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GEOMETRIC STRUCTURES OF VECTORIAL TYPE I. AGRICOLA AND T. FRIEDRICH
 

Summary: GEOMETRIC STRUCTURES OF VECTORIAL TYPE
I. AGRICOLA AND T. FRIEDRICH
Abstract. We study geometric structures of W4 ­type in the sense of A. Gray on
a Riemannian manifold. If the structure group G # SO(n) preserves a spinor or a
non­degenerate di#erential form, its intrinsic torsion # is a closed 1­form (Proposi­
tion 2.1 and Theorem 2.1). Using a G­invariant spinor we prove a splitting theorem
(Proposition 2.2). The latter result generalizes and unifies a recent result obtained
in [15], where this splitting has been proved in dimensions n = 7, 8 only. Finally
we investigate geometric structures of vectorial type and admitting a characteristic
connection # c . An interesting class of geometric structures generalizing Hopf struc­
tures are those with a # c ­parallel intrinsic torsion #. In this case, # induces a Killing
vector field (Proposition 4.1) and for some special structure groups it is even parallel.
1. Adapted connections of a geometric structure of vectorial type
Fix a subgroup G # SO(n) of the special orthogonal group and decompose the Lie
algebra so(n) = g # m into the Lie algebra g of G and its orthogonal complement m.
The di#erent geometric types of G­structures on a Riemannian manifold correspond to
the irreducible G­components of the representation R
n# m. Indeed, consider an oriented
Riemannian manifold (M n , g) and denote its Riemannian frame bundle by F(M n ). It
is a principal SO(n)­bundle over M n . A G­structure is a reduction R # F(M n ) of the

  

Source: Agricola, Ilka - Institut für Mathematik, Humboldt-Universität zu Berlin

 

Collections: Mathematics