We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi-Yau 4-orbifolds. We find that different hypersurfaces within the same family of Calabi-Yau 4-orbifolds may result in different Spin(7)-manifolds.
本文的目的是证明Finsler流形的完整性可以与黎曼流形的完整性有很大的不同。证明了维数大于2的非零常曲率正定非黎曼Finsler流形的和乐群不可能是紧李群.因此这个和乐群不会作为任何黎曼流形的和乐群出现。另外，我们给出了Heisenberg群上左不变Finsler度量的一个例子，使得它的和乐群不是（有限维）李群.这些结果肯定地回答了S. S. Chern和Z. Shen：“有没有一个Finsler流形的和乐群不是任何黎曼流形的和乐群？"
We classify the holonomy algebras of manifolds admitting an indecomposable torsion free $G_2^*$-structure, i.e. for which the holonomy representation does not leave invariant any proper non-degenerate subspace. We realize some of these Lie algebras as holonomy algebras of left-invariant metrics on Lie groups.
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-Kählerian and special pseudo-Kählerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives…
According to the Markus conjecture, closed flat affine manifolds with parallel volume should be complete. We show it is the case for three-manifolds when the holonomy centralizes an affine transformation preserving the volume. It is notably the case when the holonomy group has non trivial center or when the automorphism group is non discrete.
The classification problem for holonomy of pseudo-Riemannian manifolds is actual and open. In the present paper, holonomy algebras of Lorentz-Kähler manifolds are classified. A simple construction of a metric for each holonomy algebra is given. Complex Walker coordinates are introduced and described using the potential. Complex pp-waves are characterized in terms of the curvature, holonomy and the potential. Classification of Lorentz-Kähler symmetric spaces is reviewed.
The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the para-quaternionic-Kählerian manifolds with non-zero scalar curvature are considered. Einstein not Ricci-flat metrics of signature $(2,n)$ with all possible holonomy algebras are given.
In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of th…
In this paper we discuss some examples and general properties of holonomy groups of $\nabla^φ$ introduced by Wylie and the author, the connection corresponding to the $N=1$ Bakry-Émery Ricci curvature, and also Wylie's $\bar{\mathrm{sec}}_f$. In particular we classify all possible holonomy groups in dimension 2 and also provide two infinite families: $SL_n(\mathbb{R})$ and $SO^+(p,q)$.
Constructing compact manifolds with exceptional holonomy
例外的和乐群是7维的G2和8维的Spin（7）。具有这些和乐群的黎曼流形是Ricci平坦的。这是一篇综述性的论文，主要讨论了具有完整性G2和Spin（7）的紧致7-和8-流形的构造。最简单的构造方法是使用复几何和卡-丘分析的技巧来解决环面orbifold T^7/G或T^8/G的奇点，其中G是一个在T^7或T^8上保持平坦G2或Spin（7）-结构的有限群。也有更复杂的结构，开始于一个卡-丘流形或orbifold。本文中的所有材料在作者的书中有更详细的介绍，“Compact manifold with special holonomy”，牛津大学出版社，2000年。
Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the special orthogonal group, of dimension considerably smaller. We realize the computations of the holonomies in a unified way, by using as a main algebraic tool a nonassociative structure, that one of symplectic triple system.
In this article, we introduce and study the notion of a complete special holonomy manifold $(X,ω)$ which is given by a global perturbation potential function, i.e., there is a function $f$ on $X$ such that $ω'=ω-\mathcal{L}_{\nabla f}ω$ is sufficiently small in $L^{\infty}$-norm. We establish some vanishing theorems on the $L^{2}$ harmonic forms under some conditions on the global perturbation potential function.
This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen 7-dimensional orbifolds, with the help of asymptotically locally Euclidean spaces. Another method uses the gluing of two asymptotically cylindrical pieces and requires a certain matching condition for their cross-sections `at infinity'.
In this paper we review $G_2$ and $Spin(7)$ geometries in relation with a special type of metric structure which we call warped-like product metric. We present a general ansatz of warped-like product metric as a definition of warped-like product. Considering fiber-base decomposition, the definition of warped-like product is regarded as a generalization of multiply-warped product manifolds, by allowing the fiber metric to be non block diagonal. For some special cases, we present explicit example of $(3+3+2)$ warped-like product manifolds with $Spin(7)$ holonomy of the form $M=F\times B$, where …
A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of conformal classes on compact manifolds containing two non-homothetic Kähler metrics. The second one is the classification of compact Einstein locally conformally Kähler manifolds. The third result is the classification of the possible (restricted) Riemannian holonomy groups of compa…
We show that the $G_2$-manifolds and certain ${\rm Spin}(7)$-manifolds are endowed with natural Riemannian twistorial structures. Along the way, the exceptional holonomy representations are reviewed and other related facts are considered.
We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. Among other results, we obtain that this property is equivalent to admitting a parallel timelike vector field. We also derive some properties of the space of all such metrics on a given manifold.
The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [J. Differential Geom., 36(3):551-589, 1992], and we provide a characterization for singularities as well as a deforma…
In this article, we summarize our recent results on the study of manifolds with special holonomy via the Frölicher-Nijenhuis bracket. This bracket enables us to define the Frölicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in Kähler geometry, and assigns an $L_\infty$-algebra to each associative submanifold. We provide several concrete computations of the Frölicher-Nijenhuis cohomology.
