Manifolds with positive curvature operators are space forms
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 \(\overline{R_{p}} = \varphi \circ R_{p} \circ \varphi^{-1}\) of \(S(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\). For g \(\in O(n, \mathbb{R})\) the orthonormal basis \(\{g(e_{1}), ... , g(e_{n}) \}\) of \(T_{p}M\) gives rise to the operator \(\overline{R_{p}}(g) = \varphi_{g} \circ R_{p} \circ \varphi_{g}^{-1} = g^{-1} \circ \overline{R_{p}} \circ g = g^{-1}(\overline{R_{p}})\). The operators \(g(\overline{R_{p}}), g \in O(n, \mathbb{R})\), lie in the \(O(n, \mathbb{R})\)-invariant subspace \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) of curvature operators in \(S(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) that is defined by the Bianchi identities for \(R_{p}\). Let \(ad : \Lambda^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R})) \rightarrow \mathfrak{s} \mathfrak{o} (n, \mathbb{R})\) be the linear map defined by \(ad(R \wedge S) = [R,S]\) for \(R,S \in \mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\), and for \(p \in M\) let \(R_{p}^{\#} = ad \circ (R_{p} \wedge R_{p}) \circ ad^{*} : \mathfrak{s} \mathfrak{o} (n, \mathbb{R})) \rightarrow \mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\), where \(ad^{*}\) denotes the metric adjoint of ad and \(R_{p}\) is identified with \(\overline{R_{p}}\). Call a subset \(C\) of \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))~ \mathit{invariant}\) if it is invariant under \(O(n, \mathbb{R})\) and also invariant under the ordinary differential equation \(\frac{dR}{dt} = R^{2} + R^{\#}\). Hamilton's maximum principle says that if \(C\) is a closed, convex invariant subset of \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) and if \(R_{p} \in C\) for all p \(\in\) M, then \(R_{p}(t) \in C\) for all p \(\in\) M and all t in the domain of the Ricci flow given by \(\frac{\partial g}{\partial t} = - 2 Ric(g)\). Here \(R_{p}(t)\) is the curvature operator, or more precisely, its orbit under \(O(n, \mathbb{R})\) defined by the metric g(t). Given an appropriate closed, convex subset \(C\) of \(S_{B}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) defined by explicit conditions on the curvature operators \(R_{p}\) the authors construct a family of closed, convex subsets \(C_{a,b}\) in \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\), where \(a \geq 0\), \(b \geq 0\) are explicitly defined and \(C_{0,0} = C\). This allows the authors to construct a continuous family of invariant cones in \(S_{B}^{2}(\mathfrak{s} \mathfrak{o} (n, \mathbb{R}))\) that join the invariant cone of 2-positive curvature operators to the invariant cone of positive multiples of the identity. The authors then generalize the arguments of Hamilton for 4-manifolds to obtain the theorem stated above.
scientific article