Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions

Dmitri V. Alekseevsky; Stefano Marchiafava

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993)

  • Volume: 4, Issue: 1, page 43-52
  • ISSN: 1120-6330

Abstract

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This Note will be followed by a Note II in these Rendiconti and successively by a wider and more detailed memoir to appear next. Here six quaternionic-like structures on a manifold M (almost quaternionic, hypercomplex, unimodular quaternionic, unimodular hypercomplex, Hermitian quaternionic, Hermitian hypercomplex) are defined and interrelations between them are studied in the framework of general theory of G-structures. Special connections are associated to these structures. 1-integrability and integrability conditions are derived. Decompositions of appropriate spaces of curvature tensors are given. In Note II the automorphism groups of these quaternionic-like structures will be considered.

How to cite

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Alekseevsky, Dmitri V., and Marchiafava, Stefano. "Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.1 (1993): 43-52. <http://eudml.org/doc/244082>.

@article{Alekseevsky1993,
abstract = {This Note will be followed by a Note II in these Rendiconti and successively by a wider and more detailed memoir to appear next. Here six quaternionic-like structures on a manifold \( M \) (almost quaternionic, hypercomplex, unimodular quaternionic, unimodular hypercomplex, Hermitian quaternionic, Hermitian hypercomplex) are defined and interrelations between them are studied in the framework of general theory of G-structures. Special connections are associated to these structures. 1-integrability and integrability conditions are derived. Decompositions of appropriate spaces of curvature tensors are given. In Note II the automorphism groups of these quaternionic-like structures will be considered.},
author = {Alekseevsky, Dmitri V., Marchiafava, Stefano},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {G-structures; Quaternionic structures; Special connections; Integrability conditions; Curvature tensors; almost quaternionic; hypercomplex; unimodular quaternionic; unimodular hypercomplex; Hermitian quaternionic; Hermitian hypercomplex; integrability},
language = {eng},
month = {3},
number = {1},
pages = {43-52},
publisher = {Accademia Nazionale dei Lincei},
title = {Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions},
url = {http://eudml.org/doc/244082},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Alekseevsky, Dmitri V.
AU - Marchiafava, Stefano
TI - Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/3//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 1
SP - 43
EP - 52
AB - This Note will be followed by a Note II in these Rendiconti and successively by a wider and more detailed memoir to appear next. Here six quaternionic-like structures on a manifold \( M \) (almost quaternionic, hypercomplex, unimodular quaternionic, unimodular hypercomplex, Hermitian quaternionic, Hermitian hypercomplex) are defined and interrelations between them are studied in the framework of general theory of G-structures. Special connections are associated to these structures. 1-integrability and integrability conditions are derived. Decompositions of appropriate spaces of curvature tensors are given. In Note II the automorphism groups of these quaternionic-like structures will be considered.
LA - eng
KW - G-structures; Quaternionic structures; Special connections; Integrability conditions; Curvature tensors; almost quaternionic; hypercomplex; unimodular quaternionic; unimodular hypercomplex; Hermitian quaternionic; Hermitian hypercomplex; integrability
UR - http://eudml.org/doc/244082
ER -

References

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  2. ALEKSEEVSKY, D. V. - GRAEV, M. M., G-structures of twistor type on a manifold and underlying structures. Preprint Dip. Mat. Univ. «La Sapienza», Roma1991. MR1410066
  3. ALEKSEEVSKY, D. V. - MARCHIAFAVA, S., Quaternionic structures on a manifold and underlying structures. In preparation. Zbl0968.53033
  4. BERNARD, D., Sur la géométrie différentielle des G-structures. Ann. Inst. Fourier (Grenoble), 10, 1960, 151-270. Zbl0095.36406MR126800
  5. BONAN, E., Sur les G-structures de type quaternionien. Cahiers de topologie et géométrie différentielle, 9, 1967, 389-461. Zbl0171.20802MR233302
  6. CHERN, S. S., On a generalisation of Kähler geometry. Symposium in honor of S. Léfschetz on Algebraic geometry and topology. Princeton University Press, Princeton mathematical series, 18, 1957, 103-121. Zbl0078.14103MR87172
  7. CHERN, S. S., The geometry of G-structures. Bull. Amer. Math. Soc, 72, 1966, 167-219. Zbl0136.17804MR192436
  8. FUJIMURA, S., Q -connections and their changes on almost quaternion manifolds. Hokkaido Math. J., 5, 1976, 239-248. Zbl0333.53014MR407762
  9. KULKARNI, R. S., On the principle of uniformisation. J. Diff. Geom., 13, 1978, 109-138. Zbl0381.53023MR520605
  10. MUSSO, E., On the transformation group of a quaternionic manifold. Bollettino U.M.I., (7) 6-B, 1992, 67-78. Zbl0828.53028MR1164939
  11. OPROIU, V., Almost quaternal structures. An. st. Univ. «Al. I. Cuza» Iazi, 23, 1977, 287-298. Zbl0373.53017MR493860
  12. OPROIU, V., Integrability of almost quaternal structures. An. st. Univ. «Al. I. Cuza» Iazi, 30, 1984, 75-84. Zbl0573.53021MR800155
  13. SALAMON, S., Differential geometry of quaternionic manifolds. Ann. Scient. Ec. Norm. Sup., 4ème série, 19, 1986, 31-55. Zbl0616.53023MR860810
  14. SALAMON, S., Riemannian geometry and holonomy groups. Ed. Longman Scientific & Technical, UK, 1989. Zbl0685.53001MR1004008

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