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We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the <i>m</i>-dimensional sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="bold">S</mi><mi>m</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of constant curvature <i>c</i>. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.