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• W2030221916 abstract "In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=greater-than-or-equal-to 3> <mml:semantics> <mml:mrow> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) whose curvature operator is bounded and satisfies the pinching condition <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R greater-than-or-equal-to delta StartFraction trace left-parenthesis upper R right-parenthesis Over 2 n left-parenthesis n minus 1 right-parenthesis EndFraction normal upper I greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>tr</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>I</mml:mi> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>Rge delta frac {operatorname {tr}(R)}{2n(n-1)} mathrm {I}>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>delta >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces." @default.
• W2030221916 created "2016-06-24" @default.
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• W2030221916 date "2006-11-29" @default.
• W2030221916 modified "2023-10-18" @default.
• W2030221916 title "Complete manifolds with nonnegative curvature operator" @default.
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• W2030221916 doi "https://doi.org/10.1090/s0002-9939-06-08872-1" @default.
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