Twisted connected sums and special Riemannian holonomy. (English) Zbl 1043.53041

The author presents a new method for constructing Riemannian metrics with holonomy \(G_2\) on compact \(7\)-dimensional manifolds. Based on some work by Tian and Yau the author first constructs a class of noncompact asymptotically cylindrical Riemannian manifolds with holonomy \(SU(3)\). The Riemannian product of such a manifold and a circle \(S^1\) has again holonomy \(SU(3)\) and an end that is asymptotic to a half-cylinder. A cross-section of this half-cylinder is the Riemannian product of two circles and a complex surface of type K3 with a hyper-Kähler metric. By truncating the ends of two such asymptotically cylindrical \(7\)-manifolds, cutting off to the cylindrical metric and identifying the boundaries by an orientation-reversing isometry that interchanges the two \(S^1\) factors he obtains a closed compact Riemannian \(7\)-manifold. The twisting that is built into the isometry identifying the boundaries has the purpose to avoid an infinite fundamental group.
The construction is based on a gluing theorem for appropriate elliptic partial differential equations and yields compact \(7\)-manifolds with a \(G_2\)-structure. The author then discusses the topology of the resulting \(7\)-manifolds when one starts the construction with Fano manifolds, which leads to many new topological types of compact \(G_2\)-manifolds.


53C29 Issues of holonomy in differential geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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