Rizza manifold explained

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, 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 and, proving them in the article, 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 started calling this structure "Rizza structure", and manifolds carrying it "Rizza manifolds".^[1]

Formal definition

The content of this paragraph closely follows references and, 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;

Let M be a 2n-dimensional Finsler manifold,, and let its Finsler function. If the condition

F(x,cy) = |c|F(x,y) \forall c\in C , x\in M, y\in T _{xM}

holds true, then M is a Rizza Manifold.

See also

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

References

• .
• .
• . In this paper, Shoshichi Kobayashi acknowledges Giovanni Battista Rizza as the first one to study complex manifolds with Finsler structure, now called Rizza manifolds.
• . 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".
• . A short research announcement describing briefly the results proved in .
• . Another short presentation of the results proved in : the English translation of the title reads as:-"Finsler structures on almost complex manifolds".
• . The article giving the proofs of the results previously announced in references and : the English translation of the title reads as:-"Finsler structures of almost Hermitian type".
• . 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".

Notes and References

1. The dedication of the work reads:-"Dedicated to professor G. B. Rizza, who is the originator of the notion of Rizza manifolds."
2. See .
3. "Almost Hermitian Finsler structure": see and .
4. 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)".