464:, p. 1) himself states:-"L'esistenza di strutture di tipo diverso su una medesima varietĂ dĂ sempre luogo a problemi di confronto (The existence of structures of different kind on the same manifold always gives rise to comparison problems)".
352:
82:, "Struttura di Finsler quasi Hermitiana": his motivation for the introduction of the concept seems to be the aim of comparing two different structures existing on the same manifold. Later
219:
775:
234:
1659:
63:
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 (
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1654:
941:
965:
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663:
638:
Geometria differenziale â Analisi complessa. Convegno internazionale â Parma, 19â20 maggio 1994 in occasione del 70° compleanno di G. B. Rizza
1030:
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1309:
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635:(1994), "Omaggio a Giovanni Battista Rizza in occasione del suo 70° compleanno", in Donnini, S.; Gigante, G.; Mangione, V. (eds.),
1358:
950:
1341:
1702:
75:), nearly one year earlier than the one cited by Kobayashi. Rizza called this differential geometric structure, defined on
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The history of Rizza manifolds follows the history of the structure that such objects carry. According to
1197:
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804:
cites as the first one in the theory of Rizza manifolds: an
English translation of the title reads as:-"
17:
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698:
Rendiconti della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
641:, Serie 5 (in Italian), vol. 3, Rivista di Matematica della UniversitĂ di Parma, pp. 1â2
202:
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1460:
1331:
1135:
960:
823:
389:
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347:{\displaystyle F(x,cy)=|c|F(x,y)\qquad \forall c\in \mathbb {C} ,\quad x\in M,\quad y\in T_{x}M}
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Dedicated to professor G. B. Rizza, who is the originator of the notion of Rizza manifolds.
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750:. The article giving the proofs of the results previously announced in references (
498:, Mathematical Sciences Research Institute Publications, vol. 50, Cambridge:
110:), borrowing the scheme of notation equally from both sources. Precisely, given a
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685:. A short research announcement describing briefly the results proved in (
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1025:
846:
79:
815:
660:
Proceedings of the
International Congress of Mathematicians, Stockholm.
1241:
819:
696:(1962b), "Strutture di Finsler sulle varietĂ quasi complesse",
620:
acknowledges
Giovanni Battista Rizza as the first one to study
658:(1962a), "Finsler structures on almost complex manifolds",
102:
The content of this paragraph closely follows references (
579:"Negative vector bundles and complex Finsler structures"
706:. Another short presentation of the results proved in (
649:
Homage to
Giovanni Battista Rizza on his 70th birthday
237:
205:
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1302:
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858:
710:): the English translation of the title reads as:-"
758:: the English translation of the title reads as:-"
346:
213:
647:: an English translation of the title reads as:-"
727:Rivista di Matematica della UniversitĂ di Parma
723:"Strutture di Finsler di tipo quasi Hermitiano"
547:Rivista di Matematica della UniversitĂ di Parma
712:Finsler structures on almost complex manifolds
831:
543:"Finsler metrics on almost complex manifolds"
86:, p. 1) started calling this structure "
8:
479:"Finsler Geometry on Complex Vector Bundles"
36:: this kind of manifold is also referred as
760:Finsler structures of almost Hermitian type
1246:
838:
824:
816:
679:: CS1 maint: location missing publisher (
643:. A tribute to Rizza by his former master
594:
335:
304:
303:
270:
262:
236:
207:
206:
204:
53:
428:
411:
406:
404:
400:
222:its Finsler function. If the condition
107:
83:
755:
751:
672:
461:
445:
68:
64:
707:
686:
495:A Sampler of RiemannâFinsler Geometry
449:
103:
72:
7:
773:-forme quadratiche ed hermitiane",
442:Almost Hermitian Finsler structure
294:
14:
38:almost Hermitian Finsler manifold
71:), proving them in the article (
324:
311:
293:
878:Differentiable/Smooth manifold
448:, pp. 271, 273â274) and (
290:
278:
271:
263:
256:
241:
229:
90:", and manifolds carrying it "
1:
628:, now called Rizza manifolds.
410:The dedication of the work (
214:{\displaystyle \mathbb {R} }
1584:Classification of manifolds
700:, Serie VIII (in Italian),
583:Nagoya Mathematical Journal
541:IchijyĆ, Yoshihiro (1988),
1724:
800:. This article is the one
500:Cambridge University Press
1660:over commutative algebras
596:10.1017/S0027763000016615
1376:Riemann curvature tensor
806:Hermitian and quadratic
779:, V Serie (in Italian),
776:Rendiconti di Matematica
767:Rizza, Giovanni Battista
719:Rizza, Giovanni Battista
694:Rizza, Giovanni Battista
656:Rizza, Giovanni Battista
477:Aikou, Tadashi (2004),
375:Almost complex manifold
112:differentiable manifold
50:Shoshichi Kobayashi
30:almost complex manifold
26:Giovanni Battista Rizza
1168:Manifold with boundary
883:Differential structure
492:; et al. (eds.),
452:, pp. 83, 90â91).
