Immersions and embeddings

We can generalize and make precise the concept of a surface embedded in 3-dimensional space with the following definitions concerning a differentiable map \({\Phi\colon M^{m}\to N^{n}}\):

  • Immersion: \({\mathrm{d}\Phi}\) is injective for all \({p\in M}\); intuitively, a smooth mapping that doesn’t collapse the tangent spaces
  • Submanifold: an immersion with \({\Phi}\) injective; intuitively, an immersion that doesn’t intersect itself
  • Embedding (AKA imbedding): a submanifold with \({\Phi}\) a homeomorphism onto \({\Phi\left(M\right)}\); intuitively, a submanifold that doesn’t have intersecting limit points

40.immersions

The above depicts \({\mathbb{R}}\) immersed in \({\mathbb{R}^{2}}\); the second immersion approaches a self-intersection in the limit as the line approaches infinity.

The difference in dimension \({\left(n-m\right)}\) is called the codimension of the embedding. The Whitney embedding theorem states that any \({M^{m}}\) can be immersed in \({\mathbb{R}{}^{(2m-1)}}\) and embedded in \({\mathbb{R}^{2m}}\). Thus we can view differentiable manifolds as generalized surfaces that we study without making reference to the enclosing Euclidean space. The limiting dimension of this theorem is illustrated by noting that the real projective space \({\mathbb{R}\textrm{P}^{m}}\) cannot be embedded in \({\mathbb{R}^{(2m-1)}}\).

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