In the paper under review, the author presents some results on the basis of the Nash-Gromov theory of isometric immersions and illustrates how the same results and ideas can be extended to other structures.

A flag manifold of a compact semisimple Lie group $G$ is defined as a quotient $M=G/K$ where $K$ is the centralizer of a one-parameter subgroup $exp\left(tx\right)$ of $G$. Then $M$ can be identified with the adjoint orbit of $x$ in the Lie algebra $𝒢$ of $G$. Two flag manifolds $M=G/K$ and $M^{\text{'}}=G/K^{\text{'}}$ are equivalent if there exists an automorphism $\phi :G\to G$ such that $\phi \left(K\right)=K^{\text{'}}$ (equivalent manifolds need not be $G$-diffeomorphic since $\phi$ is not assumed to be inner). In this article, explicit formulas for decompositions of the isotropy representation for all flag manifolds...

Let $X$ a smooth finite-dimensional manifold and $W_{\Gamma }\left(X\right)$ the manifold of geodesic arcs of a symmetric linear connection $\Gamma$ on $X$. In a previous paper [Differential Geometry and Applications (Brno, 1995) 603-610 (1996; Zbl 0859.58011)] the author introduces and studies the Poisson manifolds of geodesic arcs, i.e. manifolds of geodesic arcs equipped with certain Poisson structure. In this paper the author obtains necessary and sufficient conditions for that a given Lagrange function generates a Poisson manifold...

Summary: The article is devoted to the question how to geometrically construct a 1-form on some non product preserving bundles by means of a 1-form on an original manifold $M$. First, we will deal with liftings of 1-forms to higher-order cotangent bundles. Then, we will be concerned with liftings of 1-forms to the bundles which arise as a composition of the cotangent bundle with the tangent or cotangent bundle.

The author considers the Klein-Gordon equation for $(1+1)$-dimensional flat spacetime. He is interested in those coordinate systems for which the equation is separable. These coordinate systems are explicitly known and generally do not cover the whole plane. The author constructs tensor fields which he can use to express the locus of points where the coordinates break down.

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.

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...

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 $F_1=\text{semi}{F}^{r_1}$ (semi-holonomic $r_1$-th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$-st order frame bundle), $r_2\le s$. The author uses here the so-called method of orbit reduction which provides one with a criterion for checking...

Let $Y\to M$ be a fibered manifold over a manifold $M$ and $\mu :A\to B$ 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 $T_{\text{proj}}Y\to T^{\mu }Y$, where $T_{\text{proj}}Y$ denotes the space of projective vector fields on $Y$ and $T^{\mu }$ the bundle functors associated with $\mu$.