Let $G_{n,k}$ (${\tilde{G}}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${ℝ}^n$, $d_{n,k}=k(n-k)=\text{dim}{G}_{n,k}$ and suppose $n\ge 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n,k}+1)$-fold Whitney sum $(d_{n,k}+1){\xi }_{n,k}$ of the nontrivial line bundle ${\xi }_{n,k}$ over $G_{n,k}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($=s_{n,k}$) such that the vector bundle $s{\xi }_{n,k}$ admits a nowhere vanishing section ? Obviously, $s_{n,k}\le d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1}=d_{n,1}+1$. Using results...

The notion of an implicit Hamiltonian system-an isotropic mapping H: M → (TM,ω̇) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated.

We deal with a $(1,1)$-tensor field $\alpha$ on the tangent bundle $TM$ preserving vertical vectors and such that $J\alpha =-\alpha J$ is a $(1,1)$-tensor field on $M$, where $J$ is the canonical almost tangent structure on $TM$. A connection ${\Gamma }_{\alpha }$ on $TM$ is constructed by $\alpha$. It is shown that if $\alpha$ is a $VB$-almost complex structure on $TM$ without torsion then ${\Gamma }_{\alpha }$ is a unique linear symmetric connection such that $\alpha \left({\Gamma }_{\alpha }\right)={\Gamma }_{\alpha }$ and ${\nabla }_{{\Gamma }_{\alpha }}\left(J\alpha \right)=0$.

In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of ${ℝ}^{2n}$ endowed with the standard symplectic form ${\omega }_0=dp\wedge dq$. We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous...

Two significant directions in the development of jet calculus are showed. First, jets are generalized to so-called quasijets. Second, jets of foliated and multifoliated manifold morphisms are presented. Although the paper has mainly a survey character, it also includes new results: jets modulo multifoliations are introduced and their relation to (R,S,Q)-jets is demonstrated.

The paper generalizes results of H. H. Hacisalihoglu and A. Kh. Amirov [Dokl. Akad. Nauk, Ross. Akad. Nauk 351, No. 3, 295-296 (1996; Zbl 0895.53038) and Sib. Mat. Zh. 39, No. 4, 1005-1012 (1998; Zbl 0913.53019)] on the existence and uniqueness of a Riemannian metric on a domain in ${ℝ}^n$ given prescribed values for some of the components of the Riemann curvature tensor and initial values of the metric and its partial derivatives. The authors establish the construction (existence and uniqueness) of a...

Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some “special” fibred coordinate systems on Y, and then we construct a general connection $\widetilde{ℱ}(\Gamma ,\nabla )$ on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.

We deduce a classification of all special types of nonholonomic $3$-jets. In the introductory part, we summarize the basic properties of nonholonomic $r$-jets.

First we summarize some properties of the nonholonomic $r$-jets from the functorial point of view. In particular, we describe the basic properties of our original concept of nonholonomic $r$-jet category. Then we deduce certain properties of the Weil algebras associated with nonholonomic $r$-jets. Next we describe an algorithm for finding the nonholonomic $r$-jet categories. Finally we classify all special types of semiholonomic $3$-jets.

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...