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Positive Einstein metrics with $\mathbb {S}^{4m+3}$ as the principal orbit

Part of: Global differential geometry

Published online by Cambridge University Press:  08 April 2024

Hanci Chi Open the ORCID record for Hanci Chi [Opens in a new window]
Hanci Chi*
Affiliation:
Department of Foundational Mathematics, Xi'an Jiaotong–Liverpool University, Suzhou 215123, PR China hanci.chi@xjtlu.edu.cn
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Abstract

We prove that there exists at least one positive Einstein metric on $\mathbb {HP}^{m+1}\sharp \overline {\mathbb {HP}}^{m+1}$ for $m\geq ~2$. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on $\mathbb {HP}^{m+1}\sharp \overline {\mathbb {HP}}^{m+1}$. We also investigate the existence of cohomogeneity-one positive Einstein metrics on $\mathbb {S}^{4m+4}$ and prove the existence of a non-standard Einstein metric on $\mathbb {S}^8$.

Keywords

Einstein metriccohomogeneity-one metric

MSC classification

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 5 , May 2024 , pp. 1004 - 1040
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The author is supported by NSFC (No. 12071489, No. 12301078), the Foundation for Young Scholars of Jiangsu Province, China (BK-20220282), and XJTLU Research Development Funding (RDF-21-02-083).

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