Amalendu Ghosh

Conformally flat contact metric manifolds

Abstract. A couple of classes of conformally flat contact metric manifolds have been classified. Conformally flat contact manifolds have been characterized as hypersurfaces of 4-dimensional Kaehler Einstein (in particular, Calabi-Yau) manifolds.

SASAKIAN 3-MANIFOLD AS A RICCI SOLITON REPRESENTS THE HEISENBERG GROUP

We show that, if a 3-dimensional Sasakian metric is a non-trivial Ricci soliton, then it is expanding and homothetic to the standard Sasakian metric on the Heisenberg group nil3. We have also discussed properties of the Ricci soliton potential vector field that relate to the underlying contact structure.

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.

On contact strongly pseudo-convex integrableCR manifolds

It is shown that a locally symmetric contact strongly pseudo-convex integrableCR manifold of dimension greater than 3 and other than 7 is locally isometric to a unit sphere or the Riemannian product of an (n + 1)-dimensional Euclidean space and a sphere. A conformally flat contact strongly pseudo-convex integrableCR manifold is locally isometric to a unit sphere, provided the characteristic vector field is an eigenvector of the Ricci tensor at each point.

RICCI SOLITONS AND CONTACT METRIC MANIFOLDS

We study on a contact metric manifold M2n+1(φ, ξ, η, g) such that g is a Ricci soliton with potential vector field V collinear with ξ at each point under different curvature conditions: (i) M is of pointwise constant ξ -sectional curvature, (ii) M is conformally flat. 2000 Mathematics Subject Classification. 53C15, 53C25, 53D10

Sasakian metric as a Ricci soliton and related results

Abstract We prove the following results: (i) a Sasakian metric as a non-trivial Ricci soliton is null η -Einstein, and expanding. Such a characterization permits us to identify the Sasakian metric on the Heisenberg group H 2 n + 1 as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an η -Einstein contact metric manifold M has a vector field V leaving the structure tensor and the scalar curvature invariant, then either V is an infinitesimal automorphism, or M is D -homothetically fixed K -contact.

Perfect fluid space-times whose energy-momentum tensor is conformal Killing

We show that the energy-momentum tensor T of an expanding perfect fluid space-time (M,g) is a nontrivial conformal Killing tensor if and only if M is shear-free, vorticity-free, and satisfies certain energy and force equations. In particular, if T is conformal Killing, we show that the perfect fluid is stationary and its energy-density and pressure are constant.

The critical point equation and contact geometry

AbstractIn this paper, we consider the CPE conjecture in the frame-work of

The k-almost Ricci solitons and contact geometry

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