Sorry, I meant an orthonormal frame for the normal bundle that is Lipschitz. Supposing the manifold is closed, When $d=1$ the manifold is $S^1$, so can't one just take the do parallel transport along the curve starting from a point $x$ all the way back and then use a constant constant-rate rotation to match up at $x$?

Sorry, I meant an orthonormal frame for the normal bundle that is Lipschitz. Supposing the manifold is closed, When $d=1$ the manifold is $S^1$, so can't one just take the parallel transport along the curve starting from a point $x$ all the way back and then use a constant rotation to match up at $x$?