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In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on <i>n</i>-sphere <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="bold">S</mi><mi>n</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="bold">E</mi><mi>n</mi></msup></semantics></math></inline-formula> whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="bold">E</mi><mi>n</mi></msup></semantics></math></inline-formula>, this type of torqued vector fields could not be extended globally to <inline-formula><math display="inline"><semantics><msup><mi mathvariant="bold">E</mi><mi>n</mi></msup></semantics></math></inline-formula>. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.