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...

[For the entire collection see Zbl 0742.00067.]This paper is devoted to a method permitting to determine explicitly all multilinear natural operators between vector-valued differential forms and between sections of several other natural vector bundles.

The author studies the problem how a map $L:M\to ℝ$ on an $n$-dimensional manifold $M$ can induce canonically a map $A_M\left(L\right):T^{*}{T}^{\left(r\right)}M\to ℝ$ for $r$ a fixed natural number. He proves the following result: “Let $A:T^{(0,0)}\to T^{(0,0)}\left(T^{*}{T}^{\left(r\right)}\right)$ be a natural operator for $n$-manifolds. If $n\ge 3$ then there exists a uniquely determined smooth map $H:{ℝ}^{S\left(r\right)}\times ℝ\to ℝ$ such that $A=A^{\left(H\right)}$.”The conclusion is that all natural functions on $T^{*}{T}^{\left(r\right)}$ for $n$-manifolds $(n\ge 3)$ are of the form $\{H\circ ({\lambda }_M^{\langle 0,1\rangle },\cdots ,{\lambda }_M^{\langle r,0\rangle })\}$, where $H\in C^{\infty }\left({ℝ}^r\right)$ is a function of $r$ variables.

The author proves that for a manifold $M$ of dimension greater than 2 the sets of all natural operators $TM\to (T_k^{r*}M,T_{\ell }^{q*}M)$ and $TM\to T{T}_k^{r*}M$, respectively, are free finitely generated $C^{\infty }\left({\left({ℝ}^k\right)}^r\right)$-modules. The space $T_k^{r*}M=J^r{(M,{ℝ}^k)}_0$, this is, jets with target 0 of maps from $M$ to ${ℝ}^k$, is called the space of all $(k,r)$-covelocities on $M$. Examples of such operators are shown and the bases of the modules are explicitly constructed. The definitions and methods are those of the book of I. Kolář, P. W. Michor and J. Slovák [Natural operations in differential geometry,...

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 $\mu$.

In this nice paper the author proves that all natural symplectic forms on the tangent bundle of a pseudo-Riemannian manifold are pull-backs of the canonical symplectic form on the cotangent bundle with respect to some diffeomorphisms which are naturally induced by the metric.

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor $T^A$ of A-velocities [I. Kolař, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of $T^A$ into F is finite and is less than or equal to $dim\left(F_0{ℛ}^k\right)$. The spaces of all natural transformations of Weil functors into linear...

This paper sets out to examine some of Riemann’s papers and notes left by him, in the light of the “philosophical” standpoint expounded in his writings on Naturphilosophie. There is some evidence that many of Riemann’s works, including his Habilitationsvortrag of 1854 on the foundations of geometry, may have sprung from his attempts to find a unified explanation for natural phenomena, on the basis of his model of the ether.