Gives a notion of distance along paths in a manifold, generalizing the Pythagorean theorem in Euclidean space. - Can be written as a symmetric matrix \begin{align*}\begin{pmatrix} g_{xx}&g_{xy}\\ g_{yx}&g_{yy}\end{pmatrix}\end{align*} where we then define the squared distance between two points as \begin{align*} \mathrm{d}s^2= g_{xx}\mathrm{d}x^2+g_{yy} \mathrm{d}y^2 + 2 g_{xy}\mathrm{d}x \mathrm{d}y= \sum_{\mu,\nu\in\{x,y\}}g_{\mu\nu}\mathrm{d}l^\mu \mathrm{d}l^\nu \end{align*} Can recover usual metric: by Pythagorean theorem, we have \begin{align*}\mathrm{d}s^2=\mathrm{d}x^2+\mathrm{d}y^2= \sum_{\mu,\nu\in\{x,y\}}g_{\mu\nu}\mathrm{d}l^\mu \mathrm{d}l^\nu\end{align*} and so we recover a “flat” metric \begin{align*} g_{\mu\nu}=\begin{pmatrix} g_{xx}&g_{xy}\\ g_{xy}&g_{yy}\end{pmatrix}= \begin{pmatrix} 1&0\\0&1\end{pmatrix} \end{align*}
This allows us to measure lengths of paths $\gamma$ by computing $$L = \int_\gamma \mathrm{d}s$$