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Canonical 1-forms on higher order adapted frame bundles

Jan Kurek, Włodzimierz M. Mikulski (2008)

Archivum Mathematicum

Let $(M, ℱ)$ be a foliated $m + n$ -dimensional manifold $M$ with $n$ -dimensional foliation $ℱ$ . Let $V$ be a finite dimensional vector space over $𝐑$ . We describe all canonical ( $ℱ {ol}_{m, n}$ -invariant) $V$ -valued $1$ -forms $Θ : T P^{r} (M, ℱ) \to V$ on the $r$ -th order adapted frame bundle $P^{r} (M, ℱ)$ of $(M, ℱ)$ .

Canonical nonlinear connections in the multi-time Hamilton geometry.

Atanasiu, Gheorghe, Neagu, Mircea (2009)

Balkan Journal of Geometry and its Applications (BJGA)

Canonical symplectic structures on the r-th order tangent bundle of a symplectic manifold.

Jan Kurek, Wlodzimierz M. Mikulski (2006)

Extracta Mathematicae

We describe all canonical 2-forms Λ(ω) on the r-th order tangent bundle TrM = Jr0 (R;M) of a symplectic manifold (M, ω). As a corollary we deduce that all canonical symplectic structures Λ(ω) on TrM over a symplectic manifold (M, ω) are of the form Λ(ω) = Σrk=0 αkω(k) for all real numbers αk with αr ≠ 0, where ω(k) is the (k)-lift (in the sense of A. Morimoto) of ω to TrM.

Canonical vector valued 1-forms on higher order adapted frame bundles over category of fibered squares

Anna Bednarska (2008)

Annales UMCS, Mathematica

Let Y be a fibered square of dimension (m1, m2, n1, n2). Let V be a finite dimensional vector space over. We describe all 21,m2,n1,n2 - canonical V -valued 1-form Θ TPrA (Y) → V on the r-th order adapted frame bundle PrA(Y).

Certain relation between vector fields and distributions on a differentiable manifold

Tomáš Klein (1976)

Časopis pro pěstování matematiky

Certaines identifications dans les espaces de jets.

Ceferino Ruiz (1979)

Collectanea Mathematica

Characterisations of Finetely Determined Equivariant Map Germs.

Mark Roberts (1986)

Mathematische Annalen

Charles Ehresmann's concepts in differential geometry

Paulette Libermann (2007)

Banach Center Publications

We outline some of the tools C. Ehresmann introduced in Differential Geometry (fiber bundles, connections, jets, groupoids, pseudogroups). We emphasize two aspects of C. Ehresmann's works: use of Cartan notations for the theory of connections and semi-holonomic jets.

Classification of principal connections naturally induced on $W^{2} P E$

Jan Vondra (2008)

Archivum Mathematicum

We consider a vector bundle $E \to M$ and the principal bundle $P E$ of frames of $E$ . Let $K$ be a principal connection on $P E$ and let $Λ$ be a linear connection on $M$ . We classify all principal connections on $W^{2} P E = P^{2} M \times_{M} J^{2} P E$ naturally given by $K$ and $Λ$ .

Cohomology of the variational complex in the class of exterior forms of finite jet order.

Sardanashvily, Gennadi (2002)

International Journal of Mathematics and Mathematical Sciences

Combinatorics of non-holonomous jets

Anders Kock (1985)

Czechoslovak Mathematical Journal

Connections and 1-jet fiber bundles

Pedro L. García (1972)

Rendiconti del Seminario Matematico della Università di Padova

Connections of higher order and product preserving functors

Jacek Gancarzewicz, Noureddine Rahmani, Modesto R. Salgado (2002)

Czechoslovak Mathematical Journal

In this paper we consider a product preserving functor $ℱ$ of order $r$ and a connection $Γ$ of order $r$ on a manifold $M$ . We introduce horizontal lifts of tensor fields and linear connections from $M$ to $ℱ (M)$ with respect to $Γ$ . Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.

Connections on 1-jets principal fiber bundles

Efstathios Vassiliou (1976)

Rendiconti del Seminario Matematico della Università di Padova

Connections on higher order tangent bundles

Tomáš Klein (1981)

Časopis pro pěstování matematiky

Connections on some functional bundles

Antonella Cabras, Ivan Kolář (1995)

Czechoslovak Mathematical Journal

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

We classify all $ℱ ℳ_{m, n}$ -natural operators $: J ² ⟿ J ² V^{A}$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $(Γ) : V^{A} Y \to J ² V^{A} Y$ on the vertical Weil bundle $V^{A} Y \to M$ corresponding to a Weil algebra A.

Contact and equivalence of submanifolds of homogeneous spaces

Alexandre A. M. Rodrigues (2007)

Banach Center Publications

Contact elements on fibered manifolds

Ivan Kolář, Włodzimierz M. Mikulski (2003)

Czechoslovak Mathematical Journal

For every product preserving bundle functor $T^{μ}$ on fibered manifolds, we describe the underlying functor of any order $(r, s, q), s \geq r \leq q$ . We define the bundle $K_{k, l}^{r, s, q} Y$ of $(k, l)$ -dimensional contact elements of the order $(r, s, q)$ on a fibered manifold $Y$ and we characterize its elements geometrically. Then we study the bundle of general contact elements of type $μ$ . We also determine all natural transformations of $K_{k, l}^{r, s, q} Y$ into itself and of $T (K_{k, l}^{r, s, q} Y)$ into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms...

Contact geometry of curves.

Vassiliou, Peter J. (2009)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

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