In Riemannian geometry there are many functions that have vector fields as arguments. Some (like the curvature tensor) are tensorial and some (like $<\nabla_{X}Y,Z>$) are not. Does the value of a tensor at a point taking arguments which are vector fields depend ONLY on the vector fields at that point? (As opposed to the value of the vector fields in some open subset at the point, which is the case for the non tensor quantity I just gave.)
As an extension of this question, this occurred to me as I was reading a sentence in lemma in do carmo which states "Let $M$ be a Riemannian manifold with curvature $R(X,Y)Z$. Given a parameterised surface $f(u,v)\subset M$ and $V$ a vector field on $f(u,v)$ define $R(\frac{\partial f}{\partial u}, \frac{\partial f}{\partial v} ) V $ in the obvious way. " But this is not really obvious to me. What if you can't extend the vector field locally to a vector field on $M$? Does this definition still make sense?