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Accepted

English translation of paper: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)

My library has provided a copy of the article of Bonan. Here is a summary of its contents: Section 1: The author introduces the inner product algebra of octonions (algèbra des octaves de Cayley), ...
• 106k
Accepted

Does the curvature locally determine the connection?

The answer is 'not always'. Here is a simple case where you cannot recover the connection up to gauge transformation from the curvature: Let $n=2$, let the rank of $E$ be $m$, and, since $E$ is ...
• 106k
Accepted

Are there mistakes in Kovalev's "Twisted connected sums and special Riemannian holonomy"?

The error in Kovalev's paper is described in arXiv:1206.227 (see the discussion following theorem 2.6). An alternative proof is in arXiv:1212.6929. Building on the previous work of Tian–Yau, Kovalev ...
• 178k
Accepted

How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?

I believe the following is meant: Every smooth (real) manifold $M$ has a (unique) real-analytic structure compatible with the smooth structure. So, cover $M$ with real-analytic charts, i.e. whose ...
• 6,730

Manifolds with special holonomy especially $G_2$

Joyce's book Riemannian Holonomy Groups and Calibrated Geometry is an extended version of the research monograph you are reading, with more details and background material, aimed at providing a ...
• 25.6k
Accepted

A consequence of Ambrose-Singer theorem on holonomy

Your first question is a bit ambiguous. Are you asking whether, for each $p\in U$, the matrices $S_k(p)$ span the Lie algebra of $\mathrm{Hol}^0_p(\nabla)$ or are you asking whether, after taking the ...
• 106k

Alternative (easier) Proof of Ambrose Singer Holonomy theorem

I always find it easier to work with the vector bundle induced by a linear representation of the structure group. I believe this theorem is a consequence of the following loop formula (a terse proof ...
• 27k
Accepted

Holonomy of a Ricci-flat affine connection

The answer depends on the dimension. When $n=2$, Ricci-flatness of a connection implies that it is flat, so, in that case, yes, you get holonomy reduction locally. However, when $n>2$, Ricci-...
• 106k
Accepted

Compact quaternionic Kahler manifolds of negative curvature: examples

Any (Riemannian) symmetric space admits a cocompact lattice. This is due to A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2, 1963, pp.111-122. The quaternionic hyperbolic space ...
• 28.5k
Accepted

Riemannian holonomy of generic manifolds

Here are proofs for the Riemannian and Kähler case which rely on the fact that the curvature can be seen as parallel transport around infinitesimal loops. It uses explicit deformations which are hard ...
• 4,033
Accepted

Holonomy bounded in terms of area and the curvature

There are in fact more precise versions, expressing the parallel translation around a loop as the identity map plus a curvature integral over a homotopy. References: Section 3.1 of Werner Ballmann's ...
• 271
Accepted

Killing vector fields on a compact $G_2$ manifold

A parallel vector field implies a reduction of holonomy. Any form of $G_2$ does not preserve any nonzero vector when acting in its nontrivial 7-dimensional representation. So the holonomy must be the ...
• 25.6k
Accepted

• 1,396

holonomy of connection on gerbes

$A_{\alpha\beta}=df_{\alpha\beta}+B_\beta-B_\alpha$ implies that $iA_{\alpha\beta}+iA_{\beta\gamma}+iA_{\gamma\alpha}=$ \$i(df_{\alpha\beta}+B_\beta-B_\alpha+df_{\beta\gamma}+B_\gamma-B_\beta+df_{\...
• 3,644
Accepted

Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations

No, in general the holonomy group is not solvable. Yet, this is how I'd tackle the exercise. (By the way, as it is posed the exercise cannot be solved: you need to assume that each eigenratio of both ...
• 5,289

Flatness in a neighborhood of a point condition

There must be a glitch in the formulation because the answer to the first question is an obvious "no": take a round sphere and flatten it around the North Pole. Also note that every sufficiently ...
• 13.2k