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Stable forms and special metrics. (English) Zbl 1004.53034
Fernández, Marisa (ed.) et al., Global differential geometry: the mathematical legacy of Alfred Gray. Proceedings of the international congress on differential geometry held in memory of Professor Alfred Gray, Bilbao, Spain, September 18-23, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 288, 70-89 (2001).
The author shows how certain diffeomorphism-invariant functionals in dimensions 6, 7, and 8 generate in a natural way special geometric structures in these dimensions, e.g. metrics of holonomy \(G_2\) and Spin(7), metrics with weak holonomy \(SU(3)\) and \(G_2\), and a new example in dimension 8. Contents include: an introduction, the linear algebra of stable forms, critical points, eight-manifolds with \(PSU(3)\) structure, constrained critical points, evolution equations, examples, and an appendix on the definition of volumes.
For the entire collection see [Zbl 0980.00033].

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C29 Issues of holonomy in differential geometry
58E11 Critical metrics
58A10 Differential forms in global analysis
Full Text: arXiv
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