Rigidity theorems of positively curved riemannian manifolds with symmetry

Abstract

In Riemannian geometry, classification of manifolds with positive scalar curvature is important. In this paper, manifolds are classified by using a group action: (1) $\It{nonnegatively curved}$ 4-manifolds (i.e., complete Riemannian 4-manifolds with everywhere nonnegative sectional curvature) with a finite group action, (2) $\It{positively curved n}$-manifolds (i.e., complete Riemannian n-manifolds with everywhere positive sectional curvature) with $\It{symmetry rank^1}[\frac{n-3}{2}]$ and (3) positive quaternionic K$\ddot{a}hler 4$\It{m}$-manifolds (i.e., Rimannian manifolds whose holonomy groups are contained in $\ItSp({m})Sp(1)$, for $\It{m} \ge 2$ and that have positive scalar curvature) with sym-rank $\It{M})$ \ge \frac{m}{2}+1$. Each case has a common point theoretically. For example, in each case, fixed point sets of action will be addressed (that is, the same approach will be used for each area). First, Case (1) will be addressed. Let $\It{M})$ be a closed, simply connected, nonnegatively curved 4-manifold. If $\It{M})$ admits $\It{S}^1$-action, $\It{M})$ is homeomorphic to $\It{S}^4$, $CP^2$, $\It{S}^2\times\It{S}^2$ and $CP^2# \plusmn CP^2$ (cf. [7]). In this paper, that will be extended. Assume that M admits an effective isometric $Z_m$ action for an odd integer $\It{m}\ge 41^8$ and for any $\It{g} \ne 1 \isin Z_m$-action (g) acts trivially on the homology of $\It{M}$. If ($\It{g}$) has at most one two-dimensional fixed point component, then it is first shown that $\It{M})$ is homeomorphic to $\It{S}^4$, $\It{#}^l_{i=1}\It{S}^2 \times \It{S}^2$, $\It{l}=1,2, or $\It{#}^k_{j=1}\pm \bf{CP}^2, $\It{k}=1,2,3,4,5. Moreover, the following is conjectured: if the fixed point set contains two 2-dimensional connected components, then $\It{M})$ is diffeomorphic to an $S^2$-bundle over $S^2$. Until now, High dimensional manifolds with positive sectional curvature (i.e., Case (2)) can be investigated similar to 4-dimensional manifolds. Homeomorphism class...