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

Abstract

We find necessary and sufficient conditions for the bi-Legendrian connection ∇ associated to a bi-Legendrian structure (F , G) on a contact metric manifold (M 2n+1 , φ, ξ, η, g) being a metric connection with respect to the associated metric g and then we give conditions ensuring that ∇ coincides with the (generalized) Tanaka-Webster connection of (M 2n+1 , φ, ξ, η, g). Using these results, we give some interpretations of the Tanaka-Webster connection and we study the interplays between the Tanaka-Webster, the bi-Legendrian and the Levi Civita connection in a Sasakian manifold.2000 Mathematics Subject Classification. 53B05, 53C12, 53C15.

$ N(κ)- $contact Metric Manifolds with Generalized Tanaka-Webster Connection

“…Let M be a N(κ)-contact metric manifold which is given in Example 2 with a generalized Tanaka-Webster connection. Then from (20) and by using ( 21) we get…”

“…A generalized Tanaka-Webster connection has been introduced by Tanno [14] as a generalization of Tabaka-Webster connection [15,16]. Contact manifolds with generalized Tanaka-Webster connection were studied by many researchers [17][18][19][20][21]. The concircular curvature tensor was defined by Yano [22].…”

Some Curvature Tensor Relations on Nearly Cosymplectic Manifolds with Tanaka-Webster Connection

“…Many authors have studied the Tanaka-Webster connection later. In relation to the generalized Tanaka-Webster connection Ghosh, Prakasha, Ünal and Montano have done important studies on this connection in various structures [13,[19][20][21]. We, on the other hand, tried to define some curvature tensors on nearly cosymplectic structures according to the tanaka webster connection, taking into account some previous studies.…”

N(k)-contact metric manifolds with generalized Tanaka-Webster connection

“…A generalized Tanaka-Webster connection has been introduced by Tanno [21] as a generalization of Tabaka-Webster connection [23,25]. Contact manifolds with generalized Tanaka-Webster connection were studied by many researchers [11,13,16,18,19]. The curvature tensors has been used for studying differential geometry of manifolds with structures since they determine most geometric properties of the related object.…”