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<p>We consider rigidity properties of compact symmetric spaces <i>X</i> with metric <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(g_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> of rank one. Suppose <i>g</i> is another Riemannian metric on <i>X</i> with sectional curvature <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\kappa \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> bounded by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(0 \le \kappa \le 1.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> If <i>g</i> equals <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(g_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> outside a convex proper subset of <i>X</i>,&#xa0; then <i>g</i> is isometric with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(g_0.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mn>0</mn> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We also exhibit examples of surfaces showing that the nonnegativity of the curvature is needed. Our main result complements earlier results on other symmetric spaces by Gromov and Schroeder–Ziller.</p>

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Rigidity of compact rank one symmetric spaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月28日

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  • Chris Connell,
  • Mitul Islam,
  • Thang Nguyen,
  • Ralf Spatzier

摘要

We consider rigidity properties of compact symmetric spaces X with metric \(g_0\) g 0 of rank one. Suppose g is another Riemannian metric on X with sectional curvature \(\kappa \) κ bounded by \(0 \le \kappa \le 1.\) 0 ≤ κ ≤ 1 . If g equals \(g_0\) g 0 outside a convex proper subset of X,  then g is isometric with \(g_0.\) g 0 . We also exhibit examples of surfaces showing that the nonnegativity of the curvature is needed. Our main result complements earlier results on other symmetric spaces by Gromov and Schroeder–Ziller.