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Flag manifolds with strongly positive curvature

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Abstract

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds \(W^6\), \(W^{12}\) and \(W^{24}\), which are respectively the manifolds of complete flags in \(\mathbb {C}^3\), \(\mathbb {H}^3\) and \(\mathbb {C}\mathrm {a}^3\). Together with our earlier work, this concludes the classification of simply-connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature.

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Notes

  1. Compare with the normal homogeneous case discussed in [4], Ex. 2.5], in which \(P={\text {Id}}\), \(B_+=0\) and \(B_-(\cdot ,\cdot )=[\cdot ,\cdot ]\). The curvature operator of \((\mathsf {G}/\mathsf {H},Q|_\mathfrak {m})\) is given by \(R_{\mathsf {G}/\mathsf {H}}=R_{\mathsf {G}}+3\alpha -3\mathfrak {b}(\alpha )\), see [4], (2.9)], where \(\alpha =A^*A\) is obtained from the \(A\)-tensor of the submersion \((\mathsf {G},Q)\rightarrow (\mathsf {G}/\mathsf {H},Q|_\mathfrak {m})\). Since \(R_{\mathsf {G}}=\alpha +\beta \), it follows that \(R_{\mathsf {G}/\mathsf {H}}=4R_{\mathsf {G}}-3\beta +3\mathfrak {b}(\beta )\), which is the expression given by (2.1).

  2. This implication can be proved independently of Proposition 5.2. Indeed, positive-definiteness of the second and third blocks \(\widehat{R}^2_{W^\bullet }(\vec {s}, \omega )\) and \(\widehat{R}^3_{W^\bullet }(\vec {s}, \omega )\) directly implies that , \(r=1,2,3\), cf. Remark 5.6; and simultaneous positive-definiteness of the first block \(\widehat{R}^1_{W^\bullet }(\vec {s}, \omega )\) is impossible if the \(s_r\) are all equal, since its determinant is negative.

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Acknowledgments

It is a pleasure to thank Karsten Grove and Wolfgang Ziller for valuable suggestions. We also acknowledge the hospitality of the Mathematisches Forschungsinstitut Oberwolfach, where many results in this paper were proved.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN, 46556-4618, USA

    Renato G. Bettiol

  2. Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    Ricardo A. E. Mendes

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  1. Renato G. Bettiol
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  2. Ricardo A. E. Mendes
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Correspondence to Ricardo A. E. Mendes.

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Renato G. Bettiol is partially supported by the NSF Grant DMS-1209387, USA. Ricardo A. E. Mendes is supported by SFB878 Groups, Geometry & Actions.

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Bettiol, R.G., Mendes, R.A.E. Flag manifolds with strongly positive curvature. Math. Z. 280, 1031–1046 (2015). https://doi.org/10.1007/s00209-015-1464-1

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