TITLE: Two questions on isometric embedding
QUESTION [1 upvotes]: According to the answer of the following question, I try a new version:
An special isometric embedding

Let $M$ be a Riemannian manifold and $\gamma$ be a small part of a geodesic.
Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which sends $\gamma$ into a small part of a straight line?
Assume we have a foliation of the upper half plane $\mathbb{H}^{2}$ by geodesics. Is there an embedding of $\mathbb{H}^{2}$ into some $\mathbb{R}^{n}$ which sends the family of geodesics to a family of straight lines?

REPLY [3 votes]: I don't know the answer to question #1, especially if you want a global embedding of the manifold. But if you'll settle for an isometric embedding in a neighborhood of a geodesic segment, here is one possible way to do it (I don't know if it works or not):
1) First, construct an embedding that is isometric up to infinite order along the geodesic and maps it into a straight line. This is straightforward, if you know the requisite theorems or tricks. One is to use the Cartan-Kähler theorem. The other is to use induction over the dimension of the submanifold and use Cauchy-Kovalevski to extend the near-isometric embedding one dimension at a time.
2) Now you want to use the Nash-Moser implicit function theorem to deform this embedding that is approximately isometric near the geodesic into one that is isometric near the geodesic. To do this, you need to invert the linearized operator and apply a smoothing operator and verify that you can do this without moving the geodesic.
3) If the Euclidean space has enough dimensions, the linearized operator is just a pointwise underdetermined system of linear algebraic equations (no PDE's). So what you want to do is concoct some additional linear constraints on the linearized solution that force the solution to be zero along the geodesic.
4) Finally, you need to show that there exists a 1-parameter family of smoothing operators on functions that converge to the identity and that are equal to the identity along the geodesic.
I don't have a copy handy, but I vaguely remember that maybe Gromov does things similar to this in his book Partial Differential Relations.
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