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There is also a quick abstract proof via representation theory: $S^2_0(\mathbb{R}^4)$ is a 9-dimensional representation of $\mathrm{SO}(4)/\{\pm I_4\}\simeq \mathrm{SO}(3)\times\mathrm{SO}(3)$ and, h …

Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students. However, what you are asking for is actually pretty simple: …

The method of characteristics is a bit strange here because the equation is underdetermined, so one can't expect to be able to specify a solution by fixing initial data for $u$ and $v$ along a surface …

Here is an example to think about: Let $S^6 = \mathrm{G}_2/\mathrm{SU}(3)$ be the $6$-sphere endowed with its $\mathrm{G}_2$-invariant almost Hermitian structure. There is a $\mathrm{G}_2$-invariant …

The result that you are looking for is not in Élie Cartan's 1936 book La topologie des groupes de Lie because it was not known to be true at the time the book was written. Indeed, as Cartan remarks i …

One reference is in Helgason's 1984 book Groups and Geometric Analysis. The result you want appears there as Corollary 5.12. The notation he uses is $X=G/K$ is a symmetric space where $G$ is connecte …

Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix. This follows from the well-know …

Here is an informative example that illustrates the difficulties: Consider the Lie algebra ${\frak{g}} = \mathrm{Vect}(\mathbb{S})$ of smooth vector fields on the circle $\mathbb{S}$. The flow of an …

As I suspected, the statement that the bundle $\mathrm{SO}(8)\to S^7$ is a product bundle, i.e., that $$ \mathrm{SO}(8)\simeq S^7\times\mathrm{SO}(7)\tag1 $$ as bundles over $S^7$ is in N. Steenrod's …

As YCor commented, the main point is to show that the invariant polynomials separate orbits. This follows from the compactness of $\mathrm{SO}(n)$. The point is this: Because $\mathrm{SO}(n)$ is co …

There are two aspects to this question, the global question and the local question. Also, the case $n=1$ is different from $n>1$. Basically, the answer is 'essentially yes, but with some caveats'. H …

In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$. Basically, the reason is this: If …

There are too many of these for them to have any particularly interesting structure. For example, consider any metric $g$ on the $3$-torus $\mathbb{T}^3$. By the Hodge theorem, the space of $g$-harm …

Using the reference https://arxiv.org/pdf/0903.2087.pdf, which agrees with https://arxiv.org/pdf/hep-th/0001078v1.pdf, which agrees with the reference U. Müller, C. Schubert and Anton M. E. van de Ven …

I think that you want something like this: Let $E\to X$ be a (finite rank) vector bundle over a compact, Hausdorff topological space $X$, let $\mathcal{A}\subset C(X,\mathbb{R})$ be a subalgebra that …