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We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> with potential function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> on a Lorentzian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, dim<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>></mo><mn>5</mn></mrow></semantics></math></inline-formula>, which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> that admits a time-like special torse-forming vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>, there is a function <i>f</i> called the associated function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>. It is shown that if a connected Lorentzian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, dim<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>></mo><mn>4</mn></mrow></semantics></math></inline-formula>, admits a time-like special torse-forming vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> with associated function <i>f</i> nowhere zero and satisfies the Fischer–Marsden equation, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a quasi-Einstein manifold.