The Critical Point Equation And Contact Geometry

TL;DR

This paper investigates the Critical Point Equation (CPE) conjecture within contact geometry, establishing conditions under which certain contact manifolds are Einstein, flat, or locally isometric to known geometric spaces.

Contribution

It proves that complete K-contact manifolds satisfying CPE are Einstein and isometric to spheres, and characterizes non-Sasakian (1,2)-contact manifolds satisfying CPE.

Findings

K-contact manifolds satisfying CPE are Einstein and isometric to spheres.

Non-Sasakian (1,2)-contact manifolds satisfying CPE are flat or locally isometric to Euclidean times sphere.

Provides geometric classification results related to the CPE conjecture in contact geometry.

Abstract

In this paper, we consider the CPE conjecture in the frame-work of $K$-contact and $(\kappa, \mu)$-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 $ (\kappa, \mu) $-contact metric satisfies the CPE, then $ M^{3} $ is flat and for $ n > 1 $, $ M^{2n+1} $ is locally isometric to $ E^{n+1}\times S^{n}(4)$.