All of the information about the nature of the manifold is contained in the metric ˆ𝐠. Thus: even if we don’t have (or know) the embedding, we can still work with the manifold if we know ˆ𝐠.
a) the topology is not determined by the metric b) this fact is highly non obvious and if I didnt already know some version of this it would probably just confuse me on a first reading