Smooth homotopy

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.