# 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]

The history of Rizza manifolds follows the history of the structure that such objects carry. According to Template:Harvtxt, the geometry of complex Finsler structures was first studied in the paper Template:Harv:[4] however, Rizza announced his results nearly two years before, in the short communications Template:Harv and Template:Harv, proving them in the article Template:Harv, 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 Template:Harvtxt started calling this structure "Rizza structure", and manifolds carrying it "Rizza manifolds".[1]

## Formal definition

The content of this paragraph closely follows references Template:Harv and Template:Harv, borrowing the scheme of notation equally from both sources. Precisely, given a differentiable manifold M and one of its points xM

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

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

holds true, then M is a Rizza Manifold.

## Notes

1. The dedication of the work Template:Harv reads:-"Dedicated to professor G. B. Rizza, who is the originator of the notion of Rizza manifolds."
2. See Template:Harv.
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 Template:Harv and Template:Harv.
6. Template:Harvtxt 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

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|CitationClass=citation }}. In this paper, Shoshichi Kobayashi acknowledges Giovanni Battista Rizza as the first one to study complex manifolds with Finsler structure, now called Rizza manifolds.

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|CitationClass=citation }}. 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".

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|CitationClass=citation }}. A short research announcement describing briefly the results proved in Template:Harv.

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|CitationClass=citation }}. Another short presentation of the results proved in Template:Harv: the English translation of the title reads as:-"Finsler structures on almost complex manifolds".

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|CitationClass=citation }}. The article giving the proofs of the results previously announced in references Template:Harv and Template:Harvtxt: the English translation of the title reads as:-"Finsler structures of almost Hermitian type".

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. 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".