A well-known lemma in Riemannian geometry by Klingenberg says that if x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document} is a minimum point of the distance function d(p,·)\documentclass[12pt]{minimal}
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\begin{document}$$d(p,\cdot )$$\end{document} to p in the cut locus Cp\documentclass[12pt]{minimal}
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\begin{document}$$C_p$$\end{document} of p, then either there is a minimal geodesic from p to x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document} along which they are conjugate, or there is a geodesic loop at p that smoothly goes through x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document}. In this paper, we prove that: for any point q and any local minimum point x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document} of Fq(·)=d(p,·)+d(q,·)\documentclass[12pt]{minimal}
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\begin{document}$$F_q(\cdot )=d(p,\cdot )+d(q,\cdot )$$\end{document} in Cp\documentclass[12pt]{minimal}
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\begin{document}$$C_p$$\end{document}, either x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document} is conjugate to p along each minimal geodesic connecting them, or there is a geodesic from p to q passing through x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document}. In particular, for any local minimum point x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document} of d(p,·)\documentclass[12pt]{minimal}
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\begin{document}$$d(p,\cdot )$$\end{document} in Cp\documentclass[12pt]{minimal}
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\begin{document}$$C_p$$\end{document}, either p and x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document} are conjugate along every minimal geodesic from p to x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document}, or there is a geodesic loop at p that smoothly goes through x0\documentclass[12pt]{minimal}
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\begin{document}$$x_0$$\end{document}. Earlier results based on injectivity radius estimate would hold under weaker conditions.
