Answer to Davilov

[Andree Ehresmann <ehres@u-picardie.fr>]

Davilov writes:

>Given an object M in the ``normal'' category of finitely dimensional
>smooth manifolds Man (not in SDG sense), what it the universal property
>of the tangent bundle TM?

Several authors have been interested in this problem many years ago, that
has led to the study of the functors from Man to the category of vector
bundles. Such functors are completely characterized in

        Epstein, "Natural vector bundles", Lecture Notes in Math. 99,
Springer 1969,         p. 171-195.

>From his theorems, it results in particular that Ehresmann's first order
velocity functors Tn which associate to a manifold M the vector bundle of
the 1-jets from R^n to M (in particular T for n=1) are the only such product
preserving functors.

This result is generalized in an abstract setting to characterize
connections in:
        Bowshell, "Abstract velocity functors", Cahiers de Topologie et
Geom. Diff.         XII-1 (1971), 57-82.

Later on, related (or more general) problems are considered in:

        Palais & Terng, "Natural bundles have finte order", Topology 16
(1977),       271-277,
        Epstein & Thurston, "Transformation groups and natural bundles",
Proc.  London Math. Soc. 38 (1979), 219-236.
        Eck, "Product preserving functors on smooth manifolds, J. Pure &
App. Algebra 42 (1986), 133-140.
        Kolar, An abstract characterization of the jet spaces, Cahiers de
Topologie et Geom. Diff. XXXIV-2 (1993), 121-125.
        Dupovec & Kolar, On the jets of fibered manifold morphisms, Cahiers
de Topologie et Geom. Diff. XXXIV-2 (1993), 121-125.
        
        Hoping these old references can be of some interest
                                        Best regards


                                                Andree C. Ehresmann