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Questions tagged [tangent-spaces]

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4
votes
0answers
127 views

Identification of the tangent space of a manifold and the tangent vectors to curves

I'm studying the different definitions of the tangent space for abstract manifolds, and I'm struggling to prove that these abstract concepts reduce to the classical ones when dealing with submanifolds ...
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2answers
71 views

If $(U,\varphi)$ is a coordinate chart around $p \in M$, where $M$ smooth manifold, then how does $\varphi$ induce coordinates on $T_p M$?

I am studying differential topology and I have some trouble understanding how coordinates are induced on the tangent space at any point. Let $M$ be an $n$-dimensional smooth manifold, and let $p \in ...
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vote
1answer
71 views

Lie bracket of canonical vectors on tangent space to a point on a manifold is zero.

Let M be a manifold and $T_p(M)$ be the tangent space at $p$, and $\phi$ a local chart around $p$. Let $$\left.\frac{\partial}{\partial\phi^1}\right|_{_p},\ \cdots\ ,\left.\frac{\partial}{\partial\...