| Summary: | In this article, it has been observed that a unit Killing vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> on an <i>n</i>-dimensional Riemannian 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>, influences its algebra of smooth functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>C</mi><mo>∞</mo></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. For instance, if <i>h</i> is an eigenfunction of the Laplace operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> with eigenvalue <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>(</mo><mi>h</mi><mo>)</mo></mrow></semantics></math></inline-formula> is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">H</mi><mi>h</mi></msup><mrow><mo>(</mo><mi>ξ</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a smooth function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>∈</mo><msup><mi>C</mi><mo>∞</mo></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> defines a self adjoint operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⊡</mo><mi>ξ</mi></msub></semantics></math></inline-formula> and has properties similar to most of properties of the Laplace operator on a compact Riemannian 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>. We study several properties of functions associated to the unit Killing vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>. Finally, we find characterizations of the odd dimensional sphere using properties of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⊡</mo><mi>ξ</mi></msub></semantics></math></inline-formula> and the nontrivial solution of Fischer–Marsden differential equation, respectively.
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