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 $F$ over $m$-dimensional manifolds as a functor transforming every $m$-dimensional manifold $M$ into a fibred manifold $F M \rightarrow M$ over $M$ (cf. Fibred space) and every local diffeomorphism $f : M \rightarrow N$ into a fibred manifold morphism $F f : F M \rightarrow F N$ over $f$. 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 $F$ with values in the bundle of another functor $G$ over the same base has the form of a natural transformation $F \rightarrow G$. Moreover, the $k$th order natural operators of $F$ into $G$ (cf. Natural operator in differential geometry) are in bijection with the natural transformations of the bundle functor of the $k$th jet prolongation $J ^ { k } F$ into $G$.

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

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 $T T$ and the composition $T ^ { * } T$ of the cotangent and the tangent functors. This implies that, unlike for the cotangent bundle $T ^ { * } M$, there is no natural symplectic structure on the tangent bundle $T M$ of a manifold $M$.

The complete description of all natural transformations between two product-preserving bundle functors $F$ and $G$ on the category of all manifolds and all $C ^ { \infty }$-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 $F$ or $G$ corresponds to a local algebra $A$ or $B$, respectively, and all natural transformations $F \rightarrow G$ are in bijection with the algebra homomorphisms $A \rightarrow B$, 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