Length of Curve Segment

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Given a Riemannian manifold \((M,g)\). Then the length of a curve segment \(\gamma:[a,b]\to M\) is given by

\begin{equation*} L_g(\gamma)=\int_{a}^{b} \lvert \gamma'(t)\rvert_g dt, \end{equation*}

where \(\gamma'(t)\in T_{\gamma(t)}M\) is the velocity vector of \(\gamma\) at \(t\in [a,b]\).