The purpose of this paper is to prove that if Y is a compact manifold, if Z is an Anosov vector field on Y, and if F is a flat vector bundle, there is a corresponding canonical nonzero section <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5400_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau _{\nu }\left( i_{Z}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>τ</mi> <mi>ν</mi> </msub> <mfenced close=")" open="("> <msub> <mi>i</mi> <mi>Z</mi> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of the determinant line <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5400_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu =\det H\left( Y,F\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>=</mo> <mo movablelimits="true">det</mo> <mi>H</mi> <mfenced close=")" open="("> <mi>Y</mi> <mo>,</mo> <mi>F</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. In families, this section is <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5400_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> with respect to the canonical smooth structure on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5400_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>. When F is flat on the total space of the corresponding fibration, our section is flat with respect to the Gauss-Manin connection on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5400_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>.

Anosov Vector Fields and Fried Sections

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2025年9月1日

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Jean-Michel Bismut,
Shu Shen

摘要

The purpose of this paper is to prove that if Y is a compact manifold, if Z is an Anosov vector field on Y, and if F is a flat vector bundle, there is a corresponding canonical nonzero section $\tau _{\nu }\left( i_{Z}\right) $ $τ_{ν} (i_{Z})$ of the determinant line $\nu =\det H\left( Y,F\right) $ $ν = det H (Y, F)$ . In families, this section is $C^{1}$ $C^{1}$ with respect to the canonical smooth structure on $\nu $ $ν$ . When F is flat on the total space of the corresponding fibration, our section is flat with respect to the Gauss-Manin connection on $\nu $ $ν$ .