I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me.
The first comment is made in the book by James Munkres "Analysis on Manifolds", page 249:
Any tensor field on M [M a manifold and a subset of $\mathbb{R}^n$] can be extended to a tensor field defined on an open set of $\mathbb{R}^n$ containing $M$.
The second comment is made in the book by John Lee "Riemannian Manifolds, An Introduction to Curvature", page 56:
Not every vector field along a curve [in a manifold M] need be extendible [to a neighborhood of the curve in M].
As an example, Lee mentions the case of a vector field of an intersecting curve.
Now - the notion of a tensor is quite new to me so I might go wrong here but I thought a vector field is a contravariant tensor field of rank 1. But if that is the case then the above comments contradict each other... a clarification of where I went wrong in my reasoning would be a great help, many thanks!