Spin-harmonic structures and nilmanifolds

Abstract

We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim = 4, 5), SU(3) (dim = 6) and G 2 (dim = 7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.

$S^1$-quotient of $Spin(7)$-structures

“…We also give explicit expressions for the SU(4)-structure in terms of the data on the quotient manifold, see Theorem 3.6. In the locally conformally parallel situation, we show that M has vanishing Λ 3 27 torsion component and furthermore, if the Λ 3 1 torsion component is non-zero then N = M × S 1 , see Theorem 3.7. In the balanced situation, we show that the existence of an invariant Spin(7)-structure is equivalent to the existence of a suitable section of Λ 2 14 of the quotient space, see Theorem 3.9.…”

“…Thus this gives yet another balanced Spin(7)-structure. These three examples were found in [3] denoted by N 6,22 , N 6,23 and N 6,24 .…”

“…It was discovered in [18] that balanced Spin(7)-structures are characterised by the fact that they admit harmonic spinors. In [3] the authors construct many such examples on nilmanifolds by adopting a spinorial point of view. We instead here describe, via a few simple examples, a construction of balanced Spin(7)-structures starting from suitable G 2 -structures.…”

“…In the last two sections we demonstrate how our construction can be applied to the Bryant-Salamon Spin(7)-structure on the (negative) spinor bundle of S 4 and to the flat Spin(7)-structure on R 8 . In the former case the quotient space is the anti-selfdual bundle of S 4 and in the latter it is the cone on CP 3 . We interpret the quotient of the spinor bundle as a fibrewise reverse Gibbons-Hawking ansatz.…”

“…We interpret the quotient of the spinor bundle as a fibrewise reverse Gibbons-Hawking ansatz. In both cases we also study the SU(3)-structure on the link CP 3 . Theory (The London School of Geometry and Number Theory), University College London.…”