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I am confused about the exact "type" (to borrow from computer science terminology) that tangent spaces have. Let's say I have a sphere $S$ in $\mathbb R^3$, and a tangent space $T_xS$ at some point $x$ on the sphere. Now, here is what I am wondering: is $T_xS$ a subspace of $\mathbb R^3$ containing 3-vectors, or is it $\mathbb R^2$, containing 2-vectors? Obviously both choices are isomorphic, but the difference is important if specific elements of the tangent space are to be written down on paper.

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  • I'd say definitely not $\mathbb R^2$. Some authors define such a tangent space so that it is a subspace of $\mathbb R^3$. – littleO Feb 04 '20 at 22:38
  • If $U\in\mathbb R^2$ is open and $f:U\to S\in\mathbb R^3$ is a local parametrization of $S$ with $f(u)=x$ then my definition of tangent space would be $\mathrm TS_x=\mathrm df_u(\mathbb R^2)$ which is indeed a subspace of $\mathbb R^3$ – Maximilian Janisch Feb 04 '20 at 22:41
  • The way it is usually defined formally, the answer would be neither. – Eric Wofsey Feb 04 '20 at 22:45
  • @EricWofsey What would it be, then? – Display Name Feb 04 '20 at 22:46
  • Well there are a variety of different and essentially equivalent definitions that are used, but one common one is that an element of the tangent space is a certain set of curves (all the curves through the point that have a given tangent vector). – Eric Wofsey Feb 04 '20 at 22:47
  • @EricWofsey If it was a set of curves, would it then be a vector space at all? – Display Name Feb 04 '20 at 22:48
  • Yes, though the definition of the vector space structure is a bit complicated. – Eric Wofsey Feb 04 '20 at 22:48
  • In any case, the tangent space at any point can certainly be considered as a subspace of $\mathbb{R}^3$ in a canonical way. It's not clear to me what you are looking for with your question, though. What do you mean by "write down on paper"? – Eric Wofsey Feb 04 '20 at 22:55
  • I thought the tangent space for a point $p$ on the sphere is just ${u\in \mathbb R^3:\langle u,p\rangle=0}$ which is a $2$ dimensional subspace of $\mathbb R^3$. Or more abstractly I guess, since $S=f^{-1}(0)$ where $f(x)=|x|-1,\ T_pS$ is the kernel of $d_pf$ – Matematleta Feb 04 '20 at 22:55
  • @EricWofsey is referring to the abstract definition of a smooth manifold, but if OP is reading a book such as Analysis on Manifolds by Munkres or Calculus on Manifolds by Spivak then a more concrete definition of a tangent space is usually used. – littleO Feb 04 '20 at 23:07
  • For "intuitive" definitions of the tangent space of a submanifold of $\mathbb R^n$ see https://math.stackexchange.com/q/2982185, https://math.stackexchange.com/q/3512674 and https://math.stackexchange.com/q/3503836. – Paul Frost Feb 05 '20 at 23:17

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