## Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes.(English)Zbl 0661.53027

Let 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.
Reviewer: C.S.Houh

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Zbl 0589.53048
Full Text: