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$\newcommand{\Reals}{\mathbf{R}}$I wanted to check what the difference between $(ds)^2$ (line element) and $g$ (Riemannian metric) is.

Clearly $g_p$ is defined as $T_pM \times T_pM \to \Reals$.

http://sckavassalis.blogspot.com/2009/10/bad-language-metric-vs-metric-tensor-vs.html says that $(ds)^2$ is a quadratic function of one vector. $ds^2_p:T_pM \to \Reals$.

I've read other references that state that $(ds)^2$ is the same thing as $g$. Riemannian Metric Notation

What is the difference between $(ds)^2$ and $g$?

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Customarily, the distinction between a Riemannian metric $g$ and its line element $ds^{2}$ is the same as that between a positive-definite bilinear form $B$ and the associated quadratic form $Q(v) = B(v, v)$: The line element $ds^{2}$ is the restriction of the metric $g$ to the diagonal in $TM$ (the set of pairs $(v, v)$ of tangent vectors to $M$).

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