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On formal theory of differential equations. III.

Jan Chrastina (1991)

Mathematica Bohemica

Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy...

On Galilean connections and the first jet bundle

James Grant, Bradley Lackey (2012)

Open Mathematics

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order...

On geometry of curves of flags of constant type

Boris Doubrov, Igor Zelenko (2012)

Open Mathematics

We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope...

On infinitesimal automorphisms of foliated manifolds

Jan Kurek, Włodzimierz M. Mikulski (2007)

Annales Polonici Mathematici

Let F:ℱol → ℱℳ be a product preserving bundle functor on the category ℱol of foliated manifolds (M,ℱ) without singularities and leaf respecting maps. We describe all natural operators C transforming infinitesimal automorphisms X ∈ 𝒳(M,ℱ) of foliated manifolds (M,ℱ) into vector fields C(X)∈ 𝒳(F(M,ℱ)) on F(M,ℱ).

On invariant operations on pseudo-Riemannian manifolds

Jan Slovák (1992)

Commentationes Mathematicae Universitatis Carolinae

Invariant polynomial operators on Riemannian manifolds are well understood and the knowledge of full lists of them becomes an effective tool in Riemannian geometry, [Atiyah, Bott, Patodi, 73] is a very good example. The present short paper is in fact a continuation of [Slovák, 92] where the classification problem is reconsidered under very mild assumptions and still complete classification results are derived even in some non-linear situations. Therefore, we neither repeat the detailed exposition...

On involutions of iterated bundle functors

Miroslav Doupovec, Włodzimierz M. Mikulski (2006)

Colloquium Mathematicae

We introduce the concept of an involution of iterated bundle functors. Then we study the problem of the existence of an involution for bundle functors defined on the category of fibered manifolds with m-dimensional bases and of fibered manifold morphisms covering local diffeomorphisms. We also apply our results to prolongation of connections.

On iteration of higher order jets and prolongation of connections

Miroslav Doupovec, Włodzimierz M. Mikulski (2011)

Annales Polonici Mathematici

We introduce exchange natural equivalences of iterated nonholonomic, holonomic and semiholonomic jet functors, depending on a classical linear connection on the base manifold. We also classify some natural transformations of this type. As an application we introduce prolongation of higher order connections to jet bundles.

On jets of surfaces.

Fernando Etayo Gordejuela (1991)

Collectanea Mathematica

We study the 2-jet bundle of mappings of the real plane into a manifold. We shall prove that there exists an imbedding of this 2-jet bundle into a suitable first order jet bundle, in such a way that its image is the set of fixed points of a canonical automorphism of the biggest jet bundle.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

We prove that the problem of finding all ℳ f m -natural operators B : Q ⟿ Q T A lifting classical linear connections ∇ on m-manifolds M to classical linear connections B M ( ∇ ) on the Weil bundle T A M corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all ℳ f m -natural operators C : Q ⟿ ( T ¹ p - 1 , T * ⊗ T * ⊗ T ) transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps C M ( ∇ ) : T ¹ p - 1 M = ⨁ M p - 1 T M → T * M ⊗ T * M ⊗ T M .

On lifts of projectable-projectable classical linear connections to the cotangent bundle

Anna Bednarska (2013)

Annales UMCS, Mathematica

We describe all F2Mm1,m2,n1,n2-natural operators D: Qτproj-prj ↝QT* transforming projectable-projectable classical torsion-free linear connections ∇ on fibred-fibred manifolds Y into classical linear connections D(∇) on cotangent bundles T*Y of Y . We show that this problem can be reduced to finding F2Mm1,m2,n1,n2-natural operators D: Qτproj-proj ↝ (T*,⊗pT*⊗⊗qT) for p = 2, q = 1 and p = 3, q = 0.

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