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.