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I'm work through some questions relating to connection coefficents. My question is more about the summation notation being used. Why is there no starting point (or end point) for the summation here? For the i, j, k, I take these to range from 1 to 2 for a 2 dimensional surface. Does this just imply that l does likewise?

$$ \Gamma^i_{jk} = \sum_{l} g^{il} \left( \dfrac{\partial g_{lk}}{\partial x^j} + \dfrac{\partial g_{jl}}{\partial x^k} - \dfrac{\partial g_{jk}}{\partial x^l} \right) $$

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  • $\begingroup$ $l$ probably runs through some index set, can't say much more without additional context $\endgroup$ Commented Apr 28, 2018 at 14:58
  • $\begingroup$ The sum should be over all $l$ which make sense, which needs to be interpreted in context. Without more context it is kind of hard to say, even if you know it is a surface: on an abstract surface, the sum would only be over $l = 1, 2$, but perhaps for a surface embedded in three dimensions, the sum is over $l = 1, 2, 3$. $\endgroup$
    – Joppy
    Commented Apr 28, 2018 at 14:58

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This formula comes from differential geometry, it takes place in the Cartesian coordinate space $\mathbb{R}^n$ for some value of the dimension $n$. Once $n$ has been specified, the indices $i,j,k,l$, by convention, take values in the index set $\{1,...,n\}$.

In this formula, the values $i,j,k$ are fixed and $l$ runs freely over the whole index set. So the summation symbol $\displaystyle\sum_l$, by convention, is an abbreviation for $$\sum_{l=1}^n $$ Whatever differential geometry text you read, there are likely to be hundreds of such summations, and there will be conventions for writing them in abbreviated forms.

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  • $\begingroup$ Thanks, that makes a lot of sense! I'd not seen that abbreviation in this book before, but your explanation helped a lot. The examples I've worked through were on spherical surfaces, so $ n = 2 $, but will no doubt be moving to higher values of $n$ later on. $\endgroup$
    – discojoe
    Commented Apr 29, 2018 at 11:16
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Another example of use of a similar notation which might make more sense to you is

$$\sum_{n\in \mathbb{N}}n=1+2+3+\dots$$

If the set is not specified directly under your sum, then $l$ probably runs over some set of values which make sense, maybe try reading back, that set was probably stated either explicitly or implicitly

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