Smooth homotopy

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Definition

Suppose M,N are differential manifolds, and f,g: M \to N are smooth maps. A smoooth homotopy from f to g is a smooth map from M \times I (viewed with the product manifold structure) to N such that F(x,0) = f(x) and F(x,1) = g(x) for all x.

In other words, a smooth homotopy is a homotopy from f to g (in the topological sense) which is also a smooth map when viewed with the additional structure of a manifold.