Orthogonal trajectories on an $n$-manifold embedded in $\mathbb{R}^n$

I am trying to show that there exist unique orthogonal trajectories on the $n$-manifold $L_{\lambda} \subseteq \mathbb{R}^n$ that pass through a given point. Here $L_{\lambda}$ is given by points $(x,y) \in \mathbb{R}^{\lambda} \times \mathbb{R}^{n-\lambda}$ such that $-1 \leq -|x|^2+|y|^2 \leq 1$ and $|x||y| < (\cosh 1)(\sinh 1)$. I think doing this is possible by reducing the problem to the $2$-dimensional case, but I can't quite get the formalism to work out. Any suggestions?

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 Uh... (1) An $n$-manifold embedded in $\mathbb{R}^n$ is just an open set, no? (2) To what do you want your orthogonal trajectories to be orthogonal? – Willie Wong Sep 6 at 12:29