View source for Natural transformation in differential geometry

The classical theory of differential-geometric objects was revisited from the functorial point of view by A. Nijenhuis, [[#References|[a3]]]. He defined a natural bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200601.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200602.png" />-dimensional manifolds as a [[Functor|functor]] transforming every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200603.png" />-dimensional [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200604.png" /> into a fibred manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200605.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200606.png" /> (cf. [[Fibred space|Fibred space]]) and every local [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200607.png" /> into a fibred manifold morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200608.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n1200609.png" />. 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, [[#References|[a1]]].

From this point of view, a geometric construction on the elements of one bundle of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006010.png" /> with values in the bundle of another functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006011.png" /> over the same base has the form of a natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006012.png" />. Moreover, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006014.png" />th order natural operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006015.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006016.png" /> (cf. [[Natural operator in differential geometry|Natural operator in differential geometry]]) are in bijection with the natural transformations of the bundle functor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006017.png" />th [[Jet|jet]] prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006019.png" />.

In the simplest case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006021.png" /> are two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006022.png" />th order natural bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006023.png" />-dimensional manifolds, the natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006024.png" /> are in bijection with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006025.png" />-equivariant mappings between their standard fibres, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006026.png" /> is the jet group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006027.png" /> in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006028.png" />. Several methods for finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006029.png" />-equivariant mappings in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006030.png" />-case are collected in [[#References|[a1]]]. If manifolds with an additional structure are studied, one has to consider the corresponding subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006031.png" />.

Many problems on finding natural transformations between geometrically interesting pairs of bundle functors are solved in [[#References|[a1]]]. Even a negative answer can be of geometric interest. For example, in [[#References|[a1]]] it is deduced that there is no natural equivalence between the iterated tangent functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006032.png" /> and the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006033.png" /> of the cotangent and the tangent functors. This implies that, unlike for the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006034.png" />, there is no natural [[Symplectic structure|symplectic structure]] on the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006035.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006036.png" />.

The complete description of all natural transformations between two product-preserving bundle functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006038.png" /> on the category of all manifolds and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006039.png" />-mappings was deduced in the framework of the theory of bundle functors determined by local algebras, which was established by A. Weil, [[#References|[a4]]] (cf. also [[Weil algebra|Weil algebra]]). Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006041.png" /> corresponds to a local algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006042.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006043.png" />, respectively, and all natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006044.png" /> are in bijection with the algebra homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n120/n120060/n12006045.png" />, see [[#References|[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, [[#References|[a2]]].

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