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<p>We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann–Sebastian–Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed 2<i>n</i>-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\widehat{A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation>-genus, extending recent work of Chen–Ge–Han from almost nonnegative Ricci curvature to the curvature operator setting.</p>

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Euler characteristic of closed manifolds with almost nonnegative curvature operator

  • 期刊论文

  • 发表日期：

    2026年6月23日

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  • Jingbin Cai

摘要

We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann–Sebastian–Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed 2n-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the \(\widehat{A}\) A ^ -genus, extending recent work of Chen–Ge–Han from almost nonnegative Ricci curvature to the curvature operator setting.