A study of three-dimensional paracontact (κ̃,μ̃,ν̃)-spaces

Abstract

This paper is a study of three-dimensional paracontact metric [Formula: see text]-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field [Formula: see text] is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric [Formula: see text]-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases [Formula: see text] [Formula: see text] and construct new examples of such manifolds for each case. We also show the existence of paracontact metric [Formula: see text] spaces with dimension greater than 3, such that [Formula: see text] but [Formula: see text]

Almostα-paracosymplectic manifolds

“…Next This characterization is well known in the Riemannian case ( [3,23,25]). Using same arguments in pseudo-Riemannian case, G. Calvaruso [7] proved that same result is still valid for vector fields of constant length, if it is not lightlike.…”

“…Let (M, φ, ξ, η, g) be a 3-dimensional almost α-paracosymplectic manifold .Then operator h has following types. h 1 -type) h 2 -type) Using same methods in [25] one can construct a local pseudo-orthonormal basis {e 1 , e 2 , e 3 } in a neighborhood of p where g(e 1 , e 1 ) = g(e 2 , e 2 ) = g(e 1 , e 3 ) = g(e 2 , e 3 ) = 0 and g(e 1 , e 2 ) = g(e 3 h 4 -type) Then a local pseudo-orthonormal basis {e 1 , e 2 , e 3 } is constructed in a neighborhood of p where g(e 1 , e 1 ) = g(e 2 , e 2 ) = g(e 1 , e 3 ) = g(e 2 , e 3 ) = 0 and g(e 1 , e 2 ) = g(e 3 , e 3 ) = 1. Since the tensor h is h 4 -type) (with respect to a pseudo-orthonormal basis {e 1 , e 2 , e 3 }) then he 1 = λe 1 + e 3 , he 2 = λe 2 and he 3 = e 2 + λe 3 .…”

“…Since the proof of following lemma is similar to [25] we omit proof of it. In a 3-dimensional pseudo-Riemannian manifold case, the curvature tensor can be written by (11.4)…”

GENERALIZED (Κ˜ ≠ −1, Μ˜)-Paracontact METRIC MANIFOLDS WITH Ξ(µ˜) = 0

“…We now show that while D αhomothetic deformations destroy conditions likeR XY ξ = 0, they preserve the class of paracontact (κ,μ)-spaces. Kupeli Erken and Murathan analyzed the different possibilities for the tensor fieldh in [15]. Ifh has 3.…”

Parallel almost paracontact structures on affine hypersurfaces

“…These structures play an important role in pseudo-Riemannian geometry as well as modern mathematical physics. In particular, some recent results related to paracontact geometry can be found in [15,1,6]. Moreover, recently some relations between para-complex and affine differential geometry were also studied (see [7,2] and [14]).…”