Let $X$ and $Y$ be smooth manifolds. If all smooth maps from $Y$ to $X$ are homotopic, then show that the identity map on $X$ is homotopic to some constant map(i.e that $X$ is contractible).
I have already proved the converse of this. The strategy there was to show that any arbitrary smooth map from $Y$ to $X$ are homotopic to the constant map, hence they are all homotopic to each other as homotopy is an equivalence relation. But, what strategy is to be used to prove the converse? Any hints/suggestions would be appreciated?
This is not true, take $Y$ to be a point, all maps $Y\rightarrow X$ are homotopic if $X$ is (path) connected, but $X$ is not necessarily contractible.