arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0907.1222math.DG
收藏

半平坦结构与特殊Holonomy

Half-flat Structures and Special Holonomy

Vicente Cortés, Thomas Leistner, Lars Schäfer, Fabian Schulte-Hengesbach

备注：
40 pages
发表信息：
Proc. Lond. Math. Soc. 2010
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

Hitchin证明了他的半平坦SU（3）-结构的发展方程在紧致六流形M上的任何解定义了M到七流形的扩张，并且在G_2中具有完整性。给出了一个新的证明，它不要求M是紧的。更一般地，我们证明了六流形M上的任何半平坦G-结构的演化定义了M到Ricci平坦七流形N的扩张，对于SL（3，C）的任何实形式G。若G是非紧的，则N的和乐群是G_2 ^C的非紧形式G_2^* 的子群.用近似平行的G_2-或G_2^*-结构对近似半平坦结构的扩张，以及用平行的Spin（7）-和Spin（3，4）-结构对协同校准的G_2-和G_2^*-结构的扩张，得到了类似的结果。作为应用，我们得到了任何具有不变半平坦结构的六维齐性流形都允许具有平行G_2-或G_2^*-结构的七维流形的典范扩张。对于H_3 \times H_3，其中H_3是三维Heisenberg群，我们描述了所有左不变的半平坦结构，并发展了一种方法来显式地确定由此产生的平行G_2-或G_2^*-结构而不需要积分.特别地，我们构造了三个八参数度量族，其完整性等于G_2和G_2^*。此外，我们还得到了H_3 \times H_3上满足ω（Z，Z）=0的半平坦结构（ω，ρ）诱导的度量的强刚性结果，其中Z表示中心.最后，我们描述了满足一个现实条件的稳定三形式空间的特殊几何。考虑到所有可能的现实条件，我们找到了四个不同的特殊Kähler流形和一个特殊的para-Kähler流形。

英文摘要：

It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is noncompact, then the holonomy group of N is a subgroup of the noncompact form G_2^* of G_2^C. Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G_2- or G_2^*-structures, as well as for the extension of cocalibrated G_2- and G_2^*-structures by parallel Spin(7)- and Spin(3,4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G_2- or G_2^*-structure. For the group H_3 \times H_3, where H_3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G_2- or G_2^*-structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G_2 and G_2^*. Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (ω,ρ) on H_3 \times H_3 satisfying ω(Z,Z)=0 where Z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special Kähler manifolds and one special para-Kähler manifold.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