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Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold

Josef Janyška (2001)

Archivum Mathematicum

Let M be a differentiable manifold with a pseudo-Riemannian metric g and a linear symmetric connection K . We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on T M generated by g and K . We get that all natural vector fields are of the form E ( u ) = α ( h ( u ) ) u H + β ( h ( u ) ) u V , where u V is the vertical lift of u T x M , u H is the horizontal lift of u with respect to K , h ( u ) = 1 / 2 g ( u , u ) and α , β are smooth real functions defined on R . All natural 2-vector fields are of the form Λ ( u ) = γ 1 ( h ( u ) ) Λ ( g , K ) + γ 2 ( h ( u ) ) u H u V , where γ 1 , γ 2 are smooth real functions defined...

Non-existence of some canonical constructions on connections

Włodzimierz M. Mikulski (2003)

Commentationes Mathematicae Universitatis Carolinae

For a vector bundle functor H : f 𝒱 with the point property we prove that H is product preserving if and only if for any m and n there is an m , n -natural operator D transforming connections Γ on ( m , n ) -dimensional fibered manifolds p : Y M into connections D ( Γ ) on H p : H Y H M . For a bundle functor E : m , n with some weak conditions we prove non-existence of m , n -natural operators D transforming connections Γ on ( m , n ) -dimensional fibered manifolds Y M into connections D ( Γ ) on E Y M .

Non-existence of some natural operators on connections

W. M. Mikulski (2003)

Annales Polonici Mathematici

Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators C r Q ( r e g T k r K k r ) and C r Q ( r e g T k r * K k r * ) over n-manifolds is proved. Some generalizations are obtained.

Non-holonomic ( r , s , q ) -jets

Jiří M. Tomáš (2006)

Czechoslovak Mathematical Journal

We generalize the concept of an ( r , s , q ) -jet to the concept of a non-holonomic ( r , s , q ) -jet. We define the composition of such objects and introduce a bundle functor J ˜ r , s , q k , l × defined on the product category of ( k , l ) -dimensional fibered manifolds with local fibered isomorphisms and the category of fibered manifolds with fibered maps. We give the description of such functors from the point of view of the theory of Weil functors. Further, we introduce a bundle functor J ˜ 1 r , s , q 2 - k , l defined on the category of 2 -fibered manifolds with k , l -underlying...

On ( 1 , 1 ) -tensor fields on symplectic manifolds

Anton Dekrét (1999)

Archivum Mathematicum

Two symplectic structures on a manifold M determine a (1,1)-tensor field on M . In this paper we study some properties of this field. Conversely, if A is (1,1)-tensor field on a symplectic manifold ( M , ω ) then using the natural lift theory we find conditions under which ω A , ω A ( X , Y ) = ω ( A X , Y ) , is symplectic.

On a generalization of Helmholtz conditions

Radka Malíková (2009)

Acta Mathematica Universitatis Ostraviensis

Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ( n = 1 ), and obtain a generalization of Helmholtz conditions to this case.

On curves and jets of curves on supermanifolds

Andrew James Bruce (2014)

Archivum Mathematicum

In this paper we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.

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