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I have been working through John Lee's "Introduction to Smooth Manifolds" recently. I am having trouble visualizing the unit n-sphere as a manifold in its own right, i.e. without an ambient space. It seems impossible to visualize it without embedding it in Euclidean space. I mean, the unit sphere is defined as the set of points whose (Euclidean) distance from the origin (a point of [itex] \mathbb{R}^n [/itex] that is NOT on the sphere), is equal to 1. So it seems impossible/useless to speak about the sphere without embedding it first into Euclidean space. Could someone please clarify where I may be wrong in my thinking, and also elaborate on the issue for me? Thank you.