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Let be a -dimensional foliation on an -manifold , and the -tangent bundle of . The purpose of this paper is to present some reltionship between the foliation and a natural lifting of to the bundle . Let
A classification of natural liftings of foliations to the tangent bundle is given.
The author studies the problem how a map on an -dimensional manifold can induce canonically a map for a fixed natural number. He proves the following result: “Let be a natural operator for -manifolds. If then there exists a uniquely determined smooth map such that .”The conclusion is that all natural functions on for -manifolds are of the form , where is a function of variables.
The author proves that for a manifold of dimension greater than 2 the sets of all natural operators and , respectively, are free finitely generated -modules. The space , this is, jets with target 0 of maps from to , is called the space of all -covelocities on . Examples of such operators are shown and the bases of the modules are explicitly constructed. The definitions and methods are those of the book of I. Kolář, P. W. Michor and J. Slovák [Natural operations in differential geometry,...
In this nice paper the author proves that all natural symplectic forms on the tangent bundle of a pseudo-Riemannian manifold are pull-backs of the canonical symplectic form on the cotangent bundle with respect to some diffeomorphisms which are naturally induced by the metric.
[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor of A-velocities [I. Kolař, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of into F is finite and is less than or equal to . The spaces of all natural transformations of Weil functors into linear...
For each graph G, we define a chain complex of graded modules over the ring of polynomials whose graded Euler characteristic is equal to the chromatic polynomial of G. Furthermore, we define a chain complex of doubly-graded modules whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of G. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorifications of the HOMFLYPT polynomial. We also give a simplified definition of this triply-graded...
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