# Questions tagged [holonomy]

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### Fundamental $1$-form for a Riemannian manifold?

Take a Hermitian manifold $(M,I,g)$ where $I$ is the complex structure and $g$ is the Hermitian metric. The associated fundamental $2$-form $g(\cdot,I(\cdot))$ captures a lot of the information ...
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### Incomplete Riemannian-product manifold with group holonomy strict G2

Let $M_1=\mathbb{R}^2 \times K^4$ a Ricci-flat Riemannian-product manifold, where with $K$ I mean $k3$ kummer surface ($1$-complex dimension or $2$ real dimensions), so with $K^4$ I mean real 4-...
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### Are there mistakes in Kovalev's “Twisted connected sums and special Riemannian holonomy”?

This is kind of a strange and vague question... sorry about that. I am really interested in $G_2$ Twisted Connected sums as described in this paper: https://arxiv.org/abs/math/0012189 "Twisted ...
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### Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy. My first problem is, essentially, about ...
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### Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5. Kobayashi is trying to prove that if $E$ is a vector ...
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### Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...
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### Does holonomy determine parallel transport? [duplicate]

Let $p: P \longrightarrow M$ be a smooth $G$-principal bundle endowed with a connection that determines the holonomies: $$\Phi_{\gamma}: P_{x} \overset{\cong}{\longrightarrow} P_{x}$$ for any fiber ...
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### English translation of paper: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)

I am interested in the history of $G_2$ manifolds and want to read this paper in english: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7) Does anyone know where I can find a ...
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### Holonomy of hypercomplex manifold

The following is a quote from M. Barberis, I. Dotti, M. Verbitsky, Canonical Boundles of Complex Nilmanifolds with Applications to Hypercomplex Geometry, Math. Res. Lett., 16(2), 331-347, 2009. "Not ...
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### Killing vector fields on a compact $G_2$ manifold

I am trying to show that there can not be any nonvanishing Killing vector fields on a compact $G_2$ manifold. For the definition of a $G_2$ manifold just see the Wikipedia page. I know that since ...
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### Compact quaternionic Kahler manifolds of negative curvature: examples

There is a well known problem of LeBrun-Salamon: are there any non-symmetric compact quaternionic-Kahler manifolds of positive scalar (and Ricci) curvature? It is hard and still unsolved: Quaternionic-...
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### Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...
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### Calibrated submanifolds in Spin(7) and Calabi-Yau threefold

Suppose I have a Cayley cycle in a $Spin(7)$ holonomy manifold $M$, i.e. a calibrated submanifold. In the special case that $M=CY_3\times T^2$, is it possible that the Cayley cycle reduces to a Kahler ...
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### Manifolds with special holonomy especially $G_2$

I am interested in learning about $G_2$ manifolds and am aware that one of the canonical references is Joyce's Compact Manifolds with Special Holonomy. I am certain that my background is, at this ...
I have a Riemannian manifold $(X,g)$, where $X$ is not necessarily compact or complete, and a dense open submanifold $Y \subseteq X$. In my case $X$ is a smooth quasi-projective variety over the ...