# Tangent vectors on manifolds

I have an elementary understanding of differential geometry, and I know that the concept of a vector on a manifold can be defined in several different ways. Perhaps the easiest to understand, is in terms of equivalence classes of curves $\gamma:M\rightarrow\mathbb{R}^{n}$ on a manifold $M$. In this case, vectors are defined at each point $p\in M$, as tangent vectors to curves at that point, "living" in the tangent space $T_{p}M$ to that point. A tangent vector at a point $p\in M$ is then an equivalence class of curves, mutually tangent, at that point. Another way is to construct the notion of a vector using derivations.

My question is (and apologies if it's a silly one), can there exist vectors $v$ in a given tangent space $T_{p}M$ that are not tangent to a curve passing through $p$ (essentially, are there cases where $v^{i}\neq\frac{\mathrm{d}x^{i}}{\mathrm{d}t}$)?

• If you define the tangent vectors via equivalence classes of curves, it should be obvious that there cannot be such a vector, since you need at least one curve in the equivalence class which defines this vector. – M. Winter Feb 20 '18 at 15:00
• @M.Winter Sorry, I realise what I wrote sounds a little stupid. My initial reason for asking the question was that I thought (briefly) that it was a particular quirk to the definition in terms of equivalence classes of curves, but I now realise that it would be inconsistent if this weren't also true for the other definitions. – user35305 Feb 20 '18 at 15:18