Problems in do Carmo's Riemannian Geometry I am reading do Carmo's Riemannian Geometry Chapter 7, and I want to do some exercises.
I think that I need some hints to solve the following:
Questions:


*

*How do I construct a counterexample that a local diffeomorphism does not presreve completeness?

*I want to show that any homogeneous manifold is complete, but I have no idea about the property "homogeneous".
Please give me an answer or some hints. 
 A: Your question is incomplete. Please be more precise next time. The problem is:
Let $f: M_1\rightarrow M_2$ be a local diffeomorphism of a manifold $M_1$ onto a Riemannian manifold $M_2$. Introduce on $M_1$ a Riemannian metric such that $f$ is a local isometry. Show by an example that if $M_2$ is complete, $M_1$ need not be complete.
Take $M_2$ to be the circle, $S^1$ in $\mathbb{R}^2$ with the usual Euclidean norm. Take $M_1 = (0,2)$. Wrap $(0,2)$ around $S^1$ twice (your usual covering map: $f: x\mapsto e^{2\pi i x}$), and pull back the metric from $S^1$ onto $(0,2)$. This will give you a local isometry, but $(0,2)$ with respect to this metric is not complete. The trick here is that $f$ is not one-to-one, so even though $(0,2)$ is not complete, there are enough points to cover $S^1$ and do so locally isometrically.
In the second question, a Riemannian manifold $M$ is homogeneous provided that for any pair $p, q\in M$, there exists an isometry of $M$ mapping $p$ to $q$. 
Let me give you a hint here: try using the Hopf-Rinow theorem, which is covered in do Carmo, I think in the same chapter. 
