The critical point equation and contact geometry

Abstract

In this paper, we consider the CPE conjecture in the frame-work of K-contact and (κ, µ)-contact manifolds. First, we prove that if a complete K-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere S 2n+1 . Next, we prove that if a non-Sasakian (κ, µ)-contact metric satisfies the CPE, then M 3 is flat and for n > 1, M 2n+1 is locally isometric to E n+1 × S n (4).Mathematics Subject Classification 2010: 53C25, 53C20, 53C15

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

“…In this section, we recall some basic definitions and facts on paracontact metric manifolds which we will use later. For more details and some examples, we refer to [15][16][17][18][19][20][21][22][23][24][25][26].…”

Certain Paracontact Metrics Satisfying the Critical Point Equation

“…Recently, Nato [13] obtained a necessary and sufficient condition on the norm of the gradient of the potential function for a CPE metric to be Einstein. Recently, the author considered the CPE on contact metric manifolds (see [7], [16]) and proved that the CPE conjecture is true for K-contact manifold. However, the CPE has not yet been considered on pseudo-Riemannian manifolds, for instance, paracontact metric manifolds.…”

Critical point equation on almost f-cosymplectic manifolds

“…It is very interesting to consider the CPE on odd-dimensional Riemannian manifolds. In this direction, Ghosh and Patra considered the K -contact metrics that satisfy the CPE [5], and proved that the CPE conjecture is true for this class of metric. Patra et al in [6], and De and Mandal in [7] independently considered an almost Kenmotsu manifold with CPE.…”