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I tried this question twice on math.stackexchange but got no answer so I decided to move it here. Let $M$ be a smoth manifold. Then $$C^\infty(M):=\{f:M\longrightarrow \mathbb R; f\ \textrm{is smoot …

Let $\Phi:A_M\longrightarrow A_N$ be a morphism of Lie algebroids covering $\phi:M\longrightarrow N$. Suppose $\Theta$ is a section of the pullback bundle $\phi^* A_N$. How to compute $$\langle d_{A …

Let $p_{A_M}:A_M\longrightarrow M$ be a Lie algebroid and $I:=[0, 1]$. Then $$p_{A_M}\times \textrm{id}:A_M\times I\longrightarrow M\times I,$$ is a vector bundle. There is a $C^\infty(M\times I)$-mod …

Let $A\longrightarrow M$ and $B\longrightarrow N$ be two Lie algebroides and $\Phi:A\longrightarrow B$ a morphism of Lie algebroids covering $\phi:M\longrightarrow N$. Let $\alpha, \beta\in \Gamma(A)$ …

Let $p:E\longrightarrow B$ be a surjective submersion and $\sigma: p^*(TB)\longrightarrow TE$ a complete connection. Given a path $\gamma: [a, b]\longrightarrow B$ and $s, t\in [a, b]$ such that $s<t$ …

Let $p:E\longrightarrow B$ be a smooth surjective submersion and $\sigma, \sigma^\prime: p^*(TB)\longrightarrow TE$ be two complete connections. Given a path $\gamma:I\longrightarrow B$ we can conside …

Let $A\longrightarrow M$ and $B\longrightarrow N$ be Lie algebroids with anchors $\#_A$ and $\#_B$, respectively. A morphism of Lie algebroids is a morphism of vector bundles $\Phi:A\longrightarrow …