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Of course, the spheres are compact while cotangent bundles are noncompact (unless in dimension 0). Nevertheless, a bit more interesting is the question whether the even dimensional spheres can be phas …

Maybe an even more elementary argument than the one of Tobias: The continuity of all involved operators is easy: simply all differential operators with smooth coefficients between sections of vector b …

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 …

In general, this topology is coarser than the original topology of the manifold, and, without further assumptions, strictly coarser. It coincides with the original one iff the Lorentz manifold is stro …

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 …

I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space …

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 …

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 …

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 …

There is the book by Kriegl and Michor called "Convenient setting of global analysis" published by the AMS. It goes much beyond Fréchet and really gives a big panorama. However, it is not easy reading …

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 …

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 …

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow $\ …

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 …

As it was already pointed out, on a bare vector bundle there is no intrinsic notion of "integrability". However, things change when you pass to a Lie algebroid: In this case the vector bundle $E$ is e …