Find in Library

We continue studying the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-Ricci vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</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>, which is not necessarily closed. A Riemannian manifold with Ricci operator <i>T</i>, a Coddazi-type tensor, is called a <i>T</i>-<i>manifold</i>. In the first result of this paper, we show that a complete and simply connected <i>T</i>-<i>manifold</i><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>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, of positive scalar curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, admits a closed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-Ricci vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> such that the vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">u</mi><mo>−</mo><mo>∇</mo><mi>σ</mi></mrow></semantics></math></inline-formula> is an eigenvector of <i>T</i> with eigenvalue <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><msup><mi>m</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>, if and only if it is isometric to the <i>m</i>-sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mi>α</mi></mrow><mi>m</mi></msubsup></semantics></math></inline-formula>. In the second result, we show that if a compact and connected <i>T</i>-<i>manifold</i><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>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula>, admits a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-Ricci vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature <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 mathvariant="bold">u</mi><mo>,</mo><mi mathvariant="bold">u</mi></mfenced></mrow></semantics></math></inline-formula> that has a suitable lower bound, then <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 the <i>m</i>-sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mi>α</mi></mrow><mi>m</mi></msubsup></semantics></math></inline-formula>, and the converse also holds. Finally, we show that a 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> admits a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-Ricci vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature <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 mathvariant="bold">u</mi><mo>,</mo><mi mathvariant="bold">u</mi></mfenced></mrow></semantics></math></inline-formula> has a lower bound depending on a positive constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, 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 the <i>m</i>-sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mrow><mi>α</mi></mrow><mi>m</mi></msubsup></semantics></math></inline-formula>.