On positively curved manifolds with symmetry

Abstract

Let $M$ be a closed simply connected $n$-manifold of positive sectional curvature and maximal symmetry rank $-2$ equal to $\left[\frac{n-3}{2}\right]$ for $n ≥ 9$. In this thesis, we expect to give a homeomorphism classification of $M$ under some conditions. To be precise, assume that (1) n = even ; the dimension of the fixed point set of $T^{\left[\frac{n-3}{2}\right]}$ is not equal to 2 (2) n = odd ; if there exist non-isolated circle orbits with orbit type $H$, then the dimension of the fixed point set of H is not equal to 5. Then, it turns out that M should be homeomorphic to a sphere or a complex projective space. Main tools are results from the extremal problems, analysis of the fixed point components, and the calculation of their Euler characteristics.