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Let $M$ be a manifold and $\pi : E \to M$ a rank $n$ vector bundle on $M$. We can define a connection on $E$ in two ways: We can specify the covariant derivatives $\nabla_X s$ or We can choose a co …

I was just reading the Ehresmann connection wikipedia page and noticed that it defines an Ehresmann connection to be complete if a curve in the base can be horizontally lifted over its entire domain. …

let $(M,g)$ be a Riemannian Manifold, let $ \gamma : [a,b] \rightarrow M$ be a piecewise smooth curve and let $\Omega : M \rightarrow \mathbb{R}^{n}$ be a coordinate chart. The length of $\gamma$ on t …

Suppose $M$ and $N$ are smooth manifolds. An immersion is a smooth map $f: M \rightarrow N$ whose pushforward is injective at each point. Is a smooth injective map an immersion? We can actually si …

One of the first theorems encountered in algebraic geometry is the upper semicontinuity of fiber dimension: Let $ f : X \to Y $ be a surjective regular map between irreducible varieties with irreduci …

Sorry for reviving this question. Everything Tom said is correct, but there is more to say about "coordinate-free Taylor series". It is true that arbitrary jet bundles $J^k(M,N)$ are subtle. The fib …

What you are describing is the algebraic operad Lie. More details can be found here. The Whitehouse modules are exactly what you get when you take Lie onto the other side of Schur Weyl duality. The s …