The metric of a Riemannian manifold $M^n$ is called locally Euclidean (l.E.) if for any point $p \\\\\\\\in M^n$ there exists a neighbourhood $U_p \\\\\\\\subset M^n$ isometric to a ball in the Euclidean space $E^n$ with the standart metric.
About manifolds and surfaces with l.E. metrics one can put many questions among which we will single the following ones:

1. Let $ds^2 = g_{ij}du^idu^j$, ($1 \\\\\\\\leq i,j \\\\\\\\leq n$) (1) be a given l.E. metric. How to find that local isometry between this metric and a ball in En the existence of which is guaranteed by the definition of a l.E. metric? What can we say about the smoothness of such an isometry?
2. How to verify is a given metric (1) locally Euclidean or not?
3. What are criteria for the existence of a global isometric immersion or embedding of a given n-dimensional l.E. metric in En?
4. What can we say about the existence/non-existence of isometric immersions and embeddings of given n-dimensional l.E. metrics in N-dimensional spaces ($N > n$) with constant curvature?
5. What is known about the local and global structure of surfaces with l.E. metrics?
6. Bendings and infinitesimal bendings of surfaces with l.E. metrics.