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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title = "Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits",

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.",

keywords = "Cohomogeneity one manifold, G holonomy, Noncompact Ricci-flat manifold",

author = "Hanci Chi",

note = "Publisher Copyright: {\textcopyright} 2019, Springer Nature B.V.",

year = "2019",

doi = "10.1007/s10455-019-09671-y",

language = "English",

volume = "56",

pages = "361--401",

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T1 - Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

AU - Chi, Hanci

PY - 2019

Y1 - 2019

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.

KW - Cohomogeneity one manifold

KW - G holonomy

KW - Noncompact Ricci-flat manifold

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DO - 10.1007/s10455-019-09671-y

M3 - Article

AN - SCOPUS:85068320513

SN - 0232-704X

VL - 56

SP - 361

EP - 401

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

IS - 2

ER -

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

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Keywords

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