Create lower dimensional manifolds in a larger dimension

I’m very new to Topology. Suppose I want to create a 1-dim/2-dim/3-dim manifold, which is in a 20-dim space, and I will apply some dimension reduction, like PCA to their original dim. Is there anyway to do that? I went through the website and my current code is

struct ScaledSphere{N} <: AbstractDecoratorManifold{ManifoldsBase.ℝ} where {N}
ScaledSphere(radius, n) = ScaledSphere{n}(radius), M::ScaledSphere{n}) where {n} = print(io, "ScaledSphere($(M.radius),$n)")
S = ScaledSphere(1.5, 2)

I know this is 2-dim sphere, but how to deal with 20 dimensions?