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In this What is the affine connection, and what is the intuition behind/for affine connection?, it says that in mapping of Euclidean spaces $R^n \rightarrow R^n$, we have $\nabla_Y X (p) = \lim_{t \rightarrow 0} \frac{X(t + Y(p)) - X(p)}{t}$ and we can check that it is a connection.

My question is that:

For arbitrary connection on a manifold $M$, can we construct an Euclidean embedding of arbitrary dimension, $M \rightarrow R^n$, s.t. the affine connection is compatible with the addition algebraic operation? i.e. (1) $\nabla_Y X (p) = \lim_{t \rightarrow 0} \frac{X(t + Y(p)) - X(p)}{t}$ in the Euclidean embedding induces an addition algebraic structure $t + Y(p)$ on the original manifold $M$

(2) the connections on two manifolds are the same action on the tangent bundle?

threeautumn
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  • This doesn't even make sense for submanifolds of $\Bbb R^n$ to start with. – Ted Shifrin Aug 25 '22 at 15:48
  • @TedShifrin Yes, I edited it as mapping between Euclidean space of the same dimension $R^n \rightarrow R^n$ rather than arbitrary embedding of manifolds. If it is $R^2 \rightarrow R^3$, then it has to be $\nabla_Y X(p) = \pi(\lim (\frac{X(t+Y(p)) - X(p)}{t})$ by projecting it onto the tangent space of the submanifold of $R^3$. – threeautumn Aug 26 '22 at 14:46
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    This is defined only if $X$ is defined on a tubular neighborhood of $M$ to start with. – Ted Shifrin Aug 26 '22 at 15:58

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