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The problem that I am having is that I am having quite a hard time understanding the ideas of Geometric Tangent vectors and why they are even needed - I mean one already has the usual "Tangent" of a curve, why more definitions?

We let X be an n-dimensional differentiable manifold $p \in X$, and $(U,h,V)$ is a chart for X. The set of all differentiable curves $\gamma_{1},\gamma_{2}$ through p is denoted by $K_{p}$ and $\gamma: (-\epsilon,\epsilon) \rightarrow X, \epsilon > 0$, with $\gamma(0) = p.\space$ Now $\gamma_{1} \sim \gamma_{2}$ if $\dot{\gamma_{1}}(0)_{h} = \dot{\gamma_{2}}(0)_{h}$. Where $\space \dot{\gamma}(0)_{h} \equiv d/dt(h \circ \gamma)(0) \in \mathbb{R}^{n}$

Defn: The geometric Tangent space of a space X at p is the quotient $T_{p}^{geom}X \equiv K_{p} /\sim$ where is the equivalence relation of two curves $\gamma_{1},\gamma_{2}$ through point p.

The equivalence class of $\gamma \in K_{p}$ is denoted $[\gamma] \in T_{p}^{geom}X$ and is called the geometric tangent vector of X at p.

Thanks for any help at getting a better insight.

Brian

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Yes one have the usual definition of tangent vector of a curve but the tangent of a curve is defined if the curve lies in some $\mathbb{R}^n$ i.e. if $\gamma$ the curve then $\gamma :(0,1)\rightarrow\mathbb{R}^n $. Now when we are considering the tangent vector of a point in manifold then you have to consider the curve to be in that manifold. So my point is a priori you don't know whether the curve lies in $\mathbb{R}^n$ or not. In other words you don't know whether every smooth manifold can be embedded in $\mathbb{R}^n$ or not.
P.S. The above statement is true. It is called called Whitney embedding theorem.

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  • $\begingroup$ Ok.. so we know $p \in X$. Your saying that we don't know if any $\gamma \in K_{p}$ is in the manifold $X$. So the most we can say (without any other Thrm's such as Whitney's) is that we can say that two curves are equivalent i.e. $\gamma_{1} \sim \gamma_{2}$ if $\dot{\gamma_{1}}(0)_{h} = \dot{\gamma_{2}}(0)_{h}$. So we just have an equivalence on the tangent space? So there is just a tangent plane at p in X such that each distinct vector on the tangent plane would represent a different equivalence class $[\gamma]$. I am confused as there are several "tangent spaces" e.g $T_{p}^{alg}X$ $\endgroup$ – Relative0 Jun 3 '13 at 13:43

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