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Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
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Vertical bundles of higher order tangent bundles
Let $M$ be a smooth (finite dimensional, Hausdorff and second countable) manifold. Let $T^kM$ be the manifold of equivalence class of curves that their derivates (in charts) agree up to order $k$. Let …
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Local triviality condition in vector bundles [closed]
Let $E$ and $M$ be smooth manifolds (of finite dimension, Hausdorff and second countable). Let $\pi:E\longrightarrow M$ be a surjective submersion such that:
$E_p:=\pi^{-1}(p)$ is a real vector space …
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What are the sufficient and necessary conditions for surjective submersions to be locally tr...
Let $E$, $M$ be smooth finite dimensional, Hausdorff and second-countable manifolds. Let $\pi:E \longrightarrow M$ be a surjective submersion.
$\pi$ is locally trivial if $\forall p\in M$, $\exists U …
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Higher order Leibniz rule for higher order tangent space
Let $M$ be a smooth manifold (finite dimensional, Hausdorff and second-countable) and $p\in M$ a point.
The higher cotangent space at $p$ is defined to be quotient:
$$ {T^*_p}^rM:= \eta_p/\eta_p^{r+1} …