Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 471–481 | Cite as

On Wilking’s criterion for the Ricci flow

  • H. A. Gururaja
  • Soma Maity
  • Harish SeshadriEmail author


Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators \(C(S)\), which are nonnegative in a suitable sense, to every \(Ad_{SO(n,\mathbb{C })}\) invariant subset \(S \subset \mathbf{so}(n,\mathbb{C })\). In this article we show that if \(S\) is an \(Ad_{SO(n,\mathbb{C })}\) invariant subset of \(\mathbf{so}(n,\mathbb{C })\) such that \(S\cup \{0\}\) is closed and \(C_+(S)\subset C(S)\) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in \(C_+(S)\) also admits a metric with curvature operator in \(C_+(S)\) (b) The normalized Ricci flow on any compact Riemannian manifold \(M\) with curvature operator in \(C_+(S)\) converges to a metric of constant positive sectional curvature. We also point out that if \(S\) is an arbitrary \(Ad_{SO(n,\mathbb{C })}\) subset, then \(C(S)\) is contained in the cone of curvature operators with nonnegative isotropic curvature.


Sectional Curvature Curvature Operator Compact Riemannian Manifold Invariant Subset Ricci Flow 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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