engleski

Immersions and Embeddings

All smooth manifolds can be smoothly embedded in Euclidean spaces. This justifies our habit of visualizing manifolds as subsets of R^n. To prove this, we use analysis in R^n to construct a 'local' solution, and then use partitions of unity to piece together the local solutions into a global one. First we show how any smooth map into R^n can be perturbed slightly to be an immersion, and then show how to perturb the immersion to be injective.

smooth manifold; immersion; embedding; embedded submanifold

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