Finite dimensional space of vector fields

Question

Consider a tangent bundle $TM$ and the space of sections $\mathfrak{X}(X)$. In general, for a $C^\infty$ structure, this space is an infinite dimensional vector space over $\mathbb{R}$. I'm trying to find an example in witch the space of vector fields is finite, by finding a small enough space $C^\infty/\sim$ and construct the space over this module. For example, the space of holomorphic function over a complex manifolds. In that case, I consider the tangent bundle over projective space $T^{(1,0)}(\mathbb{CP}^1)$ and the space of sections, that is, the holomorphic sections $\mathfrak{X}^{1,0}(\mathbb{CP}^1)$. Is it finite dimensional?