifolds follows the history of the structure that such objects carry. According to Shoshichi Kobayashi (1975), the geometry of complexFinsler 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-dimensionalmanifolds, "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]
Formal definition
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 manifoldM and one of its pointsx ∈ M
^ abThe dedication of the work (Ichijyō 1988, p. 1) reads:-"Dedicated to professor G. B. Rizza, who is the originator of the notion of Rizza manifolds."
^"Almost Hermitian Finsler structure": see (Rizza 1962b, pp. 271, 273–274) and (Rizza 1963, pp. 83, 90–91).
^Rizza (1962b, p. 1) himself states:-"L'esistenza di strutture di tipo diverso su una medesima varietà dà sempre luogo a problemi di confronto (The existence of structures of different kind on the same manifold always gives rise to comparison problems)".
Rizza, Giovanni Battista (1962a), "Finsler structures on almost complex manifolds", Proceedings of the International Congress of Mathematicians, Stockholm., ICM Proceedings, Stockholm{{citation}}: CS1 maint: location missing publisher (link). A short research announcement describing briefly the results proved in (Rizza 1963).
Rizza, Giovanni Battista (1962b), "Strutture di Finsler sulle varietà quasi complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian), 33 (5): 271–275. Another short presentation of the results proved in (Rizza 1963): the English translation of the title reads as:-"Finsler structures on almost complex manifolds".