## Osaka Journal of Mathematics

### On positive quaternionic Kähler manifolds with $b_{4} = 1$

#### Abstract

Let $M$ be a positive quaternionic Kähler manifold of dimension $4m$. In earlier papers, Fang and the first author showed that if the symmetry rank is greater than or equal to $[m/2]+3$, then $M$ is isometric to $\mathbf{HP}^{m}$ or $\mathit{Gr}_{2}(\mathbf{C}^{m+2})$. The goal of this paper is to give a more refined classification result for positive quaternionic Kähler manifolds (in particular, of relatively low dimension or with even $m$) whose fourth Betti number equals one. To be precise, we show in this paper that if the symmetry rank of $M$ with $b_{4}(M)=1$ is no less than $[m/2]+2$ for $m \ge 5$, then $M$ is isometric to $\mathbf{HP}^{m}$.

#### Article information

Source
Osaka J. Math., Volume 49, Number 3 (2012), 551-562.

Dates
First available in Project Euclid: 15 October 2012

https://projecteuclid.org/euclid.ojm/1350306587

Mathematical Reviews number (MathSciNet)
MR2993057

Zentralblatt MATH identifier
1268.53059

#### Citation

Kim, Jin Hong; Lee, Hee Kwon. On positive quaternionic Kähler manifolds with $b_{4} = 1$. Osaka J. Math. 49 (2012), no. 3, 551--562. https://projecteuclid.org/euclid.ojm/1350306587

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