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 |
Natural transformation in differential geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_transformation_in_differential_geometry&oldid=13906