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- W2162712890 abstract "For Riemannian manifolds<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>M</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>N</mml:mi></mml:math>, admitting a submersion<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>ϕ</mml:mi></mml:math>with compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>M</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>N</mml:mi></mml:math>, we determine conditions under which a harmonic function on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mrow><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>V</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>⊂</mml:mo><mml:mi>M</mml:mi></mml:mrow></mml:math>projects down, via its horizontal component, to a harmonic function on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>V</mml:mi><mml:mo>⊂</mml:mo><mml:mi>N</mml:mi></mml:math>." @default.
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- W2162712890 date "2012-01-01" @default.
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- W2162712890 title "Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions" @default.
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- W2162712890 doi "https://doi.org/10.1155/2012/315757" @default.
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