Fix a smooth closed manifold M. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2954_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}_M$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>M</mi> </msub> </math></EquationSource> </InlineEquation> denote the space of all pairs (g, L) such that g is a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2954_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> Riemannian metric on M and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2954_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$L\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is not the length of any closed g-geodesics. A locally constant geodesic count function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2954_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi _M:\mathcal {R}_M\rightarrow \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mi>M</mi> </msub> <mo>:</mo> <msub> <mi mathvariant="script">R</mi> <mi>M</mi> </msub> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> is constructed. For this purpose, the weight of compact open subsets of the space of closed g-geodesics is defined and investigated for an arbitrary Riemannian metric g.

Counting closed geodesics on Riemannian manifolds

期刊论文
发表日期：

2025年3月5日

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Eaman Eftekhary

摘要

Fix a smooth closed manifold M. Let $\mathcal {R}_M$ $R_{M}$ denote the space of all pairs (g, L) such that g is a $C^3$ $C^{3}$ Riemannian metric on M and $L\in \mathbb {R}$ $L \in R$ is not the length of any closed g-geodesics. A locally constant geodesic count function $\pi _M:\mathcal {R}_M\rightarrow \mathbb {Z}$ $π_{M} : R_{M} \to Z$ is constructed. For this purpose, the weight of compact open subsets of the space of closed g-geodesics is defined and investigated for an arbitrary Riemannian metric g.