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1 Introduction

Several models of real objects are given as a smooth manifold and one or more linear connections, e.g. material elasticity, see [1]. To obtain a manifold with just one characterization, one has to consider a concept of a higher order connection. In this paper, we recall the basic concepts of higher order connections from both geometric and jet–like point of view, Sects. 2 and 4. Let us note that the original ideas are those of Ehresmann, i.e. the definition of a connection by means of a horizontal distribution in a tangent space, the double fibered manifolds and holonomic and nonholonomic jets of fibered mappings. The first idea can be found in [2], the second one in [3]. The second idea was used for the case of vector bundles by Pradines, [4]. Finally, the concept of holonomic and nonholonomic jets is widely studied in [5–9]. The first idea was extended in [10], where the main formulae of higher order objects in multiple tangent spaces are derived, see also [11]. In this paper we compare the jet–like and geometric approach. We also recall a product of general connections which leads to the so called Ehresmann prolongation and show the reason why this operation is outstanding, especially concerning semiholonomic connections, Sect. 6.1. We study Ehresmann prolongation of a connection from both points of view and show the analogues in both approaches.

2 Jet Prolongation of a Fibered Manifold

Classical theory reads that \(r\)-th holonomic prolongation \(J^rY\) of \(Y\rightarrow M\) is the space of \(r\)–jets of local sections \(M\rightarrow Y\). The nonholonomic prolongation \(\widetilde{J}^{r}Y\) of \(Y\rightarrow M\) is defined by the following iteration:

  1. 1.

      \(\widetilde{J}^{1}Y=J^{1}Y,\) i.e. \(\widetilde{J}^1Y\) is a space of 1-jets of sections \(M\rightarrow Y\) over the target space \(Y\).

  2. 2.

     \(\widetilde{J}^{r}Y=J^{1}(\widetilde{J}^{r-1}Y\rightarrow M).\)

Clearly, we have an inclusion \( J^{r}Y\subset \widetilde{J}^{r}Y\) given by \(j^{r}_{x}\gamma \mapsto j^{1}_{x}(j^{r-1}\gamma ).\) Further, \(r\)-th semiholonomic prolongation \(\overline{J}^{r}Y\subset \widetilde{J}^{r}Y\) is defined by the following induction. First, by \(\beta _1=\beta _{Y}\) we denote the projection \(J^1Y\rightarrow Y\) and by \(\beta _r=\beta _{\widetilde{J}^{r-1}Y}\) the projection \(\widetilde{J}^rY=J^1\widetilde{J}^{r-1}Y\rightarrow \widetilde{J}^{r-1}Y,\ r=2,3,\ldots .\) If we set \(\overline{J}^{1}Y=J^{1}Y \) and assume we have \(\overline{J}^{r-1}Y\subset \widetilde{J}^{r-1}Y\) such that the restriction of the projection \(\beta _{r-1}:\widetilde{J}^{r-1}Y\rightarrow \widetilde{J}^{r-2}Y\) maps \(\overline{J}^{r-1}Y\) into \(\overline{J}^{r-2}Y,\) we can construct \(J^{1}\beta _{r-1}:J^{1}\overline{J}^{r-1}Y\rightarrow J^{1}\overline{J}^{r-2}Y\) and define

$$ \overline{J}^{r}Y=\{A\in J^{1}\overline{J}^{r-1}Y; \ \beta _{r}(A)=J^{1}\beta _{r-1}(A)\in \overline{J}^{r-1}Y\}. $$

If we denote by \(\fancyscript{F}\fancyscript{M}_{m,n}\) the category with objects composed of fibered manifolds with \(m\)-dimensional bases and \(n\)-dimensional fibres and morphisms formed by locally invertible fiber-preserving mappings, then, obviously, \(J^r,\overline{J}^r\) and \(\widetilde{J}^r\) are bundle functors on \(\fancyscript{F}\fancyscript{M}_{m,n}\).

Alternatively, one can define the \(r\)-th order semiholonomic prolongation \(\overline{J}^rY\) by means of natural target projections of nonholonomic jets, see [9]. For \(r\ge q\ge 0\) let us denote by \(\pi ^r_{q}\) the target surjection \(\pi ^r_{q}:\widetilde{J}^rY\rightarrow \widetilde{J}^qY\) with \(\pi ^r_r\) being the identity on \(\widetilde{J}^rY.\) We note that the restriction of these projections to the subspace of semiholonomic jet prolongations will be denoted by the same symbol. By applying the functor \(J^k\) we have also the surjections \(J^k\pi ^{r-k}_{q-k}:\widetilde{J}^r Y\rightarrow \widetilde{J}^q Y\) and, consequently, the element \(X\in \widetilde{J}^r Y\) is semiholonomic if and only if

$$\begin{aligned} (J^k\pi ^{r-k}_{q-k})(X)=\pi ^r_q(X)\ \text {for any integers}\ 1\le k\le q\le r. \end{aligned}$$
(1)

In [9], the proof of this property can be found and the author finds it useful when handling semiholonomic connections and their prolongations.

