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<p>We prove that there exist <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2073_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textsf{SU}_{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="sans-serif">SU</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-invariant metrics on Aloff–Wallach spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2073_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(W^7_{k_1, k_2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>W</mi> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> <mn>7</mn> </msubsup> </math></EquationSource> </InlineEquation>, as well as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2073_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textsf{SU}_{5}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="sans-serif">SU</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation>-invariant metrics on the Berger space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2073_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(B^{13}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>B</mi> <mn>13</mn> </msup> </math></EquationSource> </InlineEquation>, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved 2-planes.</p>

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Positive Sectional Curvature is not Preserved under The Ricci Flow in Dimensions Seven and Thirteen

  • 期刊论文

  • 发表日期：

    2025年6月28日

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  • David González-Álvaro,
  • Masoumeh Zarei

摘要

We prove that there exist \(\textsf{SU}_{3}\) SU 3 -invariant metrics on Aloff–Wallach spaces \(W^7_{k_1, k_2}\) W k 1 , k 2 7 , as well as \(\textsf{SU}_{5}\) SU 5 -invariant metrics on the Berger space \(B^{13}\) B 13 , which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved 2-planes.