Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes.

*(English)*Zbl 0661.53027Let M be an n-dimensional Riemannian manifold with tangent space \(T_ pM\) at the point \(p\in M\). An element \(z\in T_ pM\otimes {\mathbb{C}},\) the complexified tangent space, is said to be isotropic if \((z,z)=0\). A complex linear subspace \(V\subset T_ pM\otimes {\mathbb{C}},\) is totally isotropic if \(z\in V\) then \((z,z)=0\). By definition, the curvature of a Riemannian manifold M is positive on totally isotropic two-planes if the complex sectional curvature \(K(\sigma)>0\) whenever \(\sigma \subseteq T_ pM\otimes {\mathbb{C}}\) is a totally isotropic two-plane situated at any point \(p\in M\). The authors prove the following main theorem: Let M be a compact simply connected n-dimensional Riemannian manifold which has positive curvature on totally isotropic two-planes where \(n\geq 4\). Then M is homeomorphic to a sphere. This theorem was announced in [the second author, Bull. Am. Math. Soc., New Ser. 14, 279-282 (1986; Zbl 0589.53048)].

Let \({\mathcal R}: \Lambda^ 2T_ pM\to \Lambda^ 2T_ pM\) be the curvature operator at p. \({\mathcal R}\) is said to be (k,1)-positive if \(\{\sum^{k}_{i=1}<{\mathcal R}(\omega_ i),\omega_ i>\}>0\) for any orthonormal set \(\{\omega_ 1,...,\omega_ k\}\) of k elements from \(\Lambda^ 2T_ pM\), each of which has rank \(\leq 1\). The following corollary is proved: Let M be a compact simply connected n-dimensional Riemannian manifold, where \(n\geq 2\). If the curvature of M satisfies one of the following conditions, then M is homeomorphic to a sphere: (i) M has positive curvature operator, (ii) M has (2,2)-positive curvature operators, or (iii) M has strictly pointwise (1/4)-pinched sectional curvatures. Also a theorem regarding the structure of stable minimal two- spheres in Riemannian manifolds with nonnegative curvature is proved.

As a corollary it is proved that an odd-dimensional compact Riemannian manifold whose sectional curvatures are nonstrictly pointwise (1/4)- pinched must have vanishing second homotopy group. Finally it is remarked that if M satisfies all the hypotheses of the main theorem except for being simply connected, it is not necessarily true that the universal cover of M is a sphere, but still \(\pi_ k(M)=0\) for \(2\leq k\leq n/2\) can be concluded.

Let \({\mathcal R}: \Lambda^ 2T_ pM\to \Lambda^ 2T_ pM\) be the curvature operator at p. \({\mathcal R}\) is said to be (k,1)-positive if \(\{\sum^{k}_{i=1}<{\mathcal R}(\omega_ i),\omega_ i>\}>0\) for any orthonormal set \(\{\omega_ 1,...,\omega_ k\}\) of k elements from \(\Lambda^ 2T_ pM\), each of which has rank \(\leq 1\). The following corollary is proved: Let M be a compact simply connected n-dimensional Riemannian manifold, where \(n\geq 2\). If the curvature of M satisfies one of the following conditions, then M is homeomorphic to a sphere: (i) M has positive curvature operator, (ii) M has (2,2)-positive curvature operators, or (iii) M has strictly pointwise (1/4)-pinched sectional curvatures. Also a theorem regarding the structure of stable minimal two- spheres in Riemannian manifolds with nonnegative curvature is proved.

As a corollary it is proved that an odd-dimensional compact Riemannian manifold whose sectional curvatures are nonstrictly pointwise (1/4)- pinched must have vanishing second homotopy group. Finally it is remarked that if M satisfies all the hypotheses of the main theorem except for being simply connected, it is not necessarily true that the universal cover of M is a sphere, but still \(\pi_ k(M)=0\) for \(2\leq k\leq n/2\) can be concluded.

Reviewer: C.S.Houh

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |