I have been doing some reading on tori. What I can make out of it is that a torus can be equipped with different metrics -- locally Euclidean or as an embedded surface. It is said however that the torus with the locally Euclidean metric cannot be realized as an embedded surface. Why is this true and what is the metric as an embedded surface like? Why would we want the latter metric, since it seems to me the former is more natural?
Thanks.