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The $E_6$ and $E_7$ decompositions you list are explained in Cartan's 1894 thesis (see pages 89–92 for these formulae). For $E_8$, Cartan instead gives a decomposition (a $\mathbb{Z}_3$-grading) of t …

Note: I'm updating my answer to give a better (i.e., simpler) example plus a little more information about how to derive the example from Douglas' results (which may not be entirely clear upon first …

Yes, it is well-known that a $6$-manifold has an $\mathrm{SU}(3)$-structure if and only if it is orientable and spinnable (i.e., it has a spin structure). The necessity of these two conditions is c …

In the OP's particular case, the situation is somehwat simpler than the general case that José discusses. That's because the family of left-invariant metrics on $\mathrm{SU}(4)$ that the OP wants to …

This depends on what you mean by 'Darboux-like'. It is certainly not true that a closed nondegenerate 3-form on a 6-manifold is necessarily locally equivalent to one of the 'flat' models, so there is …

In any case, the proof is very simple. Consider the $3$-form $$ \phi_0 = dx^1\wedge dx^2\wedge dx^3 + dx^4\wedge dx^5\wedge dx^6. $$ I claim that the subgroup $G\subset\mathrm{GL}(6,\mathbb{R})$ that …

Cartan showed that the lowest dimensional (nontrivial) $E_8$-module is ${\frak{e}}_8$ itself, i.e., the adjoint representation, which has dimension $248$. The next smallest nontrivial irreducible mod …

Another couple of examples are the sphere rolling on the plane (or any surface, for that matter) without twisting or slipping, which is described by a connection on a principal SO(3)-bundle over the s …

Your confusion is revealed in this sentence "Or one defines something called the exterior covariant derivative D (see wiki) and then the curvature is simply the exterior covariant derivative of the co …

I may be misreading the sources that you list for the definition of Wick-rotatable, but, I believe that the following construction does fit that definition: According to the Wikipedia page that the O …

To understand the local geometry of this equation, I think one should apply the Calabi resolution of the Killing equation. (See E. Calabi, On compact, Riemannian manifolds with constant curvature. I, …

There are several methods, but let me describe (what is perhaps the simplest) one: On $\mathbb{R}^4$ with coordinates $(t,x,u,p)$, consider the pair of $2$-forms \begin{aligned} \Upsilon_0 &= (du-p\ …

As I understand it, the equation you are imposing on the function $\theta(x,y)$, defined on a domain $D\subset\mathbb{R}^2$ in the $xy$-plane is that, for some positive constants $\lambda_1\not=\lambd …

The answer to your question is 'yes', that is the general solution. This is one of the basic results in the theory of the variational bicomplex. It is a statement of the vanishing of a certain cohom …

New answer: I now have an answer for the subgroup case that the OP originally asked about. In fact, one has the following result: Let $G$ be a connected compact Lie group and let $p = (p_1,\ldots,p …