$\eta $-Einstein contact metric manifolds with purely transversal Bach tensor

Remarkable Classes of Almost 3-Contact Metric Manifolds

Axioms ◽

10.3390/axioms10010008 ◽

2021 ◽

Vol 10 (1) ◽

pp. 8

Author(s):

Giulia Dileo

Keyword(s):

Geometric Properties ◽

New Class ◽

Contact Metric Manifolds ◽

Cosymplectic Manifolds ◽

Sasaki Manifolds

We introduce a new class of almost 3-contact metric manifolds, called 3-(0,δ)-Sasaki manifolds. We show fundamental geometric properties of these manifolds, analyzing analogies and differences with the known classes of 3-(α,δ)-Sasaki (α≠0) and 3-δ-cosymplectic manifolds.

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Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500397 ◽

2019 ◽

Vol 16 (03) ◽

pp. 1950039 ◽

Cited By ~ 8

Author(s):

V. Venkatesha ◽

Devaraja Mallesha Naik

Keyword(s):

Vector Field ◽

Scalar Curvature ◽

Constant Scalar Curvature ◽

Sasakian Manifold ◽

Contact Metric Manifold ◽

Reeb Vector Field ◽

3 Dimensional ◽

Flow Vector ◽

Contact Metric Manifolds ◽

Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

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BACH FLOWS OF PRODUCT MANIFOLDS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500399 ◽

2012 ◽

Vol 09 (05) ◽

pp. 1250039 ◽

Cited By ~ 2

Author(s):

SANJIT DAS ◽

SAYAN KAR

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Ricci Flow ◽

Flow Characteristics ◽

Geometric Flow ◽

Flow Equations ◽

First Order ◽

Bach Tensor ◽

Point Condition ◽

Future Work

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.

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