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We introduce a special vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> on a Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that the Lie derivative of the metric <i>g</i> with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> is equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mi>R</mi><mi>i</mi><mi>c</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi><mi>c</mi></mrow></semantics></math></inline-formula> is the Ricci tensor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is a smooth function on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>N</mi><mi>m</mi></msup></semantics></math></inline-formula>. We call this vector field a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Ricci vector field. We use the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Ricci vector field on a Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> and find two characterizations of the <i>m</i>-sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mi>m</mi></msup><mfenced open="(" close=")"><mi>α</mi></mfenced></mrow></semantics></math></inline-formula>. In the first result, we show that an <i>m</i>-dimensional compact and connected Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> with nonzero scalar curvature admits a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Ricci vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is a nonconstant function and the integral of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi><mi>c</mi><mfenced separators="" open="(" close=")"><mi>ω</mi><mo>,</mo><mi>ω</mi></mfenced></mrow></semantics></math></inline-formula> has a suitable lower bound that is necessary and sufficient for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to <i>m</i>-sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mi>m</mi></msup><mfenced open="(" close=")"><mi>α</mi></mfenced></mrow></semantics></math></inline-formula>. In the second result, we show that an <i>m</i>-dimensional complete and simply connected Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> of positive scalar curvature admits a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Ricci vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> has an appropriate upper bound, if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>m</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> is isometric to <i>m</i>-sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi><mi>m</mi></msup><mfenced open="(" close=")"><mi>α</mi></mfenced></mrow></semantics></math></inline-formula>.