CONTACT METRIC (κ,μ)-SPACES AS BI-LEGENDRIAN MANIFOLDS

Abstract

We describe a contact metric manifold whose Reeb vector field belongs to the (κ, µ)-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric (κ, µ)-spaces in terms of a canonical connection which can be naturally defined on them.2000 Mathematics subject classification: 53C12, 53C15, 53C25.

Some Remarks on the Generalized Tanaka-Webster Connection of a Contact Metric Manifold

“…Note also that since M 2n+1 is Sasakian and in particular K-contact, by Lemma 3.6, also Q is flat. Finally, we prove that * ∇ coincides with the bi-Legendrian connection corresponding to (L, Q), that is * ∇ verifies (ii) and (iii) in (6). The relations * T (X, ξ) = [ξ, X] Q for X ∈ Γ (L) and * T (Y, ξ) = [ξ, Y ] L for Y ∈ Γ (Q) hold because L and Q are flat and, on the other hand, * T (X, ξ) = φhX = 0, * T (Y, ξ) = φhY = 0.…”

“…Moreover, these Legendrian foliations are totally geodesic, hence they verify (i)-(iv) of Proposition 2.9. This bi-Legendrian structure and the corresponding bi-Legendrian connection has been studied in detail in [6] where in particular it is proved that D (λ) and D (−λ) are never both flat.…”

On the classification of contact metric ‐spaces via tangent hyperquadric bundles

“…In our case the explicit expressions of and are (see [, p. 127] or ): Using the previous equations one gets (see ): Theorem Let be a non‐Sasakian contact metric ‐manifold. Then one of the following must hold: both and are positive definite; is positive definite and is negative definite; both and …”

“…The Legendre foliation  is called positive, negative or flat according to the circumstance that the bilinear form  is positive definite, negative definite or vanishes identically, respectively. In our case the explicit expressions of ( ) and (− ) are (see [1, p. 127] or [7]):…”

On Submanifolds of (kappa, mu) – Contact Metric Manifolds

“…The first class consists of the unit tangent sphere bundles of spaces of constant curvature, equipped with their natural contact metric structure and the second class contains all the threedimensional unimodular Lie groups, except the commutative one, admitting the structure of a left invariant ) ( ,   -contact metric manifolds [4], [5], [13]. One of the peculiarities of these manifolds is that its tangent space give rise to three mutually orthogonal distributions corresponding to the eigenspaces of M so that these manifolds are endowed with a bi-Legendrian structure [11]. This is why the distributions of ) , (   -contact metric manifolds are seemed to the interesting as a field of study.…”