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Let $M$ be a smooth manifold. If $\nabla$ is a linear connection on $M$, I would like to induce a unique linear connection on an open subset $U\subseteq M$. I know that for all $p\in U$ there is a natural isomorphism $T_pU\cong T_pM$, so I can restrict global vector fields to local vector fields on $U$. Unfortunately there are some local vector fields on $U$ that don't came from a restriction of global vector fields.

For this reason I can't find a reasonable linear connection $\nabla^U$ over $U$ induced by $\nabla$. I need help.

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The connection $\nabla$ on a manifold $M$ is a local operator. The value of $\nabla_X(Y)$ at a point $p \in M$ depends only on $X_p$ and the value of $Y$ in an arbitrary small neighborhood around $p$. This is enough to define the connection on $TU$ when $U \subset M$ is an open subset, without extending the vector fields involved to the whole of $M$. More generally, you may want to read about the pullback of a connection which allows you to restrict a connection to more general submanifolds and even more.

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    The proposition about the local behavior of the connection says: If Y and Z are GLOBAL vector fields that coincide on an open neighborhood of $p$ then $\nabla_XY=\nabla_XZ$ (The same thing is true for $X$). I don't understand in which way I can apply this statement to the problem.2012-12-08
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Take an open cover $\{U_j\}$ of $U$ given by relatively compact open sets in $U$ and a partition of unity $\{\phi_j\}$ subordinated to $\{U_j\}$ (i.e. $\mathrm{supp}\phi_j\subset U_j$).

For any vector-field $X$ on $U$, $X=\sum \phi_j\cdot X$ and $\phi_j X$ is a vector-field on $M$. Therefore we can define $\nabla^U_XY=\sum\nabla_{\phi_j X}Y$ and $\nabla^U_YX=\sum\nabla_Y\phi_j X$ The definition is locally meaningful, because the covering is locally finite, and so are the sums.

Hope it helps.

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    Ok now it is clear. Thanks2012-12-08