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The length metrics used in physics are usually a squared norm, like the following: $$ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu}. \tag{1}$$

What other kinds of continuous metrics could we define? Why not the following multi-linear candidates? \begin{align} ds &= g_{\mu} \, dx^{\mu}, \tag{2} \\[1ex] ds^3 &= g_{\mu \nu \lambda} \, dx^{\mu} \, dx^{\nu} \, dx^{\lambda}, \tag{3} \\[1ex] \vdots \\[1ex] ds^n &= g_{\mu_1 \mu_2 \mu_3 \dots \mu_n} \, dx^{\mu_1} \, dx^{\mu_2} \, dx^{\mu_3} \dots dx^{\mu_n}. \tag{4} \end{align} What are the basic rules that "forces" physics to use the "squared" metric (1)?

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