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]

History

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]

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 ${\displaystyle M}$ and one of its points ${\displaystyle x\in M}$,

• ${\displaystyle TM}$ is the tangent bundle of ${\displaystyle M}$;
• ${\displaystyle T_{x}M}$ is the tangent space at ${\displaystyle x}$;

Definition 1. Let ${\displaystyle M}$ be a ${\displaystyle 2n}$-dimensional Finsler manifold, ${\displaystyle n\geq 1}$, and let ${\displaystyle F:TM\to \mathbb {R} }$ 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 ${\displaystyle M}$ 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. "Almost Hermitian Finsler structure": see ( Rizza 1962b , pp. 271, 273–274) and ( Rizza 1963 , pp. 83, 90–91).
4. 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, vol. 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), vol. 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".