Answer by Peter Michor for Functoriality of the cotangent bundle
I do not know a larger class of smooth mappings; and I considered this question intensively when co-writing the book "Natural operations in differential geometry, Springer-Verlag, 1993"(pdf).See also...
View ArticleFunctoriality of the cotangent bundle
Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle construction,...
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