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 Shoshichi Kobayashi ( 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 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

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{{ 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".
• 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".