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Hi could someone please explain the concept of a tangent space clearly and in simple terms? I've been really confused for ages about this. Also, the definition of an immersion follows from tangent spaces definition so I'm confused with that too. I was told that $f:X\to Y$ is an immersion ($X,Y$ manifolds) if the rank of $T_x f$ is the dimension of $X$ but I'm not sure how you go about calculating the rank of such an abstractly defined function. Please can someone explain this, with the use of examples?

Thanks!

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There is (at least) two ways of seeing the tangent space of a manifold $M$ at a point $p$ :

  • As an equivalence class of "velocity vectors". Consider two curves $\gamma_1, \gamma_2$ equivalents if $ \frac{d}{dt}\left(\varphi \circ \gamma_1 \right) \big|_{t=0} = \frac{d}{dt}\left(\varphi \circ \gamma_2 \right) \big|_{t=0}$ where $\varphi$ is a chart around $p$, and $\gamma_1(0) = \gamma_2(0) = p$.
  • In an algebraic way. You define a vector at $p$ as a derivation, which is a linear map $X_p : \mathcal C^{\infty}(M) \to \mathbb R$ which satisfy Leibniz rule : $X_p(fg) = g(p)X_p(f) + f(p)X_p(g)$.

From the first definition, you can imagine the tangent space as all the space of " all possible velocity vectors at $p$" for a particle moving only in $M$ and passing by $p$.

The second definition is more abstract but the meaning is the same. What happened in $\mathbb R^n$ ? We can see that ponctual derivations are simply derivations of a fonction $f : \mathbb R^n \to \mathbb R$, i.e simply the application $ f \to \sum_k \lambda_k \frac{\partial f}{\partial x_k}$ which is exactly directionnal derivative along the vector $v = (\lambda_1, \dots, \lambda_n)$.

Now, an immersion is simply a smooth map $f : M \to N$ such that the tangent map $T_pf : T_pM \to T_{f(p)}N$ is injective. Since $\dim M = \dim T_pM$ for any smooth manifolds (it follows from the derivation viewpoint, since a basis for $T_pM$ is $(\frac{\partial }{\partial x_1}, \dots, \frac{\partial }{\partial x_1})$, it follows easily that $f$ is an immersion if for every $p$, $\text{rank} T_pf= \dim M $.

The link between $T_pM$ as derivation space and as equivalence class of curves is not immediate. I think a good book for this kind of questions is the excellent "Introduction to Smooth Manifolds" by Lee. Lot of details inside, and really clear writing.

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