# Tag Info

Accepted

### Why aren't tangent spaces simply defined as vector spaces with same dimension as the manifold?

We don't just want to have a vector space to call the "tangent space". We want to do geometric things with the tangent space, and we can't do those things if it's just an arbitrary vector space of ...
• 329k
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### Lie algebra: intuition of "Lie Algebra is tangent space of corresponding Lie Group"?

Intuitively, you can think of the tangent space of a surface at a point as the space of all "directions" you can move from that point (with all velocities) while staying on the surface. That ...
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### Why are the partial derivatives a basis of the tangent space?

To distinguish between points and tangent vectors, let $p=(p_1,...,p_n)\in \mathbb{R}^n$ a point of $\mathbb{R}^n$ and $v=(v_1,...,v_n)\in T_p(\mathbb{R}^n)$ a point of the tangent space $\mathbb{R}^n$...

### Why aren't tangent spaces simply defined as vector spaces with same dimension as the manifold?

If they are “just vector spaces” at each point, then every set would be a manifold. The whole point is to link the topological structure of the manifold (using coordinate patches that fit together ...
• 153k
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• 54.2k
Accepted

### Understanding vector field(s) on $\mathbb{S}^3$.

$S^3$ is often defined as a subset (submanifold) of $\Bbb{R}^4$, so at a point $p\in S^3\subset \Bbb{R}^4$, the tangent space $T_pS^3$ is a subset of $T_p\Bbb{R}^4$ (or atleast there is a very natural ...
• 54.2k

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