Manifolds with positive curvature operators are space forms
type: Item
instance of: scholarly article
title: Manifolds with positive curvature operators are space forms
zbMATH Open document ID: 1185.53073
author: Christoph Böhm
author: Burkhard Wilking
published in: Annals of Mathematics. Second Series
publication date: 2009-07-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0606187
full work available at URL: p10.xhtml
review text: For each point \(p\) of a Riemannian manifold \(M\) the curvature tensor defines a self adjoint curvature operator \(R_{p} : \Lambda^{2}(T_{p}M) \rightarrow \Lambda^{2}(T_{p}M)\) such that \(\langle R_{p}(v \wedge w), x \wedge y \rangle = \langle R(v,w) x, y \rangle\) for \(x,y,z,w \in T_{p}M\). Here \(\Lambda^{2}(T_{p}M)\) is equipped with the inner product induced from the inner product on \(T_{p}M\). The curvature operator is said to be 2-\textit{positive} if the sum of the two smallest eigenvalues is positive. In this article the authors use Hamilton's maximum principle and an injectivity radius estimate of Klingenberg to obtain the following Theorem: On a compact Riemannian manifold of dimension \(n \geq 3\) the normalized Ricci flow evolves a Riemannian metric with 2-positive curvature operator at every point to a limit metric with constant sectional curvature. This result, previously known in dimensions \(\leq 5\), carries over to orbifolds of dimension \(\geq 3\). Let \(O(n,\mathbb{R})\) denote the group of real orthogonal \(n\times n\) matrices, and let \(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) denote its Lie algebra of real \(n\times n\) skew symmetric matrices. To generalize the arguments of \textit{R. S. Hamilton} [J. Differ. Geom. 24, 153--179 (1986; Zbl 0628.53042)] the authors begin by considering the vector space \(S(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) of linear operators on \(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) that are self adjoint with respect to the inner product given by \(\langle A , B \rangle = - \frac{1}{2}~\text{trace}(AB)\) for A,B \(\in \mathfrak{s} \mathfrak{o} (n, \mathbb{R}\). For a point p \(\in\) M an orthonormal basis \(\{e_{1},\dots , e_{n} \}\) of \(T_{p}M\) defines a natural linear isometry \(\varphi : \Lambda^{2}(T_{p}M) \rightarrow \mathfrak{s} \mathfrak{o} (n, \mathbb{R})\), and the self adjoint curvature operator \(R_{p} : \Lambda^{2}(T_{p}M) \rightarrow \Lambda^{2}(T_{p}M)\) gives rise to an element \(\ove…
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 5578712
zbMATH Keywords: 2-positive curvature operator
zbMATH Keywords: normalized Ricci flow
zbMATH Keywords: Hamilton maximum principle
zbMATH Keywords: constant sectional curvature
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
OpenAlex ID: W2024908347
Wikidata QID: Q102218458
arXiv ID: math/0606187
DOI: 10.4007/ANNALS.2008.167.1079
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description: scientific article