gt.geometric topology - Lipschitz orthonormal frames on submanifolds of $mathbf{R}^n$ ?

You will probably need to require $C^2$ smoothness of your submanifold. Take a simple example of $d=1$ and the manifold the graph of the function $f(x) = x^alpha$ for $x > 0$ and $f(x) = 0$ if $x le 0$. Then for $1 < alpha < 2$, the normal bundle is only Hölder continuous, so no $L$ exists at $x = 0$. Or would you exclude this counterexample for the reason that the exponential map from the normal bundle is not a diffeomorphism onto any ball of radius $r > 0$ at $x = 0$?

Generally, you will probably need to have a bound on the extrinsic curvature, for which $C^2$ and compactness would suffice.

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