Let $M$ be a smooth manifold (with boundary). Suppose I have a smooth vector field $T$ defined on the complement of a compact subset $K$ of $M$ and I wish to extend $T$ to the whole of $M$. What are the obstructions to doing so? Does $M$ need to be compact? Does $K$ need to be a submanifold?

I have a feeling that an argument involving a partition of unity of $K$ would suffice for extending $T$ into $K$, but have not been able to find a decent reference.

Follow up question: Suppose I have a (pseudo)-Riemannian metric on $M$ and two vector fields now, $l$ and $n$, such that $l^{a} l_a = 1 = n^{a} n_a$ and $l^a n_a = 0$. Are these orthonormality properties preserved under the extension into $K$?