3
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.