Integral formulas in Riemannian geometry

In the present paper, we initiate the study of $$*$$
 -
 $$\eta $$
 -Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a $$*$$
 -
 $$\eta $$
 -Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $$*$$
 -
 $$\eta $$
 -Ricci soliton. Next, we deliberate $$*$$
 -
 $$\eta $$
 -Ricci soliton admitting $$(\kappa ,\mu )^\prime $$
 -almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$
 . Finally we present some examples to decorate the existence of $$*$$
 -
 $$\eta $$
 -Ricci soliton, gradient almost $$*$$
 -
 $$\eta $$
 -Ricci soliton on Kenmotsu manifold.

$$*$$ ∗ - $$\eta $$ η -Ricci soliton and contact geometry

In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$
 -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$
 -Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$
 -almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$
 -almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.

Certain types of metrics on almost coKähler manifolds

<jats:p>In this paper, we study α-cosymplectic manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> admitting <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Ricci tensor. First, it is shown that a <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Ricci semisymmetric manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> is <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Ricci flat and a <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>ϕ</mi>
                     </math>
                  </jats:inline-formula>-conformally flat manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> is an <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
                        <mi>η</mi>
                     </math>
                  </jats:inline-formula>-Einstein manifold. Furthermore, the <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Weyl curvature tensor <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
                        <msup>
                           <mrow>
                              <mi mathvariant="script">W</mi>
                           </mrow>
                           <mi>∗</mi>
                        </msup>
                     </math>
                  </jats:inline-formula> on <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> has been considered. Particularly, we show that a manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> with vanishing <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Weyl curvature tensor is a weak <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16">
                        <mi>ϕ</mi>
                     </math>
                  </jats:inline-formula>-Einstein and a manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> fulfilling the condition <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18">
                        <mi>R</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <msub>
                                 <mrow>
                                    <mi>E</mi>
                                 </mrow>
                                 <mrow>
                                    <mn>1</mn>
                                 </mrow>
                              </msub>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>E</mi>
                                 </mrow>
                                 <mrow>
                                    <mn>2</mn>
                                 </mrow>
                              </msub>
                           </mrow>
                        </mfenced>
                        <mo>⋅</mo>
                        <msup>
                           <mrow>
                              <mi mathvariant="script">W</mi>
                           </mrow>
                           <mi>∗</mi>
                        </msup>
                        <mo>=</mo>
                        <mn>0</mn>
                     </math>
                  </jats:inline-formula> is <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M19">
                        <mi>η</mi>
                     </math>
                  </jats:inline-formula>-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M20">
                        <mi>M</mi>
                     </math>
                  </jats:inline-formula> admitting <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M21">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M22">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Ricci soliton and almost <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M23">
                        <mi>∗</mi>
                     </math>
                  </jats:inline-formula>-Ricci soliton are drawn.</jats:p>

/pdf/ricci-tensor-on-a-cosymplectic-manifolds-1lt78tq2.pdf

∗
 -Ricci Tensor on 
 α
 -Cosymplectic Manifolds

This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure (g,ω,X). First, we show that a complete SM equipped with an almost ∗-RS with ω≠ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat.

https://www.mdpi.com/2504-3110/7/2/156/pdf?version=1675584283

Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons

In this short note, we prove a non-existence result for $$*$$
 -Ricci solitons on non-cosymplectic $$(\kappa ,\mu )$$
 -almost cosymplectic manifolds.

Xinxin Dai

Papers

Non-existence of $$*$$ ∗ -Ricci solitons on $$(\kappa ,\mu )$$ ( κ , μ ) -almost cosymplectic manifolds