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Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic. Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line? The second question: Assume $M$ is foliated with a family of non closed geodesics. Is there an isometric embedding of $M$ into $\mathbb{R}^{n}$ which send the family of geodesics to a familly of geodesics of $\mathbb{R}^{n}$?

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Concerning the first question, what about an irrational geodesic in a torus? Since it has accumulation points, it cannot be mapped isometrically to a straight line.

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