Second order parallel tensors on some paracontact manifolds

Abstract

The object of the present paper is to study the symmetric and skew-symmetric properties of a second order parallel tensor on paracontact metric (k, µ)-spaces and almost β-para-Kenmotsu (k, µ)-spaces. In this paper, we prove that if there exists a second order symmetric parallel tensor on a paracontact metric (k, µ)-space M, then either M is locally isometric to a product of a flat n + 1-dimensional manifold and an n-dimensional manifold of constant sectional curvature −4, or the second order parallel tensor is a constant multiple of the associated metric tensor g of M²ⁿ⁺¹. If there is a second order parallel tensor on an almost β-para-Kenmotsu (k, µ)-space with k ≠ 0, then it is a constant multiple of the associated metric tensor g of M²ⁿ⁺¹.

Mathematics Subject Classification (2010):

• 53C15
• 53C25
• 53D10
• 53D15

Key words:

• Second order parallel tensor
• paracontact metric (k, µ)-space
• almost β-para-Kenmotsu (k, µ)-space