What properties define the tangent bundle up to diffeomorphism?

Question

In the theory of smooth manifolds there are many ways in which the tangent bundle can be defined, begging the question: what set of properties define the tangent bundle 'up to diffeomorphism'?

These properties would be such that, given a manifold $M$,

1. $TM$, whether defined by derivations or tangent curves or something else, complies with these properties.

2. Any two manifolds complying with such properties are diffeomorphic.

3. The theory of the tangent bundle can be develop from these properties alone

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As an example, the following properties define $\mathbb{R}$ up to isomorphism:

• $\mathbb{R}$ is an ordered field.
• $\mathbb{R}$ has the least upper bound property.

These properties are such that

1. $\mathbb{R}$, whether defined by Dedekind Cuts or Cauchy sequences, complies with these properties.

2. Any sets complying with such properties are isomorphic.

3. The theory of $\mathbb{R}$ can be develop from these properties alone.

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Edit: I've been considering the following:

Result (?): Let $M$ be a smooth manifold of dimension $m$, then there is a unique manifold $N$ up to diffeomorphism such that:

1. $\dim N = 2m$.

2. There is a smooth submersion $N\hookrightarrow M$.

3. Any $p\in N$ belongs to a chart $(U,\phi)$ where $\phi$ is a diffeomorphism $U\to M\times \mathbb{R}^m$.

4. ...

At least a fourth condition is needed, perhaps one that would keep the 'size' of $N$ in check, as $M\times \mathbb{R}^m$ (too 'small') complies with the first three conditions.

Answer 1

Your question is a reasonable one, but I don't see any easy answer. Your example is a rather simple object with, as you say, only two properties that uniquely define it. Note, however, that the properties have to be stated in terms of mathematical terms (ordered, field, least upper bound) that have to be defined.

More complex mathematical objects such as a manifold or its tangent bundle require a longer and more complex list of properties. In particular, a crucial aspect of manifolds and their tangent bundles are their so-called local properties. Locality is a powerful concept that pervades almost every area of mathematics including even algebra. But it makes the definitions more involved.

If you assume you know what smooth functions defined on a neighborhood of $p \in M$ are, then the definition of $T_pM$ as the space of derivations on smooth functions defined in a neighborhood of $p$ is essentially a definition using properties only. It just relies on more terminology and the definitions of the terms used. But the spirit is the same.

There's another way to proceed. The idea is to associate to each point $p \in M$ a vector space. The cotangent space at $p$ can be defined to be $$ T^*_pM = \mathcal{I}/\mathcal{I^2}, $$ where $f \in \mathcal{I}$ if it is a smooth function whose domain is a open neighborhood of $p$ and that vanishes at $p$. And $\mathcal{I}$ is the ideal generated by squares of functions in $\mathcal{I}$. $T^*_pM$ is obviously a vector space. The tangent space at $p$ is the dual vector space $$ T_pM = (T_p^*M)^*. $$

All of the better known properties of $T_*M$ can be derived from this definition. But this is overly complicated, which is why we never use it. It's used in algebraic geometry, because the spaces there are not as simply defined.