Natural transformation in differential geometry

The classical theory of differential-geometric objects was revisited from the functorial point of view by A. Nijenhuis, [a3]. He defined a natural bundle over -dimensional manifolds as a functor transforming every -dimensional manifold into a fibred manifold over (cf. Fibred space) and every local diffeomorphism into a fibred manifold morphism over . Later it was taken into consideration that certain geometric objects can be constructed on certain special types of manifolds only. This led to an analogous concept of bundle functor on a category over manifolds, [a1].

From this point of view, a geometric construction on the elements of one bundle of a functor with values in the bundle of another functor over the same base has the form of a natural transformation . Moreover, the th order natural operators of into (cf. Natural operator in differential geometry) are in bijection with the natural transformations of the bundle functor of the th jet prolongation into .

In the simplest case, if and are two th order natural bundles over -dimensional manifolds, the natural transformations are in bijection with the -equivariant mappings between their standard fibres, where is the jet group of order in dimension . Several methods for finding -equivariant mappings in the -case are collected in [a1]. If manifolds with an additional structure are studied, one has to consider the corresponding subgroup of .

Many problems on finding natural transformations between geometrically interesting pairs of bundle functors are solved in [a1]. Even a negative answer can be of geometric interest. For example, in [a1] it is deduced that there is no natural equivalence between the iterated tangent functor and the composition of the cotangent and the tangent functors. This implies that, unlike for the cotangent bundle , there is no natural symplectic structure on the tangent bundle of a manifold .

The complete description of all natural transformations between two product-preserving bundle functors and on the category of all manifolds and all -mappings was deduced in the framework of the theory of bundle functors determined by local algebras, which was established by A. Weil, [a4] (cf. also Weil algebra). Each or corresponds to a local algebra or , respectively, and all natural transformations are in bijection with the algebra homomorphisms , see [a1] for a survey. An analogous characterization of all natural transformations between two product preserving bundle functors on the category of fibred manifolds was deduced by W. Mikulski, [a2].

References

[a1]	I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993)
[a2]	W. Mikulski, "Product preserving bundle functors on fibered manifolds" Archivum Math. (Brno) , 32 (1996) pp. 307–316
[a3]	A. Nijenhuis, "Natural bundles and their general properties" , Diff. Geom. in Honor of K. Yano , Kinokuniya (1972) pp. 317–334
[a4]	A. Weil, "Théorie des points proches sur les variétés différentiables" Colloq. C.N.R.S., Strasbourg (1953) pp. 111–117