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• W1999278843 endingPage "1414" @default.
• W1999278843 startingPage "1405" @default.
• W1999278843 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=application/x-tex>M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a closed manifold which admits a foliation structure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper F> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of codimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>qgeq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a bundle-like metric <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g 0> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket g 0 right-bracket Subscript upper B> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msub> <mml:mo stretchy=false>]</mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>[g_0]_B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the space of bundle-like metrics which differ from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g 0> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>g_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only along the horizontal directions by a multiple of a positive basic function. Assume <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a transverse conformal vector field and the mean curvature of the leaves of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper M comma script upper F comma g 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(M,mathcal {F},g_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanishes. We show that the integral <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript upper M Endscripts upper Y left-parenthesis upper R Subscript g Sub Superscript upper T Subscript Superscript upper T Baseline right-parenthesis d mu Subscript g> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>int _MY(R^T_{g^T})dmu _g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is independent of the choice of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g element-of left-bracket g 0 right-bracket Subscript upper B> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msub> <mml:mo stretchy=false>]</mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>gin [g_0]_B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g Superscript upper T> <mml:semantics> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>T</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>g^T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the transverse metric induced by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=application/x-tex>g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R Superscript upper T> <mml:semantics> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>T</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>R^T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the transverse scalar curvature. Moreover if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q greater-than-or-equal-to 3> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>qgeq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we have <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript upper M Endscripts upper Y left-parenthesis upper R Subscript g Sub Superscript upper T Subscript Superscript upper T Baseline right-parenthesis d mu Subscript g Baseline equals 0> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>int _MY(R^T_{g^T})dmu _g=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g element-of left-bracket g 0 right-bracket Subscript upper B> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msub> <mml:mo stretchy=false>]</mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>gin [g_0]_B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. However there exist codimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> minimal Riemannian foliations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper M comma script upper F comma g right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(M,mathcal {F},g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and transverse conformal vector fields <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript upper M Endscripts upper Y left-parenthesis upper R Subscript g Sub Superscript upper T Subscript Superscript upper T Baseline right-parenthesis d mu Subscript g Baseline not-equals 0> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>int _MY(R^T_{g^T})dmu _gneq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Therefore, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript upper M Endscripts upper Y left-parenthesis upper R Subscript g Sub Superscript upper T Subscript Superscript upper T Baseline right-parenthesis d mu Subscript g> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>int _MY(R^T_{g^T})dmu _g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1999278843 created "2016-06-24" @default.
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• W1999278843 date "2012-09-07" @default.
• W1999278843 modified "2023-09-27" @default.
• W1999278843 title "A conformal integral invariant on Riemannian foliations" @default.
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