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A time ago, I was asking if there exists an universal property of the tangent space and what it says about any construction of it. I've found the definition maded in Tammo Dieck's book of Algebraic Topology: A tangent space at $p$ is a vector space $T_pM$ such that any chart $(U,x)$ on $M$ induces a chart $(T_pM, \text{d}x_p)$ on the tangent space, where $\text{d}x_p\colon T_pM\to\mathbb{R}^n$ is an isomorphism and for any two charts $(U,x)$ and $(V,y)$ the coordinate change $\text{d}y_p\circ\text{d}x_p^{-1}$ is equals to the derivative of the coordinate change $y\circ x^{-1}$ at $x(p)$. For me, at least, this approach of defining the tangent space do justice to the idea of "the best linear approximation of a manifold near a point", since all coordinate system on $M$ has it's own approximation and the coordinate change is approximately the one of the charts on $M$.

My question is about the construction of the tangent space. I've already seen that the tangent space via curves satisfies all these properties, but I can't show the same for the tangent space via derivations at $p$. I ask for a proof that the tangent space defined via derivations also satisfies this universal property.

Alex
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Pauli
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1 Answers1

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Let $M$ be a manifold and $p\in M$. Define $$ T_pM\colon = Der_p(\mathcal C^\infty M)=\{D\colon\mathcal C^\infty M\to \mathbb R \vert D \text{ is derivative at }p\}. $$ Now we check that the universal property holds for $T_p M$ defined above. A chart $(U,x)$ of $M$ indeed induces an isomorphism via $$ \mathrm d x_p\colon T_pM \to \mathbb R^n,\,D\mapsto (D(x^1),\,D(x^2),\,\dots,D(x^n)). $$ Now consider two charts $(U,x),\,(V,y)$ on $M$ with $U\cap V \neq\emptyset$. For $v\in \mathbb R^n$ we have $(\mathrm d x_p)^{-1}(v)=D_v$ where $$ D_v(f)=D_{x(p)}(f\circ x^{-1})(v). $$ Thus $$ (\mathrm d y_p \circ (\mathrm d x_p)^{-1})(v)=\mathrm d y_p(D_v)=(D_v(y^i))_i=(D_{x(p)}(y^i\circ x^{-1})(v))_i=D_{x(p)}(y\circ x^{-1})(v) $$ which means that indeed $\mathrm d y_p \circ (\mathrm d x_p)^{-1}=D_{x(p)}(y\circ x^{-1})$.

Alex
  • 943