Find in Library

This article generalizes some geometric structures on warped product manifolds equipped with a Poisson structure to doubly warped products of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. First, we introduce the notion of Poisson doubly warped product manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mrow></mrow><mi>f</mi></msub><mi>B</mi><msub><mo>×</mo><mi>b</mi></msub><mi>F</mi><mo>,</mo><mo>Π</mo><mo>=</mo><msup><mi>μ</mi><mi>v</mi></msup><msubsup><mo>Π</mo><mrow><mi>B</mi></mrow><mi>h</mi></msubsup><mo>+</mo><msup><mi>ν</mi><mi>h</mi></msup><msubsup><mo>Π</mo><mrow><mi>F</mi></mrow><mi>v</mi></msubsup><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and express the Levi-Civita contravariant connection, curvature and metacurvature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mrow></mrow><mi>f</mi></msub><mi>B</mi><msub><mo>×</mo><mi>b</mi></msub><mi>F</mi><mo>,</mo><mo>Π</mo><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in terms of Levi-Civita connections, curvatures and metacurvatures of components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msub><mo>Π</mo><mi>B</mi></msub><mo>,</mo><msub><mi>g</mi><mi>B</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><msub><mo>Π</mo><mi>F</mi></msub><mo>,</mo><msub><mi>g</mi><mi>F</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We also study compatibility conditions related to the Poisson structure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Π</mo></semantics></math></inline-formula> and the contravariant metric <i>g</i> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>f</mi></msub><mi>B</mi><msub><mo>×</mo><mi>b</mi></msub><mi>F</mi></mrow></semantics></math></inline-formula>, so that the compatibility conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msub><mo>Π</mo><mi>B</mi></msub><mo>,</mo><msub><mi>g</mi><mi>B</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><msub><mo>Π</mo><mi>F</mi></msub><mo>,</mo><msub><mi>g</mi><mi>F</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> remain consistent in the Poisson doubly warped product manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mrow></mrow><mi>f</mi></msub><mi>B</mi><msub><mo>×</mo><mi>b</mi></msub><mi>F</mi><mo>,</mo><mo>Π</mo><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>.