Left-invariant hypercontact structures on three-dimensional Lie groups
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Left-invariant hypercontact structures on three-dimensional Lie groups Giovanni Calvaruso · Antonella Perrone
© Akadémiai Kiadó, Budapest, Hungary 2014
Abstract We characterize three-dimensional manifolds admitting an almost contact metric 3-structure and completely classify left-invariant hypercontact structures on threedimensional Lie groups. Keywords
Almost contact (metric) 3-structures · Hypercontact structures
Mathematics Subject Classification
53C15 · 53C25 · 53B05 · 53D15
1 Introduction The study of 3-Sasakian manifolds has proven to be relevant both in mathematics and in physics. As it is well known, contact metric 3-structures are not a natural generalization of 3-Sasakian structures, since a contact metric 3-structure is necessarily 3-Sasakian [9]. Also for this reason, in recent years, special classes of almost contact (metric) 3-structures attracted the interest of a growing number of researchers. Generally speaking, the contact metric condition, which on the one hand permits to deduce many classification results, on the other hand precludes the possibility to consider several remarkable cases. In analogy with hyper-Kähler structures, hypercontact manifolds [6] were introduced as the quaternionic generalization of contact manifolds. A hypercontact manifold is a (4n + 3)dimensional (smooth, connected) manifold M, equipped with an almost contact metric 3structure (ϕi , ξi , ηi , g), and a triple of contact forms (α1 , α2 , α3 ), satisfying the compatibility condition g(X, ϕi Y ) = dαi (X, Y ), for all vector fields X, Y on M and for all indices i = 1, 2, 3. It is a natural problem to determine which are the homogeneous models of hyper-
G. Calvaruso (B) · A. Perrone Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy e-mail: [email protected] A. Perrone e-mail: [email protected]
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contact manifolds, starting from the lowest possible dimension, i.e., from three-dimensional examples. In this paper, after reporting some basic information on almost contact 3-structures and hypercontact structures, in Sect. 3 we shall address the issue of the existence of almost contact metric 3-structures on a three-dimensional manifold (Theorem 3.1). Then, in Sect. 4 we shall give the classification of left-invariant hypercontact structures on three-dimensional Lie groups (Theorem 4.1).
2 Preliminaries We first report some definitions and properties about contact and hypercontact metric manifolds. All manifolds are supposed to be connected and smooth. Let M denote a (2n + 1)-dimensional manifold. An almost contact structure on M is a triple formed by a 1-form η, a vector field ξ and a (1, 1)-tensor field ϕ, satisfying η(ξ ) = 1,
ϕ 2 = −I + η ⊗ ξ.
To note that these conditions imply ϕ(ξ ) = 0 and ηϕ = 0. A Riemannian metric g on M is said to be compatible with the almost contact structure if g(ϕ X, ϕY ) = g(X, Y ) − η(X )η(Y ), for any tangent vector fields X, Y . In such a case, η = g(·, ξ ),
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