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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 $Usubseteq M$. I know that for all $pin U$ there is a natural isomorphism $T_pUcong T_pM$, 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.

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.

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. $mathrm{supp}phi_jsubset U_j$).

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

Hope it helps.

For a connection $nabla$ on a vector bundle $E$ over $M$ one can show two things:

1. If two sections $X,Y$ of $TM$ agree at a point $p$ then $(nabla_Xpsi)_p=(nabla_Ypsi)_p$ for all sections $psi$ of $E$.




2. If two sections $phi,psi$ agree locally arround $p$ then $(nabla_Xphi)_p=(nabla_Xpsi)_p$ for all sections $X$ of $TM$.

Furthermore:

3. Any $X_pin T_pM$ can be extended to a section of $TM$.




4. For any open neighbourhood $U$ of $p$ and any $psiin E_{|U}$ there exists a section $tilde{psi}$ of $E$ which locally arround $p$ agrees with $psi$ (just take a chart multiply by a suitable bump function and set it $0$ elsewhere).

Now you can define the induced connection $nabla'$ as follows:

$$(nabla'_Xpsi)_p=(nabla_tilde{X}tilde psi)_p $$

where $tilde{X}$ is any extension of $X_p$ and $tilde psi$ is any local extension of $psi$. By the above this is independent of the choice of $tilde{X}$ and $tilde psi$.