Killing forms on G2 and Spin7 manifolds
Killing forms on G2 and Spin7 manifolds
Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that on a compact manifold with holonomy G2 or Spin7 any Killing form has to be parallel. The main tool is a universal Weitzenboeck formula. We show how such a formula can be obtained for any given holonomy group and any representation defining a vector bundle.
15 pages
Issues of holonomy in differential geometry, Mathematics - Differential Geometry, Spectral problems; spectral geometry; scattering theory on manifolds, Spin and Spin\({}^c\) geometry, Global differential geometry of Hermitian and Kählerian manifolds, 53C55, Differential Geometry (math.DG), FOS: Mathematics, special holonomy manifolds, Killing forms, \(G\)-structures
Issues of holonomy in differential geometry, Mathematics - Differential Geometry, Spectral problems; spectral geometry; scattering theory on manifolds, Spin and Spin\({}^c\) geometry, Global differential geometry of Hermitian and Kählerian manifolds, 53C55, Differential Geometry (math.DG), FOS: Mathematics, special holonomy manifolds, Killing forms, \(G\)-structures
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