induced connection on open sets 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.
 A: 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.
A: 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.
A: For a connection $\nabla$ on a vector bundle $E$ over $M$ one can show two things:


*

*If two sections $X,Y$ of $TM$  agree at a point $p$ then $(\nabla_X\psi)_p=(\nabla_Y\psi)_p$ for all sections $\psi$ of $E$.

*If two sections $\phi,\psi$   agree locally arround $p$ then $(\nabla_X\phi)_p=(\nabla_X\psi)_p$ for all sections $X$ of $TM$.
Furthermore:


*Any $X_p\in T_pM$ can be extended to a section of $TM$.

*For any  open neighbourhood $U$ of $p$ and any $\psi\in E_{|U}$ 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:
$$(\nabla'_X\psi)_p=(\nabla_\tilde{X}\tilde \psi)_p $$
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 $\tilde \psi$.
