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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} …

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 …

Let $\pi:E\longrightarrow M$ a smooth fibre bundle. A connection is a linear bundle homomorphism $\Phi:TE\longrightarrow TE$ such that $\Phi$ is a projection to the vertical bundle $VE\subset TE$. I r …

Let $p:E \longrightarrow M$ be a smooth fibre bundle, with standard fibre space $F$ and $G$ a Lie group acting effectively on $F$ as a structure group. Then, are the transition functions always smoot …

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 …