Linear perturbations of metrics with holonomy Spin(7)
Introduction
Riemannian metrics with holonomy have been studied in differential geometry since the celebrated theorem of Berger [3], listing the possible holonomy groups of an irreducible, nonsymmetric simply connected Riemannian manifold. Metrics with holonomy contained in are known to be Ricci-flat [4], and they imply the presence of a parallel spinor [24]. They are also relevant for string theory (see [14]).
The first local examples of metrics with holonomy were constructed in [5], and the first complete metric was obtained in [6]; the latter takes the form of an explicit -invariant metric on the spinor bundle over . It was later shown in [10] that this metric belongs to a one-parameter family of invariant metrics.
We note that the metrics of [6] are of cohomogeneity one; other cohomogeneity one metrics with holonomy contained in have been constructed in [11], [18], [14], [23], [9], [7], [1], [2]. Outside of the cohomogeneity one setting other constructions exist, but the metrics they determine are not explicit (see [16], [17], [13], [20]).
As observed in [5], a metric with holonomy contained in is defined by a closed form Ω which is pointwise linearly equivalent to a reference 4-form on with stabilizer . It is then possible to define perturbations of a -metric by replacing Ω with a perturbed form which remains pointwise linearly equivalent to Ω. Notice that for the parallel 3-forms φ arising in the context of holonomy the form is always linearly equivalent to φ for δ sufficiently small; in other terms, φ is stable in the sense of [15]. The form Ω is not stable, however, so more work is needed in order to obtain a perturbation.
One possible approach was considered in [19, Section 5.2] by taking for vector fields on M. In terms of the infinitesimal action ρ of on , this amounts to setting , where A is the skew-symmetric endomorphism . We recall that under the bundle of four-forms splits as the skew-symmetric A determines a perturbation term δ in . Whilst this construction gives nontrivial perturbations of the original metric in the case of (mutatis mutandis: the relevant decomposition is and the perturbation δ an element of ), it turns out that in the case the perturbed form never defines a -structure ([19]).
A different ansatz was considered in [8] in the context of -structures, which amounts to imposing that A be nilpotent, rather than skew-symmetric. The key observation, working at a point, is that when the form is always in the same -orbit as Ω for any t; one then says that δ is a linear perturbation of Ω. It turns out (see [8]) that one can assume A to be nilpotent without loss of generality.
In this paper we study nilpotent perturbations of the -form Ω. By a case-by-case analysis of the possible Jordan forms of a nilpotent matrix in , and making use of -invariance of (3), we prove that any linear perturbation of the form Ω is defined by a rank one nilpotent matrix, i.e. it has the form with orthogonal vector fields. In terms of (2), the resulting perturbations of the form turn out to be elements of .
We apply the method of linear perturbations to the Bryant-Salamon metric; we construct a family of linear perturbations parameterized by three functions of one variable. However, it turns out that the resulting metrics are isometric; due to the fact that nilpotent perturbations preserve volumes, we do not recover the squashed deformations of [10].
Our result complements the result of [21], stating that the Bryant-Salamon is rigid in the class of asymptotically conical metrics.
Acknowledgments. This work is partly based on the second author's master thesis [22]. We thank Thomas Madsen for useful discussions.
Section snippets
Linear perturbations
In this section we classify linear perturbation at a point of 4-forms defining a -structure, proving that they are in one-to-one correspondence with nilpotent matrices of rank one in .
We first recall some results from [8]. For a lighter notation, we shall write instead of . Denote by the natural action of on . We shall write for . Proposition 1.1 Fix and a solution of Then lies in the same [8]
A cohomogeneity one description of the Bryant-Salamon metric
Recall from [6] that the spinor bundle S over carries a cohomogeneity one metric with holonomy ; this metric has cohomogeneity one under the action of . In this section we give a description of these metrics in terms of cohomogeneity one actions which will be needed in order to study the linear perturbations.
Explicitly, the Lie group contains two copies of , i.e. At the Lie algebra level,
Linear perturbations of the Bryant-Salamon metric
In this section we study -invariant linear perturbations of the Bryant-Salamon metric.
By Theorem 1.7, a linear perturbation of a -structure is obtained by the choice of a rank one nilpotent endomorphism of the tangent bundle at each point. Thus, the global data for a linear perturbation is given by the choice of a vector field X and a one form α with . Since we work in the -invariant setting, we will require both vector field and form to be invariant.
Thus, the first step
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