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Natural affinors on the extended r -th order tangent bundles

Gancarzewicz, Jacek, Kolář, Ivan (1993)

Proceedings of the Winter School "Geometry and Physics"

The authors prove that all natural affinors (i.e. tensor fields of type (1,1) on the extended r -th order tangent bundle E r M over a manifold M ) are linear combinations (the coefficients of which are smooth functions on ) of four natural affinors defined in this work.

Natural lifting of connections to vertical bundles

Kolář, Ivan, Mikulski, Włodzimierz M. (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

One studies the flow prolongation of projectable vector fields with respect to a bundle functor of order ( r , s , q ) on the category of fibered manifolds. As a result, one constructs an operator transforming connections on a fibered manifold Y into connections on an arbitrary vertical bundle over Y . It is deduced that this operator is the only natural one of finite order and one presents a condition on vertical bundles over Y under which every natural operator in question has finite order.

Natural liftings of foliations to the r -tangent bunde

Mikulski, Włodzimierz M. (1994)

Proceedings of the Winter School "Geometry and Physics"

Let F be a p -dimensional foliation on an n -manifold M , and T r M the r -tangent bundle of M . The purpose of this paper is to present some reltionship between the foliation F and a natural lifting of F to the bundle T r M . Let L q r ( F ) ( q = 0 ,...

Natural operations of Hamiltonian type on the cotangent bundle

Doupovec, Miroslav, Kurek, Jan (1997)

Proceedings of the 16th Winter School "Geometry and Physics"

The authors study some geometrical constructions on the cotangent bundle T * M from the viewpoint of natural operations. First they deduce that all natural operators transforming functions on T * M into vector fields on T * M are linearly generated by the Hamiltonian vector field with respect to the canonical symplectic structure of T * M and by the Liouville vector field of T * M . Then they determine all natural operators transforming pairs of functions on T * M into functions on T * M . In this case, the main generator is...

Natural operators between vector valued differential forms

Cap, Andreas (1991)

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0742.00067.]This paper is devoted to a method permitting to determine explicitly all multilinear natural operators between vector-valued differential forms and between sections of several other natural vector bundles.

Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles

Mikulski, W. M. (1996)

Proceedings of the 15th Winter School "Geometry and Physics"

The author studies the problem how a map L : M on an n -dimensional manifold M can induce canonically a map A M ( L ) : T * T ( r ) M for r a fixed natural number. He proves the following result: “Let A : T ( 0 , 0 ) T ( 0 , 0 ) ( T * T ( r ) ) be a natural operator for n -manifolds. If n 3 then there exists a uniquely determined smooth map H : S ( r ) × such that A = A ( H ) .”The conclusion is that all natural functions on T * T ( r ) for n -manifolds ( n 3 ) are of the form { H ( λ M 0 , 1 , , λ M r , 0 ) } , where H C ( r ) is a function of r variables.

Natural operators lifting vector fields on manifolds to the bundles of covelocities

Mikulski, W. M. (1996)

Proceedings of the Winter School "Geometry and Physics"

The author proves that for a manifold M of dimension greater than 2 the sets of all natural operators T M ( T k r * M , T q * M ) and T M T T k r * M , respectively, are free finitely generated C ( ( k ) r ) -modules. The space T k r * M = J r ( M , k ) 0 , this is, jets with target 0 of maps from M to k , is called the space of all ( k , r ) -covelocities on M . 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,...

Natural operators on frame bundles

Krupka, Michal (2000)

Proceedings of the 19th Winter School "Geometry and Physics"

Let F 1 be a natural bundle of order r 1 ; a basis of the s -th order differential operators of F 1 with values in r 2 -th order bundles is an operator D of that type such that any other one is obtained by composing D with a suitable zero-order operator. In this article a basis is found in the following two cases: for F 1 = semi F r 1 (semi-holonomic r 1 -th order frame bundle), s = 0 , r 2 < r 1 and F 1 = F 1 ( 1 -st order frame bundle), r 2 s . The author uses here the so-called method of orbit reduction which provides one with a criterion for checking...

Natural operators transforming projectable vector fields to product preserving bundles

Tomáš, Jiří (1999)

Proceedings of the 18th Winter School "Geometry and Physics"

Let Y M be a fibered manifold over a manifold M and μ : A B be a homomorphism between Weil algebras A and B . Using the results of Mikulski and others, which classify product preserving bundle functors on the category of fibered manifolds, the author classifies all natural operators T proj Y T μ Y , where T proj Y denotes the space of projective vector fields on Y and T μ the bundle functors associated with μ .

Natural symplectic structures on the tangent bundle of a space-time

Janyška, Josef (1996)

Proceedings of the 15th Winter School "Geometry and Physics"

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.

Natural transformations of Weil functors into bundle functors

Mikulski, Włodzimierz M. (1990)

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor T A 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 T A into F is finite and is less than or equal to dim ( F 0 k ) . The spaces of all natural transformations of Weil functors into linear...

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