# Tag Info

• 83k
Accepted

### Coordinate patches attached to a 2-manifold

These coordinate patches as seen in the picture are open sets $U_j\subset M$ subordinate to the atlas $(\varphi_i,U_j)$ of the manifold $M$ equipped with a coordinate grid that is induced by the ...
• 576
Accepted

### Utility of the coordinate free definition of the derivative on manifolds.

The importance comes in when you replace your spaces by manifolds which are not (open subsets of) $\mathbb{R}^n$. Then, you are by definition of a manifold always able to choose a local coordinate ...
• 1,086

### Utility of the coordinate free definition of the derivative on manifolds.

In addition to nictor000's answer, let me address your quarrels with Property 1. Indeed, there is no canonical map $T_pM\to M$, but there is something that takes its place. For any $v\in T_pM$ there ...
• 5,568
Accepted

• 2,781
1 vote
Accepted

### Isotropy subgroup of $SL(2,\mathbb{C})$ action.

We can also do this by using the extended complex plane for $\mathbb{P}^1$ and Mobius transformations $(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix})z=\frac{az+b}{cz+d}$. We are ...
• 14.6k
1 vote

### Metric tensor for resulting manifold defined at coordinate $x$ of n-dimensional manifold $\mathcal{M}$

Let $ds^2$ be the Riemannian metric on $M$ (which is an $n$-dimensional manifold). Then for the map $F(x)=(x,f(x))$, the pull-back $F^*(ds^2 + dt^2)$ of the product metric $ds^2 + dt^2$ on \$M\times {\...
• 83k

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