We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: - A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. - A map between quaternionic manifolds endowed with the nonintegrable almost twistorial structures is twistorial if and only if it is quaternionic and totally-geodesic. As an application, we describe all the quaternionic maps between open sets of quaternionic projective spaces.
@article{ASNSP_2010_5_9_1_47_0,
author = {Ianu\c{s}, Stere and Marchiafava, Stefano and Ornea, Liviu and Pantilie, Radu},
title = {Twistorial maps between quaternionic manifolds},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {47--67},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {1},
year = {2010},
mrnumber = {2668873},
zbl = {1193.53121},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_1_47_0/}
}
TY - JOUR AU - Ianuş, Stere AU - Marchiafava, Stefano AU - Ornea, Liviu AU - Pantilie, Radu TI - Twistorial maps between quaternionic manifolds JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 47 EP - 67 VL - 9 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_1_47_0/ LA - en ID - ASNSP_2010_5_9_1_47_0 ER -
%0 Journal Article %A Ianuş, Stere %A Marchiafava, Stefano %A Ornea, Liviu %A Pantilie, Radu %T Twistorial maps between quaternionic manifolds %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 47-67 %V 9 %N 1 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_1_47_0/ %G en %F ASNSP_2010_5_9_1_47_0
Ianuş, Stere; Marchiafava, Stefano; Ornea, Liviu; Pantilie, Radu. Twistorial maps between quaternionic manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 47-67. http://www.numdam.org/item/ASNSP_2010_5_9_1_47_0/
[1] and , Quaternionic-like structures on a manifold: Note 2. Automorphism groups and their interrelations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 4 (1993), 53–61. | EuDML | MR
[2] and , A report on quaternionic-like structures on a manifold, In: “Proceedings of the International Workshop on Differential Geometry and its Applications” (Bucharest, 1993), Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 55 (1993), 9–34. | MR | Zbl
[3] and , Quaternionic structures on a manifold and subordinated structures, Ann. Mat. Pura Appl. 171 (1996), 205–273. | MR | Zbl
[4] , and , Compatible complex structures on almost quaternionic manifolds, Trans. Amer. Math. Soc. 351 (1999), 997–1014. | MR | Zbl
[5] , Sur les -structures de type quaternionien, Cahiers Topologie Géom. Différentielle 9 (1967), 389–461. | EuDML | Numdam | MR | Zbl
[6] and , Quaternionic maps between hyperkähler manifolds, J. Differential Geom. 55 (2000), 355–384. | MR | Zbl
[7] and , Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), 589–640. | EuDML | Numdam | MR | Zbl
[8] , Die Funktionentheorie der Differentialgleichungen und mit vier reellen Variablen, Comment. Math. Helv. 7 (1935), 307–330. | EuDML | MR | Zbl
[9] and , “Principles of Algebraic Geometry”, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1978. | MR | Zbl
[10] , Nonlinear Dirac operator and quaternionic analysis, Comm. Math. Phys. 281 (2008), 251–261. | MR | Zbl
[11] , and , Harmonic maps between quaternionic Kähler manifolds, J. Nonlinear Math. Phys. 15 (2008), 1–8. | MR | Zbl
[12] , Hypercomplex algebraic geometry, Quart. J. Math. Oxford Ser. (2) 49 (1998), 129–162. | MR | Zbl
[13] and , Quaternionic maps between quaternionic Kähler manifolds, Math. Z. 250 (2005), 523–537. | MR | Zbl
[14] and , Harmonic morphisms between Weyl spaces and twistorial maps II, Ann. Inst. Fourier (Grenoble), to appear. | EuDML | Numdam | MR | Zbl
[15] , Sulle varietà a struttura quaternionale generalizzata, Rend. Mat. 3 (1970), 529–545. | MR | Zbl
[16] , Almost quaternal structures, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. 23 (1977), 287–298. | MR | Zbl
[17] , Integrability of almost quaternal structures, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. 30 (1984), 75–84. | MR | Zbl
[18] , On a class of twistorial maps, Differential Geom. Appl. 26 (2008), 366–376. | MR | Zbl
[19] , and , Hypercomplex structures associated to quaternionic manifolds, Differential Geom. Appl. 9 (1998), 273–292. | MR | Zbl
[20] , Quaternionic algebra and sheaves on the Riemann sphere, Quart. J. Math. Oxford Ser. (2) 49 (1998), 163–198. | MR | Zbl
[21] , Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. 19 (1986), 31–55. | EuDML | Numdam | MR | Zbl
[22] , Special structures on four-manifolds, Riv. Mat. Univ. Parma 17* (1991), 109–123. | MR | Zbl
[23] , Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–224. | MR | Zbl
[24] , HyperKähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421–450. | EuDML | MR | Zbl
[25] , Quaternionic submanifolds in quaternionic symmetric spaces, Tohoku Math. J. 38 (1986), 513–538. | MR | Zbl
