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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].

1 vote
0 answers
59 views

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 …
0 votes
1 answer
134 views

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 …
5 votes
2 answers
1k views

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 …
1 vote
0 answers
67 views

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} …