Wikipedia defines the vector field at a point on a manifold to lie in its tangent space. But is this general enough? Consider a surface traction vector on some manifold, for example. It will have a component in the normal direction. Is this not considered a vector field then since it doesn't lie in the tangent space?
2026-04-29 15:27:14.1777476434
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Definition of a vector field on a differentiable manifold.
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More generally this could be described in terms of vector bundles,
A vector bundle ... makes precise the idea of a family of vector spaces parameterized by another space X ...: to every point x of the space X we associate (or "attach") a vector space V(x) ..., which is then called a vector bundle over X.
... and sections,
Essentially, a section assigns to every point ... a vector from the attached vector space, in a continuous manner.
Indeed, it is not a vector field on the surface, for the reason you wrote. Such a thing is called a vector field along the inclusion of the surface in the ambient manifold. What it basically means is that at any point on the surface there is a vector tangent to the ambient space (and not necessarily to the surface).