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When writing down a chart $\phi$ from an open subset $U$ of a differentiable manifold $M$ to ${\mathbb R}^n$, should I write $\phi=(x^1, \dots,x^n)$ or $\phi=(x_1, \dots, x_n)$? I have not been able to find a clear answer looking around, but I would like to learn the correct notation for coordinates.

Luca
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  • When stating and proving theorems, I use upper indices, because 95% of the time, it works out nicely with all the other notational conventions. When concretely working things out in lower dimensions I either use things like $(x,y,z),(r,\theta,\phi)$ or $(\xi,\eta,\zeta)$ or depending on my mood (read how many times I have to write exponents like $(x^2)^2$) I might then decide between upper/lower indices. – peek-a-boo Jan 20 '24 at 20:29

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There is no universal agreement on this. Here is what I do:

It all starts with tensors on an abstract vector space. Here is an explanation of my notation. This is consistent with the notation commonly used by applied mathematicians.

Turning to differential geometry, we start with the tangent space $V = T_xM$, which is an abstract vector space. According to the conventions described, we would write a basis of $V$ as a row matrix of tangent vectors, $$ \begin{bmatrix} e_1 & \cdots & e_n \end{bmatrix} $$ and its dual basis as a column matrix of $1$-forma $$ \begin{bmatrix} \epsilon^1 \\ \vdots \\ \epsilon^n \end{bmatrix}. $$ In particular, given coordinates $(x^1, \dots, x^n)$, the coordinate basis can be written as $$ \begin{bmatrix} \partial_1 & \cdots & \partial_n \end{bmatrix}, $$ and tangent vector $v$ can be written as $$ v = a^k\partial_k = \begin{bmatrix} \partial_1 & \cdots & \partial_n \end{bmatrix}\begin{bmatrix} a^1 \\ \vdots \\ a^n \end{bmatrix}. $$ The dual basis would then be written as a column matrix, so the indices should be superscripts: $$ \begin{bmatrix} dx^1 \\ \vdots \\ dx^n \end{bmatrix}. $$ For consistency, this means we should write the coordinates using superscripts.

This allows you, if you want, to write a lot of formulas involving the coefficients of tangent vectors and $1$-forms with respect to bases in terms of matrices and matrix multiplication without writing indices explicitly. This is consistent with how one translates between abstract notation and notation using coordinates or frames.

Deane
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