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We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field <inline-formula><math display="inline"><semantics><mi mathvariant="bold">w</mi></semantics></math></inline-formula> on a connected Riemannian manifold <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> we show that for each non-constant smooth function <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi><mo>∞</mo></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> there exists a non-zero vector field <inline-formula><math display="inline"><semantics><msup><mi mathvariant="bold">w</mi><mi>f</mi></msup></semantics></math></inline-formula> associated with <i>f</i>. In particular, we show that for an eigenfunction <i>f</i> of the Laplace operator on an <i>n</i>-dimensional compact Riemannian manifold <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> with an appropriate lower bound on the integral of the Ricci curvature <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><msup><mi mathvariant="bold">w</mi><mi>f</mi></msup><mo>,</mo><msup><mi mathvariant="bold">w</mi><mi>f</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> gives a characterization of the odd-dimensional unit sphere <inline-formula><math display="inline"><semantics><msup><mi mathvariant="bold">S</mi><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>. Also, we show on an <i>n</i>-dimensional compact Riemannian manifold <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> that if there exists a positive constant <i>c</i> and non-constant smooth function <i>f</i> that is eigenfunction of the Laplace operator with eigenvalue <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mi>c</mi></mrow></semantics></math></inline-formula> and the unit Killing vector field <inline-formula><math display="inline"><semantics><mi mathvariant="bold">w</mi></semantics></math></inline-formula> satisfying <inline-formula><math display="inline"><semantics><mrow><msup><mfenced separators="" open="∥" close="∥"><mo>∇</mo><mi mathvariant="bold">w</mi></mfenced><mn>2</mn></msup><mo>≤</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>c</mi></mrow></semantics></math></inline-formula> and Ricci curvature in the direction of the vector field <inline-formula><math display="inline"><semantics><mrow><mo>∇</mo><mi>f</mi><mo>−</mo><mi mathvariant="bold">w</mi></mrow></semantics></math></inline-formula> is bounded below by <inline-formula><math display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mi>n</mi><mo>−</mo><mn>1</mn></mfenced><mi>c</mi></mrow></semantics></math></inline-formula> is necessary and sufficient for <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to the sphere <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="bold">S</mi><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Finally, we show that the presence of a unit Killing vector field <inline-formula><math display="inline"><semantics><mi mathvariant="bold">w</mi></semantics></math></inline-formula> on an <i>n</i>-dimensional Riemannian manifold <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> with sectional curvatures of plane sections containing <inline-formula><math display="inline"><semantics><mi mathvariant="bold">w</mi></semantics></math></inline-formula> equal to 1 forces dimension <i>n</i> to be odd and that the Riemannian manifold <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> becomes a K-contact manifold. We also show that if in addition <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> is complete and the Ricci operator satisfies Codazzi-type equation, then <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an Einstein Sasakian manifold.