I am reading Shape of Space by Weeks. The author gives a philosophical comment:
Even if the universe connects up with itself in funny ways (if it's a three-torus, for example), this doesn't mean that it curves around in some four-dimensional space. The essence of the intrinsic point of view is that a manifold exists in and of itself, and needn't lie in any higher-dimensional space.
What exactly does this mean? I have learned what a differentiable manifold is in differential geometry, but I do not think this knowledge is necessary in understanding this comment. Sorry if this comment is too philosophical. I would appreciate intuitive answers.