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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.

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You're right about exercise 2.3. This is one of the corrections posted on my webpage; you should download that list and keep it nearby when you read.

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.

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