Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

Hanci Chi

doi:10.1007/s10455-019-09671-y

Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

Hanci Chi^*

^*Corresponding author for this work

McMaster University

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over CP², HP² and OP² with respective principal orbits G / K the Wallach spaces SU(3) / T², Sp(3) / (Sp(1)Sp(1)Sp(1)) and F₄/ Spin (8). Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete G₂ metric discovered in Bryant and Salamon (Duke Math J 58(3):829–850, 1989) and Gibbons et al. (Commun Math Phys 127(3):529–553, 1990). It lies in the interior of the 1-parameter family on ⋀-2CP2. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on G / K.

Original language	English
Pages (from-to)	361-401
Number of pages	41
Journal	Annals of Global Analysis and Geometry
Volume	56
Issue number	2
DOIs	https://doi.org/10.1007/s10455-019-09671-y
Publication status	Published - 2019
Externally published	Yes

Keywords

Cohomogeneity one manifold
G holonomy
Noncompact Ricci-flat manifold

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10.1007/s10455-019-09671-y

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abstract = "We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over CP2, HP2 and OP2 with respective principal orbits G / K the Wallach spaces SU(3) / T2, Sp(3) / (Sp(1)Sp(1)Sp(1)) and F4/ Spin (8). Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete G2 metric discovered in Bryant and Salamon (Duke Math J 58(3):829–850, 1989) and Gibbons et al. (Commun Math Phys 127(3):529–553, 1990). It lies in the interior of the 1-parameter family on ⋀-2CP2. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on G / K.",

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AU - Chi, Hanci

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N2 - We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over CP2, HP2 and OP2 with respective principal orbits G / K the Wallach spaces SU(3) / T2, Sp(3) / (Sp(1)Sp(1)Sp(1)) and F4/ Spin (8). Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete G2 metric discovered in Bryant and Salamon (Duke Math J 58(3):829–850, 1989) and Gibbons et al. (Commun Math Phys 127(3):529–553, 1990). It lies in the interior of the 1-parameter family on ⋀-2CP2. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on G / K.

AB - We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over CP2, HP2 and OP2 with respective principal orbits G / K the Wallach spaces SU(3) / T2, Sp(3) / (Sp(1)Sp(1)Sp(1)) and F4/ Spin (8). Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete G2 metric discovered in Bryant and Salamon (Duke Math J 58(3):829–850, 1989) and Gibbons et al. (Commun Math Phys 127(3):529–553, 1990). It lies in the interior of the 1-parameter family on ⋀-2CP2. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on G / K.

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KW - G holonomy

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JO - Annals of Global Analysis and Geometry

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Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

Abstract

Keywords

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