Question

Please help me understand in simple terms what the Nash embedding theorem is and does.

Answer 1

the Nash embedding theorem explains how it is possible to embed isometrically a manifold (object containing various forms and components) in a Euclidean space by using differential calculus instead of algebra and geometry. Here "isometrically" means that the lengths of the curves are preserved. Note that this theorem is only how to embed Riemannian manifold which is a differentiable manifold where the tangent vectors about each point have an inner product defined in a way that allows a generalized study of distance and orthogonality.
Therefore this theorem states that any Riemannian manifold can be visualized as a submanifold of Euclidean space.

Answer 2

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