348:
215:
1703:Differential geometry
414:, p. 1) reads:-"
349:
216:
18:differential geometry
1315:Covariant derivative
866:Topological manifold
729:, (2) (in Italian),
575:Kobayashi, Shoshichi
235:
203:
1349:Exterior derivative
951:AtiyahâSinger index
900:Riemannian manifold
802:Shoshichi Kobayashi
618:Shoshichi Kobayashi
508:2004srfg.book.....B
502:, pp. 83â105,
56:), the geometry of
1655:Secondary calculus
1609:Singularity theory
1564:Parallel transport
1332:De Rham cohomology
971:Generalized Stokes
490:Chern, Shiing-Shen
390:Hermitian manifold
344:
211:
61:Finsler structures
32:also supporting a
1690:
1689:
1572:
1571:
1337:Differential form
991:Whitney embedding
925:Differential form
626:Finsler structure
622:complex manifolds
616:. In this paper,
486:Bryant, Robert L.
484:, in Bao, David;
357:holds true, then
98:Formal definition
34:Finsler structure
1715:
1708:Smooth manifolds
1682:Stratified space
1640:Fréchet manifold
1354:Interior product
1247:
944:
840:
833:
826:
817:
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783:(1â2): 221â249,
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385:Finsler manifold
380:Complex manifold
353:
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179:Finsler manifold
166:
77:even-dimensional
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1718:
1717:
1716:
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1713:
1712:
1693:
1692:
1691:
1686:
1625:Banach manifold
1618:Generalizations
1613:
1568:
1505:
1402:
1364:Ricci curvature
1320:Cotangent space
1298:
1236:
1078:
1072:
1031:Exponential map
995:
940:
934:
854:
844:
765:
717:
692:
671:
664:ICM Proceedings
654:
645:Enzo Martinelli
631:
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117:and one of its
100:
92:Rizza manifolds
88:Rizza structure
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1178:Parallelizable
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1058:Lie derivative
1055:
1053:Integral curve
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1014:Diffeomorphism
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859:Basic concepts
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633:Martinelli, E.
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363:Rizza Manifold
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136:tangent bundle
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24:, named after
22:Rizza manifold
13:
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2:
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1677:Supermanifold
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1398:Wedge product
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1342:Vector-valued
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1272:Tangent space
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1044:
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1037:
1036:in Lie theory
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1024:
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939:Main results
937:
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920:Tangent space
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756:Rizza (1962b)
753:
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517:0-521-83181-4
513:
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431:, p. 6).
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165:Definition 1.
159:
155:
154:tangent space
151:
144:
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116:
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97:
95:
93:
89:
85:
84:IchijyĆ (1988
81:
78:
74:
70:
66:
62:
59:
55:
51:
43:
41:
39:
35:
31:
27:
23:
19:
1604:Moving frame
1599:Morse theory
1589:Gauge theory
1381:Tensor field
1310:Closed/Exact
1289:Vector field
1257:Distribution
1216:
1198:Hypercomplex
1193:Quaternionic
930:Vector field
888:Smooth atlas
807:
805:
780:
774:
770:
759:
730:
726:
711:
704:(5): 271â275
701:
697:
659:
648:
637:
586:
582:
550:
546:
494:
462:Rizza (1962b
457:
441:
436:
429:IchijyĆ 1988
423:
415:
412:IchijyĆ 1988
362:
358:
356:
196:
191:
190:
183:
176:-dimensional
173:
168:
163:
157:
145:
139:
131:
125:
121:
114:
108:IchijyĆ 1988
101:
91:
87:
47:
37:
21:
15:
1549:Levi-Civita
1539:Generalized
1511:Connections
1461:Lie algebra
1393:Volume form
1294:Vector flow
1267:Pushforward
1262:Lie bracket
1161:Lie algebra
1126:G-structure
915:Pushforward
895:Submanifold
752:Rizza 1962a
589:: 153â166,
446:Rizza 1962b
69:Rizza 1962b
65:Rizza 1962a
1697:Categories
1672:Stratifold
1630:Diffeology
1426:Associated
1227:Symplectic
1212:Riemannian
1141:Hyperbolic
1068:Submersion
976:HopfâRinow
910:Submersion
905:Smooth map
797:0123.15203
747:0129.14101
733:: 83â106,
708:Rizza 1963
687:Rizza 1963
613:0326.32016
567:0885.53031
534:1073.53093
471:References
450:Rizza 1963
188:, and let
104:Rizza 1963
73:Rizza 1963
1554:Principal
1529:Ehresmann
1486:Subbundle
1476:Principal
1451:Fibration
1431:Cotangent
1303:Covectors
1156:Lie group
1136:Hermitian
1079:manifolds
1048:Immersion
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