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Perhaps even simpler than the examples from electromagnetism in $\mathbb{R}^3$ minus some points is the following: The angle "function" $\varphi\colon S^1 \longrightarrow \mathbb{R}$ is not really gl …

There are well-known normal form theorems for all kind of constant rank submanifolds in symplectic manifolds, even global statements (and only such a thing seems to be of interest if you want to talk …

Johannes' answer can be upgraded to the following statement: Let $M$ be a second countable and Hausdorff manifold (who cares about others?) and $\pi_i\colon E_i \longrightarrow M$ vector bundles for …

The standard connection is the Levi-Civita connection of the flat metric. So if you have an embedding such that the given connection is the (projection of the) flat connection then you can also induce …

I wonder whether it is possible to have a smooth action of the $ax+b$ Lie group with compactly supported fundamental vector fields on $\mathbb{R}^2$ in such a way that it is non-trivial at least at on …

This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the infini …

Denote by $p_M: M \times N \longrightarrow M$ and $p_N: M \times N \longrightarrow N$ the canonical projections. Then you get an induced bilinear map from $\Omega^i(M) \times \Omega^j(N) \longrightarr …

Depending on you sign convention, this goes as follows. First you denote the bundle projection by $pr\colon E^* \longrightarrow X$. For a section $s \in \Gamma^\infty(E)$ you have a linear function $J …

I guess you want the topological equivalence to preserve the bundle structure of $TM \longrightarrow M$ otherwise it becomes a bit arbitrary, right? In this case a non-zero vertical vector field will …

Well, not to all of them, but nevertheless a nice approach: in the differential topology book by Bröcker and Jänich, they discuss various applications of the implicit function theorem and the theorem …

I'm not quite sure if these are the accepted names, but you find the list of irreducible Hermitian symmetric spaces (Cartan's list) in e.g. Helgason's "Differential geometry and symmetric spaces" (Cha …

Let me add a few remarks on Theo's answer. For a compact connected symplectic manifold, it is known that the integration with respect to the Liouville volume form (whatever normaliyation you prefer) i …

Non-Hausdorffness shows up in several contexts when dealing with Lie groupoids: the integration (Lie's 3rd Theorem) for Lie algebroids to Lie groupoids will typically produce a non-Hausdorff one, if i …

This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with …

This is a sort of standard construction you can find in several places. I don't know where this was done first though... OK: first you can extend your horizontal lift from vector fields to all (symm …