Isometries on almost Ricci–Yamabe solitons

Abstract

Abstract The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere $$S^n(r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h . Next, we studied complete gradient almost Ricci–Yamabe soliton with $$\alpha \ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space $$E^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> or Euclidean sphere $$S^n.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

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