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I am confused with the idea of tangent vector or tangent space. First of all, I learned that there is an isomorphism from $ \mathbb{R}_a^n$ onto $T_a( \mathbb{R} ^n)$ from John M.Lee' book Introduction to Smooth Manifolds. Although we have the perspective of regarding tangent vectors as an operator defined on $\mathbb{R} ^n$ or more generally, a manifolds, I am still have trouble with it. Again, on Lee's book,

For example, any geomantic tangent vector $v_a \in \mathbb{R} _a^n$ yields a map $D_{v,a}:C^\infty ( \mathbb{R} ^n)\to \mathbb{R} $, which takes the directional derivative in the direction $v$ at $a$: $$D_{v,a}f = D_v f(a) = \frac{d}{dt} f(a+tv)$$

Here are my questions: Now considering a special manifold, a surface embedded in $\mathbb{R}^3$ and the tangent of the surface. I know we have to define a smooth function $f$ on the (special) manifold and we must define a function $f$ with three dimensions in order to take directional derivative by a three dimensional vector $v_a$. However, our manifold is a two dimensional surface embedded in $\mathbb{R} ^3$. Unfortunately, $2\neq 3$. So, what's dimension of the domain of $f$ with respect to the special manifold?

Edit:@Jack Lee: He points out that tangent vectors to the sphere are defined more abstractly as derivations. At first we have Euclidean space,$\mathbb{R}^3$, and the space $\mathbb{R}_a^n$, so we have its tangent space $T_p(\mathbb{R}^3)$ and we prove that these two linear vector space are isomorphic. We then define the general tangent space w.r.t manifolds.

Let $M$ be a smooth manifold with or without boundary, and let $p$ be a point of $M$. A linear map $v:C^\infty(M)\to R$ is called a derivation at $p$ if it satisfies $$v(fg) = f(p)vg + g(p)vg$$ for all $f,g \in C^\infty(M)$. The set of all derivation of $C^\infty(M)$ at $p$, denoted by $T_p M$, is a vector space called the tangent space to $M$ at $P$. An element of $T_p M$ is called a tangent vector at $p$.

The author then discuss the sub-manifolds and the relation between the tangent space of the embedding space and its ambient space.

Let $M$ be a smooth manifold with or without boundary, and let $S\subseteq M$ be an immersed or embedded submanifold. Since the inclusion map $\iota:\hookrightarrow M$ is a smooth immersion, at each point $p\in S$ we have in injective linear map $d_{\iota_p}:T_p S\to T_p M$. In terms of derivations, this injection works in the following way: for any vector $v\in T_p S$, the pimage vector $\tilde{v} = d_{\iota_p}(v)\in T_p M$ acts on smooth functions on $M$ by $$\tilde{v}f = d_{\iota_p}(v)f = v(f\cdot\iota) = v(f\big|_S)$$

Finally, there is also a picture illustrated on the book help a lot. enter image description here

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  • $\begingroup$ Could you please clarify what exactly is your doubt? If possible, depending minimally on the image? $\endgroup$ – Aloizio Macedo Dec 31 '15 at 2:30
  • $\begingroup$ The notation $C^{\infty} (\mathbb{R}^n)$ means the space of all infinitely differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$. $\endgroup$ – Mark Fantini Dec 31 '15 at 2:31
  • $\begingroup$ @AloizioMacedo I delete the picture for short. my question is the last question. what's the dimension of the domain of f. $\endgroup$ – Brooks Dec 31 '15 at 2:48
  • $\begingroup$ @MarkFantini tanks a lot. i get the idea of this definition, but f is defined on the surface and the surface is not $R^3$.... isn't it restrict the domain to the surface? whatever, i don't know $\endgroup$ – Brooks Dec 31 '15 at 2:51
  • $\begingroup$ Is your question What is the definition of a smooth function from one manifold to another?", or "What is the definition of the derivative of one smooth function from one manifold to another?". The question is still not clear to me. For instance, I don't know what relation you tried to establish between your questions and what are above them. $\endgroup$ – Aloizio Macedo Dec 31 '15 at 3:22
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I think you're trying to read more into this definition than is there. The definition you quoted is only talking about tangent vectors to $\mathbb R^n$, not to submanifolds of $\mathbb R^n$ such as the sphere. Tangent vectors to the sphere are defined more abstractly as derivations (see p. 54). The relationship between tangent vectors to $\mathbb R^n$ and tangent vectors to a submanifold like the sphere isn't developed until Chapter 5.

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  • $\begingroup$ thank you so much. you really help me a lot. It is the second edition of this book, isn't it. I will read those material. I choose your answer for the best answer. thx $\endgroup$ – Brooks Dec 31 '15 at 23:58
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    $\begingroup$ Yes, I was referring to the second edition. If you have the first edition, tangent vectors to manifolds are defined on p. 65, and the relationship with tangent vectors to $\mathbb R^n$ is discussed in Chapter 8. $\endgroup$ – Jack Lee Jan 1 '16 at 0:04

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