Geometric classifications of <i>k</i>-almost Ricci solitons admitting paracontact metrices
Yanlin Li, Dhriti Sundar Patra, Nadia Alluhaibi, Fatemah Mofarreh, Akram Ali
Abstract
Abstract The prime objective of the approach is to give geometric classifications of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci solitons associated with paracontact manifolds. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo>,</m:mo> <m:mi>η</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {M}^{2n+1}\left(\varphi ,\xi ,\eta ,g) be a paracontact metric manifold, and if a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K -paracontact metric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g represents a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(g,V,k,\lambda ) and the potential vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V is Jacobi field along the Reeb vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ξ</m:mi> </m:math> \xi , then either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:math> k=\lambda -2n , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -Ricci soliton. Next, we consider <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K -paracontact manifold as a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton with the potential vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V is infinitesimal paracontact transformation or collinear with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ξ</m:mi> </m:math> \xi . We have proved that if a paracontact metric as a <jats:inline-graphic xmlns:xli