| Summary: | We aim to examine the influence of the existence of a nonzero eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of the de-Rham operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> on a <i>k</i>-dimensional Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>. If the vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, there are two operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>ζ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ψ</mi><mi>ζ</mi></msub></semantics></math></inline-formula> associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>, called the basic operator and the associated operator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>, respectively. We show that the existence of an eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> on a compact manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that the integral of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ric</mi><mo>(</mo><mi>ζ</mi><mo>,</mo><mi>ζ</mi><mo>)</mo></mrow></semantics></math></inline-formula> admits a certain lower bound, forces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to a <i>k</i>-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> on a connected and complete Riemannian space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>div</mi><mfenced open="(" close=")"><mi>ζ</mi></mfenced><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> and annihilating the associated operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ψ</mi><mi>ζ</mi></msub></semantics></math></inline-formula>, forces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to the <i>k</i>-dimensional Euclidean space, provided that the squared length of the covariant derivative of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> possesses a certain lower bound.
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