A curve γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} lies on an oriented hypersurface S of M, we say that γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} is three-dimensional and closed. We show that if γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2π\documentclass[12pt]{minimal}
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\begin{document}$$2\pi $$\end{document}; and that, conversely, if the total torsion of γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} is an integer multiple of 2π\documentclass[12pt]{minimal}
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\begin{document}$$2\pi $$\end{document}, then there exists an oriented hypersurface of M in which γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.
