Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations

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 53-61
  • ISSN: 1120-6330

Abstract

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We consider different types of quaternionic-like structures. The interrelations between automorphism groups of the subordinated structures and of some admissible connections are studied. A characterization of automorphisms of a quaternionic structure as some kind of projective transformations is given. General results on harmonicity of an automorphism of some -structure are obtained and applied to the case of an almost Hermitian quaternionic structure. Different noteworthy transformations groups of quaternionic Kähler or hyperKähler manifolds and their interrelations are studied. In particular, new characterizations of quaternionic Kähler manifolds that admit a quaternionic automorphism different from an isometry are given. This Note follows a Note I with the same general title, published in these Rendiconti, [2], and it is preliminary to a wider memoir just mentioned in previous one.

How to cite

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Alekseevsky, Dmitri V., and Marchiafava, Stefano. "Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.1 (1993): 53-61. <http://eudml.org/doc/244299>.

@article{Alekseevsky1993,
abstract = {We consider different types of quaternionic-like structures. The interrelations between automorphism groups of the subordinated structures and of some admissible connections are studied. A characterization of automorphisms of a quaternionic structure as some kind of projective transformations is given. General results on harmonicity of an automorphism of some \( G \)-structure are obtained and applied to the case of an almost Hermitian quaternionic structure. Different noteworthy transformations groups of quaternionic Kähler or hyperKähler manifolds and their interrelations are studied. In particular, new characterizations of quaternionic Kähler manifolds that admit a quaternionic automorphism \( \varphi \) different from an isometry are given. This Note follows a Note I with the same general title, published in these Rendiconti, [2], and it is preliminary to a wider memoir just mentioned in previous one.},
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; Automorphism groups of G-structures; Quaternionic transformations; Harmonic automorphisms; quaternionic structures; automorphism groups; harmonicity},
language = {eng},
month = {3},
number = {1},
pages = {53-61},
publisher = {Accademia Nazionale dei Lincei},
title = {Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations},
url = {http://eudml.org/doc/244299},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Alekseevsky, Dmitri V.
AU - Marchiafava, Stefano
TI - Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations
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 - 53
EP - 61
AB - We consider different types of quaternionic-like structures. The interrelations between automorphism groups of the subordinated structures and of some admissible connections are studied. A characterization of automorphisms of a quaternionic structure as some kind of projective transformations is given. General results on harmonicity of an automorphism of some \( G \)-structure are obtained and applied to the case of an almost Hermitian quaternionic structure. Different noteworthy transformations groups of quaternionic Kähler or hyperKähler manifolds and their interrelations are studied. In particular, new characterizations of quaternionic Kähler manifolds that admit a quaternionic automorphism \( \varphi \) different from an isometry are given. This Note follows a Note I with the same general title, published in these Rendiconti, [2], and it is preliminary to a wider memoir just mentioned in previous one.
LA - eng
KW - G-structures; Quaternionic structures; Automorphism groups of G-structures; Quaternionic transformations; Harmonic automorphisms; quaternionic structures; automorphism groups; harmonicity
UR - http://eudml.org/doc/244299
ER -

References

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  1. ALEKSEEVSKY, D. V. - GRAEV, M. M., Calabi-Yau metric on the Fermat surface. Isometries and totally geodesic submanifolds. JGP, vol. 7, n. 1, 1990, 21-43. Zbl0749.53028MR1094729DOI10.1016/0393-0440(90)90018-X
  2. ALEKSEEVSKY, D. V. - MARCHIAFAVA, S., Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions. Rend. Mat. Acc. Lincei, s. 9, v. 4, 1993, 43-52. Zbl0781.53023MR1225886
  3. ALEKSEEVSKY, D. V. - MARCHIAFAVA, S., Quaternionic structures on a manifold and underlying structures. In preparation. Zbl0968.53033
  4. ALEKSEEVSKY, D. V. - MARCHIAFAVA, S., Transformation groups of quaternionic manifolds. In preparation. Zbl0878.53029
  5. EELLS, J. - LEMAIRE, L., Selected topics in harmonic maps. Conference board of the mathematical sciences regional conference series in mathematics, n. 50. Ed. American Mathematical Society, Providence, Rhode Island1983. Zbl0515.58011MR703510
  6. FUJIMURA, S., -connections and their changes on almost quaternion manifolds. Hokkaido Math. J., 5, 1976, 239-248. Zbl0333.53014MR407762
  7. KOBAYASHI, S., Transformation groups in differential geometry. Springer-Verlag, Berlin-Heidelberg-New York1972. Zbl0829.53023MR355886
  8. MUSSO, E., On the transformation group of a quaternionic manifold. Bollettino U.M.I., (7) 6-B, 1992, 67-78. Zbl0828.53028MR1164939
  9. PICCINNI, P., On the infinitesimal automorphisms of quaternionic structures. J. de Math. Pures et Appl., to appear. Zbl0829.53029MR1249411
  10. POON, Y. S. - SALAMON, S., Quaternionic Kähler 8-manifolds with positive scalar curvature. J. Diff. Geom., 33, 1991, 363-378. Zbl0733.53035MR1094461

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