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I have seen on wikipedia that an affine space in $\mathbb{R}^n$, with $V$ as its vector space of translation, is a smooth manifold but don't know the explanation. I have to know about the tangent spaces at a point of an affine space.

Can someone explain me little bit about its manifold structure and its tangent spaces.

Can its tangent spaces be identified with the tangent spaces of $V$?

Thanks in advance!

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A smooth manifold is just a second countable Hausdorff topological space with a smooth atlas. Since translation in $\mathbb{R}^n$ is a homeomorphism, an affine space $\tau + V \subset \mathbb{R}^n$ for $\tau \in \mathbb{R}^n$ and $V$ a $k$-dimensional linear subspace of $\mathbb{R}^n$ is naturally homeomorphic to $\mathbb{R}^k \cong V \subset \mathbb{R}^n$. So $\tau + V$ is a second countable Hausdorff topological space for the induced topology and it only remains to put a smooth manifold structure on it, i.e. a smooth atlas. Since $\tau + V$ is globally homeomorphic to $\mathbb{R}^k$, we only need one chart $(\tau + V, \varphi)$ for the atlas, for example translation by $-\tau$: $$\varphi : \tau + V \to \mathbb{R}^k$$ defined by $\varphi(w) = w - \tau \in V$­. Since there is only one chart, smooth compatibility is automatically satisfied and this gives $\tau + V$ a smooth manifold structure, with translation by $-\tau$ a diffeomorphism between $\tau + V$ and $V \cong \mathbb{R}^k$.

Now since by construction translating by $-\tau$ is a diffeomorphism between $\tau + V$ and $V$, we can naturally identify their tangent spaces by the differential of this translation. Concretely, identifying the tangent spaces of $\mathbb{R}^n$ with $\mathbb{R}^n$ itself as usual, if $\gamma$ is a curve in $\tau + V$ with $\gamma(0) = \tau + v$ and tangent vector $\dot{\gamma}(0) = u \in T_{\tau + v}(\tau + V)$, then u gets identified to the tangent vector at $0$ of the translated curve $\gamma(t) - \tau$ which is $(d/dt)(\gamma(t) - \tau)|_{t=0} = \dot{\gamma}(0) = u$.

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