# Various definitions of the tangent space

Let $$M$$ be a finite-dimensional smooth manifold with atlas $$\mathcal A$$ and $$p\in M$$. There are various equivalent definitions of the tangent space $$T_pM$$:

1. approach via derivations: the space of derivations of smooth functions: $$C^\infty(M)\to \mathbb R$$,
2. approach via germs: the space of derivations of the germs at $$p$$, $$C^\infty_p\to\mathbb R$$,
3. velocity vector approach: the set of curves $$\sigma : (-\varepsilon, \varepsilon) \to M$$ such that $$\sigma(0)=p$$, divided by an appropiate equivalence relation,
4. physicists' approach: set of functions $$X_p: \mathcal A\to \mathbb R^n$$ that take a coordinate chart and give coordinates relative to this chart, assigning them according to the transformation rule.
5. algebraic geometers' approach: the dual of $$\mathfrak m_p/\mathfrak m_p^2$$, where $$\mathfrak m_p$$ is the ideal of germs vanishing at $$p$$,

I wonder how one defines tangent space for different kinds of manifolds than smooth, especially:

• analytic/complex manifolds (i.e. one doesn't want to use partitions of unity as in 1.),
• $$C^k$$ manifolds for $$1\le k<\infty$$ (i.e. 5. is infinite dimensional),
• infinite-dimensional Banach manifolds.

Could you please provide references in which the above definitions are discussed? I would like to know which definitions are preferred (and to which are equivalent) in which contexts.

• John Lee's Introduction to Smooth Manifolds does all the approaches except (as far as I've seen) not 5. – Chris Mar 5 at 2:42
• As far as i remember Lee's smooth manifold only did for algebraic approach and the just mention few others. However, i found a decent explanation of all approach in Jeff Lee's manifolds – Sou Mar 5 at 4:17
• @Sou I agree - John Lee's book covers the algebraic approach and mentions others. It also mentions that the definition with germs works in the analytic/complex case. I'm looking for something explaining how these definitions tie together (or become irrelevant) in other categories than just $C^\infty$ manifolds. Unfortunately Jeff Lee's book doesn't cover other categories either... – Paweł Czyż Mar 5 at 19:41
• I didn't know much about that but probably Lang's book or KMS may help. – Sou Mar 6 at 7:41