# What are the allowed dimensions for vector fields?

When one first encounters the concept of vector field, especially in physics, it is often presented just as n-tuple of numbers $(x_1, x_2, \ldots , x_n)$ prescribed to each point. In this manner $n$ is allowed to accept arbitrary value.

However, when one proceeds with vector fields in differential geometry, the dimension of a manifold dictates the dimension of a tangent space:

$$\dim M = \dim T_x(M)$$

There are other ways to attach a vector space to each point of manifolds --- tensor products of tangent spaces, p-forms, etc.

Now I ask, given an n-dimensional manifold, what are allowable dimensions for vector spaces on it? Are all of them allowed, or there is no way to naturally construct a k-dimensional vector field (in a broad sense) for certain k?

Another (stronger) version for this question, given n-dimenstional vector space, can we naturally construct k-dimensional vector space from it for arbitrary k?

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$M\times \mathbb{R}^k$ is a $k$ dimensional vector bundle over an $\dim M$ dimensional manifold. This construction always works, for any manifold $M$ and any dimension $k$. You have to specify what "natural" means. This has a precise mathematical meaning, and might be what you are after, but I am not sure... – Thomas Rot Jul 17 '12 at 8:04
Can you help me with this naturality, but without severe category theory? I would call natural what can be encountered in physics --- you have a manifold (and that's all you have), vector bundles on it with what dimensions are physically plausible? – Yrogirg Jul 23 '12 at 7:23