Rizza manifold

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]

Historical notes

In particolare Rizza ha introdotto, in modo efficace, la nozione di varietà di Finsler quasi hermitiana. Come ha osservato Kobayashi, Rizza è stato il primo a proporre tale tipo di struttura, poi studiata da vari autori in particolare della scuola giapponese, alcuni dei quali chiamano le varietà considerate Rizza manifolds.[3]

— Enzo Martinelli, (Martinelli 1994, p. 2)

The history of Rizza manifolds follows the history of the structure that such objects carry. According to Shoshichi Kobayashi (1975), the geometry of complex Finsler structures was first studied in the paper (Rizza 1964):[4] however, 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":[5] his motivation for the introduction of the concept seems to be the aim of comparing two different structures existing on the same manifold.[6] 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 manifold M and one of its points x ∈ M

• TM is the tangent bundle of M;
• T_xM is the tangent space at x;

Definition 1. Let M be a 2n-dimensional Finsler manifold, n ≥ 1, and let F : TM → ℝ its Finsler function. If the condition

(1)     ${\displaystyle F(x,cy)=|c|F(x,y)\qquad \forall c\in \mathbb {C} ,\quad x\in M,\quad y\in T_{x}M}$

holds true, then M is a Rizza Manifold.

See also

• Almost complex manifold
• Complex manifold
• Finsler manifold
• Hermitian manifold

Notes

1. The 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."
2. See (Ichijyō 1988, p. 6).
3. The italic emphasis is due to Enzo Martinelli himself. An English translation reads as:-"In particular, Rizza introduced, in an effective way, the notion of almost hermitian Finsler manifold. As Kobayashi observed, Rizza was the first to propose such kind of structure, later studied by various authors, belonging in particular to the Japanese school (T.n.: of differential geometry), some of them calling the considered varieties Rizza manifolds".
4. Note that there is a typo in the bibliography given by Kobayashi: it is incorrectly stated that Rizza's article was published in 1965.
5. "Almost Hermitian Finsler structure": see (Rizza 1962b, pp. 271, 273–274) and (Rizza 1963, pp. 83, 90–91).
6. 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)".

References

• Aikou, Tadashi (2004), "Finsler Geometry on Complex Vector Bundles" (PDF), in Bao, David; Bryant, Robert L.; Chern, Shiing-Shen; et al. (eds.), A Sampler of Riemann–Finsler Geometry, Mathematical Sciences Research Institute Publications, 50, Cambridge: Cambridge University Press, pp. 83–105, Bibcode:2004srfg.book.....B, ISBN 0-521-83181-4, MR 2132658, Zbl 1073.53093.
• Ichijyō, Yoshihiro (1988), "Finsler metrics on almost complex manifolds", Rivista di Matematica della Università di Parma, (IV), 14*: 1–28, MR 1045035, Zbl 0885.53031.
• Kobayashi, Shoshichi (1975), "Negative vector bundles and complex Finsler structures", Nagoya Mathematical Journal, 57: 153–166, doi:10.1017/S0027763000016615, MR 0377126, Zbl 0326.32016. In this paper, Shoshichi Kobayashi acknowledges Giovanni Battista Rizza as the first one to study complex manifolds with Finsler structure, now called Rizza manifolds.
• Martinelli, E. (1994), "Omaggio a Giovanni Battista Rizza in occasione del suo 70° compleanno", in Donnini, S.; Gigante, G.; Mangione, V. (eds.), Geometria differenziale – Analisi complessa. Convegno internazionale – Parma, 19–20 maggio 1994 in occasione del 70° compleanno di G. B. Rizza, Serie 5 (in Italian), 3, Rivista di Matematica della Università di Parma, pp. 1–2. A tribute to Rizza by his former master Enzo Martinelli: an English translation of the title reads as:-"Homage to Giovanni Battista Rizza on his 70th birthday".
• Rizza, Giovanni Battista (1962a), "Finsler structures on almost complex manifolds", Proceedings of the International Congress of Mathematicians, Stockholm., ICM Proceedings, Stockholm. 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".
• Rizza, Giovanni Battista (1963), "Strutture di Finsler di tipo quasi Hermitiano", Rivista di Matematica della Università di Parma, (2) (in Italian), 4: 83–106, MR 0166742, Zbl 0129.14101. The article giving the proofs of the results previously announced in references (Rizza 1962a) and Rizza (1962b): the English translation of the title reads as:-"Finsler structures of almost Hermitian type".
• Rizza, Giovanni Battista (1964), "F-forme quadratiche ed hermitiane", Rendiconti di Matematica, V Serie (in Italian), 23 (1–2): 221–249, MR 0211370, Zbl 0123.15203. This article is the one Shoshichi Kobayashi cites as the first one in the theory of Rizza manifolds: an English translation of the title reads as:-"Hermitian and quadratic F-forms".