Tohoku Mathematical Journal

On the quaternionic manifolds whose twistor spaces are Fano manifolds

Radu Pantilie

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Let $M$ be a quaternionic manifold, $\dim M=4k$, whose twistor space is a Fano manifold. We prove the following:

  (a) $M$ admits a reduction to ${\rm Sp}(1)\times{\rm GL}(k,\mathbb{H})$ if and only if $M=\mathbb{H} P^k$,

  (b) either $b_2(M)=0$ or $M={\rm Gr}_2(k+2,\mathbb{C})$.

This generalizes results of S. Salamon and C. R. LeBrun, respectively, who obtained the same conclusions under the assumption that $M$ is a complete quaternionic-Kähler manifold with positive scalar curvature.

Article information

Tohoku Math. J. (2), Volume 67, Number 4 (2015), 507-511.

First available in Project Euclid: 22 December 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C28: Twistor methods [See also 32L25]
Secondary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

Quaternionic manifolds


Pantilie, Radu. On the quaternionic manifolds whose twistor spaces are Fano manifolds. Tohoku Math. J. (2) 67 (2015), no. 4, 507--511. doi:10.2748/tmj/1450798069.

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