We study triviality of Nash families of proper Nash submersions or, in a more general setting, the triviality of pairs of proper Nash submersions. We work with Nash manifolds and mappings defined over an arbitrary real closed field $R$. To substitute the integration of vector fields, we study the fibers of such families on points of the real spectrum $\widetilde{R^p}$ and we construct models of proper Nash submersions over smaller real closed fields. Finally we obtain results on finiteness of topological types in...

All natural affinors on the $r$-th order cotangent bundle $T^{r*}M$ are determined. Basic affinors of this type are the identity affinor id of $T{T}^{r*}M$ and the $s$-th power affinors $Q_M^s:T{T}^{r*}M\to V{T}^{r*}M$ with $s=1,\cdots ,r$ defined by the $s$-th power transformations $A_s^{r,r}$ of $T^{r*}M$. An arbitrary natural affinor is a linear combination of the basic ones.

Let $r,s,q,m,n\in \mathbb{N}$ be such that $s\ge r\le q$. Let $Y$ be a fibered manifold with $m$-dimensional basis and $n$-dimensional fibers. All natural affinors on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are classified. It is deduced that there is no natural generalized connection on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$. Similar problems with ${\left(J^{r,s}{(Y,ℝ)}_0\right)}^{*}$ instead of ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are solved.

We deduce that for $n\ge 2$ and $r\ge 1$, every natural affinor on $J^{r}T$ over $n$-manifolds is of the form $\lambda \delta$ for a real number $\lambda$, where $\delta$ is the identity affinor on $J^{r}T$.

We describe all F2Mm1,m2,n1,n2-natural affinors on the r-th order adapted frame bundle PrAY over (m1,m2, n1, n2)-dimensional fibered-fibered manifolds Y.

For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle $T^{r,s,q*}Y$ over an (m,n)-dimensional fibered manifold Y.

Let $F$ and $G$ be two natural bundles over $n$-manifolds. We prove that if $F$ is of type (I) and $G$ is of type (II), then any natural differential operator of $F$ into $G$ is of order 0. We give examples of natural bundles of type (I) or of type (II). As an application of the main theorem we determine all natural differential operators between some natural bundles.

We define natural first order Lagrangians for immersions of Riemannian manifolds and we prove a bijective correspondence between such Lagrangians and the symmetric functions on an open subset of m-dimensional Euclidean space.

We determine all natural functions on $T^{*}{T}^{\left(r\right)}$ and $T^{*}{T}^{r*}$.

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.

We determine all first order natural operators transforming $(0,2)$–tensor fields on a manifold $M$ into $(0,2)$–tensor fields on $TM$.