Some geometric flows their solitons and associated partial differential equations

Abstract

A manifold of odd dimension is termed as almost contact mertic newlinemanifod (ACMM) if we find a linear transformation T , a vector newlinefield and#961;, a 1-form and#969; and a Riemannian metric l agreeing with the newlinefollowing equations: newlineT 2 X = and#8722;X + and#969;(X)and#961;, and#969;(and#961;) = 1. newline(1) newlineFor such manifolds, we also have the follwing : newlineT and#961; = 0, l(X, and#961;) = and#969;(X), and#969;(T X) = 0. (2) newlinel(T X, T Y ) = l(X, Y ) and#8722; and#969;(X)and#969;(Y ). (3) newlinel(T X, Y ) = and#8722;l(X, T Y ), g(T X, X) = 0. (4) newline(and#8711; X and#969;)Y = l(and#8711; X and#961;, Y ). (5) newlineAn ACMM is CMM when newlinedand#969;(X, Y ) = T (X, Y ) = l(X, T Y ). T is fundamental two form of newlinethe manifold. For aTDGSSF we know the following [1] newlineG(X, Y )Z = newline+ newline+ newlineand#8722; newline+ newlineh 1 {l(Y, Z)X and#8722; l(X, Z)Y } newlineh 2 {l(X, T Z)T Y and#8722; l(Y, T Z)T X newline2l(X, T Y )T Z} + h 3 {and#969;(X)and#969;(Z)Y newlineand#969;(Y )and#969;(Z)X newlinel(X, Z)and#969;(Y )and#961; and#8722; g(Y, Z)and#969;(X)and#961;}. newline1 newline(6) newline(7) newline(8)2 newlineRic(X, Y ) = (2h 1 + 3h 2 and#8722; h 3 )l(X, Y ) newlineand#8722; (3h 2 + h 3 )and#969;(X)and#969;(Y ). newliner = 6h 1 + 6h 2 and#8722; 4h 3 . newlineIf the space form has QSM [11] then newlineand#8711; X and#961; = and#8722;and#946;T X. newline(9) newline(10) newline(11) newline(and#8711; X and#969;)Y = l(and#8711; X and#961;, Y ) = and#8722;and#946;l(T X, Y ). newline(12) newline(and#8711; X and#969;)and#961; = and#8722;and#946;and#969;(T X) = 0. newline(13) newlineA ACMM is called (and#945;, and#946;)trans-Sasakian manifold (and#945;, and#946;TSM) if newlinefor the linear connection and#8711; newlineand#8711; X and#961; = and#8722;and#945;X + and#946;(X and#8722; and#969;(X)and#961;). newline(14) newlineIn 1990, Blair and Oubina [16] constructed such a metric manifold. newlineIn 1995, (and#954;, and#956;)-contact metric manifold was constructed by Blair newlineand colaborators [14]. A special kind of such manifold is N (and#954;)- newlinemanifold with and#956; = 0. Zamkovoy [78] framed the para analogue of newline(and#954;, and#956;)CMM newline