One way to define the tangent space of a manifold at a point $p\in M$ is the following: We define an equivalent relation on the space of curves passing $p$ as follows: Two curves $\alpha, \beta$ are equivalents iff they have tangencity of order at least one that is $\parallel \alpha(t)-\beta(t)\parallel=o(|t|)$, in a local smooth coordinate.
Then the tangent bundle is the disjoint union of these tangent spaces.
One can generalize this by increasing the order of tangencity, that is $\parallel \alpha(t) -\beta (t)\parallel=o(|t|^{k})$.
Does the later gives us the $k-th$ tangent bundle? In this way, what would we get when $k$ is not an integer. For non integer $k$, do we get a well known structure(as a fiber bundle or some thing like this)?
