HOLONOMY AND 3-SASAKIAN HOMOGENEOUS MANIFOLDS VERSUS SYMPLECTIC TRIPLE SYSTEMS
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Springer Science+Business Media New York (2020)
HOLONOMY AND 3-SASAKIAN HOMOGENEOUS MANIFOLDS VERSUS SYMPLECTIC TRIPLE SYSTEMS C. DRAPER∗ Department of Applied Mathematics University of M´alaga, 29071 Spain [email protected]
Dedicated to the memory of Professor Thomas Friedrich
Abstract. Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the special orthogonal group, of dimension considerably smaller. We realize the computations of the holonomies in a unified way, by using as a main algebraic tool a nonassociative structure, that of a symplectic triple system.
1. Introduction 3-Sasakian geometry is a natural generalization of Sasakian geometry introduced independently by Kuo and Udriste in 1970 [23], [25]. 3-Sasakian manifolds are very interesting objects. In fact, any 3-Sasakian manifold has three orthonormal Killing vector fields which span an integrable 3-dimensional distribution. Under some regularity conditions on the corresponding foliation, the space of leaves is a quaternionic K¨ ahler manifold with positive scalar curvature. And conversely, over any quaternionic K¨ahler manifold with positive scalar curvature, there is a principal SO(3)-bundle that admits a 3-Sasakian structure [10]. The canonical example is the sphere S4n+3 realized as a hypersurface in Hn+1 . Besides, any 3Sasakian manifold is an Einstein space of positive scalar curvature [22]. In spite of that, during the period from 1975 to 1990, approximately, 3-Sasakian manifolds lived in relative obscurity, probably due to the fact that, according to DOI: 10.1007/S00031-020-09609-w Supported by Junta de Andaluc´ıa through projects FQM-336 and UMA18FEDERJA-119 and by the Spanish Ministerio de Ciencia e Innovaci´ on through project PID2019-104236GB-I00, all of them with FEDER funds. Received March 18, 2019. Accepted April 22, 2020. Corresponding Author: C. Draper, e-mail: [email protected] 2000 Mathematics Subject Classification. Primary: 53C29. Secondary: 53C05; 53C30; 53C25. Keywords: Invariant affine connection, homogeneous manifold, 3-Sasakian structure, holonomy algebra, symplectic triple system, curvature. ∗
C. DRAPER
the authors of the monograph [10], Boyer and Galicki, the holonomy group of a 3-Sasakian manifold never reduces to a proper subgroup of the special orthogonal group. There was the idea that manifolds should be divided into different classes according to their holonomy group, being special geometries: those with holonomy group not of general type. The revival of 3-Sasakian manifolds occurred in the nineties. On one hand, Boyer and Galicki began to study 3-Sasakian geometry because it appeared as a natural object in their quotient construction for certain hyperk¨ahler manifolds [12]. This construction helped to find new collections of examples, and, since then, the topology and geometry of these manifolds have been continuously studied. On the other hand,
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