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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 …

As asked, the answer to the question is 'no'. The simply-connected cover $f:\mathbb{R}^2\to S$ of Sherck's first surface $S$ (which is defined in $\mathbb{R}^3$ by the equation $\mathrm{e}^{z} \cos x …

Note: I am making the standard assumptions in Finsler geometry, i.e., that $F:TM\to\mathbb{R}$ is smooth away from the zero section and $F$ is 'strictly convex'. Although $\partial_i\partial_j\exp^k_ …

Here's a perhaps more serious example: There is a compact $4$-manifold that fibers over both $\mathbb{RP}^3$ and $(S^1\times S^2)/\mathbb{Z}_2$ where, in each case, the fibers are circles. (The $\ma …

The group $H$ acts transitively and primitively on $\mathbb{C}=\mathbb{R}^2$. ('Primitive' means that $H$ preserves no nontrivial foliation.) It's a consequence of the classification of transitive pr …

The $\sigma_i$ as defined above satisfy $\mathrm{d}\sigma_i = -2\,\sigma_j\wedge\sigma_k$ when $(i,j,k)$ is an even permutation of $(1,2,3)$. For later use, let $E_i$ be the dual (left-invariant) fram …

Here's an expansion of my comment that the natural formulation of this problem as an EDS is on the coframe bundle $\pi: P\to M$, which, I hope, will be helpful. Also, because it will match my usual n …

Remark: I assume that you want $A$ to be a non-trivial bundle. Otherwise, of course, any parallelizable compact manifold would be an example. In particular, any compact Lie group would be an exampl …

There are probably too many such $(M,g,I_+,I_-)$ to really expect a 'classification'. For instance, consider the case when a complex manifold $(M,I_+)$ has real dimension $4$, and the $I_+$-holomorphi …

The answer is 'not always', in particular, not when $(n,r)=(2,3)$. The image is obviously is a smooth manifold when $n=0$, for then the image in $\mathbb{R}^{(n+1)r}=\mathbb{R}^r$ is the sphere $\Sigm …

Yes, there is a standard procedure to analyze such systems, essentially, it is Cartan's method of prolongation combined with his theory of involutive systems. There are other approaches as well, but …

Your equations are equivalent to the $1$-form equations $$ \mathrm{d}f = X_0(f,g)\,\mathrm{d}s + Y_0(f,g)\,\mathrm{d}t \quad \text{and}\quad \mathrm{d}g = X_1(f,g)\,\mathrm{d}s + Y_1(f,g)\,\mathrm{ …

General $G$-structures are not usually treated in textbooks, other than the basic definitions, so it's not surprising that there would be few if any textbooks that treat the intrinsic torsion of $G$-s …

If the second covariant derivative of every vector field $Z$ is symmetric in the sense that $\nabla(\nabla Z)$ (which is a section of $TM\otimes T^*M\otimes T^*M$) is a section of the sub bundle $TM\o …

In their comments to my first answer, the OP has clarified that they did not mean to regard the metric $h$ as a given, but, rather, an output of the problem of prescribing coframings by specifying the …