# confusion on exercises from LEE's Book on Riemannian Manifold

I am reading John Lee's "Riemannian Manifolds an Introduction to Curvature" . At page 15 exercise 2.3 asks to prove that there exist a smooth extension of a function defined on a embedded submanifold. The exercise is wrong for example we can take f(x)= 1/x defined on positive reals which is a embedded submanifold of $\mathbb{R}$ . I think for a global extension we need the submanifold to be closed. Though this does not affect the later development because connection on a Riemannian manifold depends on the local behaviour of the vector fields. Next at the page 157 I didn't get the definition of exterior angle which is to take the "oriented angle" on the closed interval [$- \pi, \pi$ ]. Shouldn't be [ $0, \pi$]. And then determine the sign where they preserve the orientation or not. Otherwise there will be a ambiguity in the definition as there will be two possible choice of the angle in the given interval. Please do clarify . I am struck from further progression.

As for the exterior angle, I really did mean $[-\pi,\pi]$. If the vertex is concave, the rotation from $\dot\gamma(a_i^-)$ to $\dot\gamma(a_i^+)$ will be counterclockwise.