Deformations of G_2 and Spin(7) Structures on Manifolds

Details Deformations of G_2 and Spin(7) Structures on Manifolds Deformations of G_2 and Spin(7) Structures on Manifolds

2003-01-20

arxiv.org/abs/math/0301218v3

Canadian Journal of Mathematics 57, 1012-1055 (2005)

Mathematics

Differential Geometry

52 pages, no figures, part of the author's Ph.D. Thesis. For revised version: Many proofs algebraically simplified in the G_2

Scientific paper

We consider some infinitesmal and global deformations of G_2 structures on 7-manifolds. We discover a canonical way to deform a G_2 structure by a vector field in which the associated metric gets "twisted" in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field whose solution would yield a manifold with holonomy G_2. Similarly we consider analogous constructions for Spin(7) structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased non-linearity of the Spin(7) case.

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