Now let us recall local coordinates on higher order jet prolongations of a fibered manifold \(Y\rightarrow M\). Let us denote by \(x^{i},\ i=1,\,\ldots ,\,m\) the local coordinates on \(M\) and \(y^{p},\ p=1,\,\ldots ,\,n\) the fiber coordinates of \(Y\rightarrow M\). We recall that the induced coordinates on the holonomic prolongation \(J^{r}Y\) are given by \((x^{i},y^{p}_{\alpha }),\) where \(\alpha \) is a multiindex of range \(m\) satisfying \(|\alpha |\le r.\) Clearly, the coordinates \(y^{p}_{\alpha }\) on \(J^{r}Y\) are characterized by the complete symmetry in the indices of \(\alpha \). Having the nonholonomic prolongation \(\widetilde{J}^{r}Y\) constructed by the iteration, we define the local coordinates inductively as follows:

  1. (1)

    Suppose that the induced coordinates on \(\widetilde{J}^{r-1}Y\) are of the form

    $$ (x^i,y^p_{k_1\ldots k_{r-1}}),\ k_1,\,\ldots ,\,k_{r-1}=0,1,\,\ldots ,\,m. $$
  2. (2)

    We define the induced coordinates on \(\widetilde{J}^r Y\) by

    $$ (x^i,y^p_{k_1\ldots k_{r-1}0}=y^p_{k_1\ldots k_{r-1}},y^p_{k_1\ldots k_{r-1}i}=\frac{\partial }{\partial x^i}y^p_{k_1\ldots k_{r-1}}), $$

    i.e. induced coordinates are partial derivatives are obtained as partial derivatives of fiber coordinates with respect to the base coordinates.

It remains to describe coordinates on the semiholonomic prolongation \(\overline{J}^{r}Y\). Let \((k_{1},\,\ldots ,\,k_{r})\), \(\ k_{1},\,\ldots ,\,k_{r} =0,1,\,\ldots ,\,m\) be a sequence of indices and denote by \(\langle k_{1},\,\ldots ,\,k_{s}\rangle \), \(s\le r\) the sequence of non-zero indices in \((k_{1},\,\ldots ,\,k_{r})\) respecting the order. Then the definition of \(\overline{J}^{r}Y\) reads that the point \((x^{i},y^{p}_{k_{1}\ldots k_{r}})\in \widetilde{J}^{r}Y\) belongs to \(\overline{J}^{r}Y\) if and only if \(y^{p}_{k_{1} \ldots k_{r}}=y^{p}_{l_{1} \ldots l_{r}}\) whenever \(\langle k_{1},\,\ldots ,\,k_{r}\rangle = \langle l_{1},\,\ldots ,\,l_{r}\rangle \)

3 Iterated Tangents

Another concept, in this paper called geometric, of a connection rises from the theory of iterated tangent spaces. Let us recall that the bundle \(T^kM\rightarrow T^{k-1}M\) is equipped with the structure of a \(k\)-fold vector bundle. Particularly, \(T^kM\) admits \(k\) different projections to \(T^{k-1}M\) ,

$$ \,\,\rho _s:= T^{k-s}\pi _s:T^{k}M\rightarrow T^{k-1}M,\,\, $$

where \(\pi _s\) is the natural projection \(T^sM\rightarrow T^{s-1}M\,, s=1,2,\,\ldots ,\, k\). Each projection defines a vector bundle with basis \(T^{k-1}M\) and the total space is composed of \(2^{k-1}n\)-dimensional vector spaces as fibers. The local coordinates on the neighborhoods

$$ T^sU\subset T^{s}M, \quad \text {where}\quad T^{s-1}U=\pi _s(T^sU),\,\,s=1,2,\,\ldots ,\, k, $$

are derived from coordinates, or coordinate mappings, \((u^i)\), which are given on the neighborhood \(U\subset M\) :

\(U\):      \((u^i),\,\,i=1,2,\dots ,\, n,\)

\(TU\):     \((u^i, u_1^i),\)   where   \(u^i:= u^i\circ \pi _1,\,\,u^i_1:= du^i,\)

\(T^2U\):   \((u^i, u_1^i,u^i_2,u^i_{12}),\)

where   \(u^i:= u^i\circ \pi _1\pi _2,\,\,u^i_1:= du^i\circ \pi _2,\,\,\, u^i_2:= d(u^i\circ \pi _1),\,\,\,u^i_{12}:= d^2u^i,\)

etc.

Proposition 3.1

Coordinate mappings given on the neighborhood \(T^{s-1}U\) induce coordinate mappings on the neighborhood \(T^sU\) with respect to the projection \(\pi _s\) by adding the differentials of these mappings.

Local coordinates are obtained by the following principle: to the coordinates of a point of a manifold we attach the coordinates of the vector tangent to the manifold at that point. We use the following notation :   the coordinates of a neighborhood \(T^kU\) consist of two copies of local coordinates on \(T^{k-1}U\) where the second copy is equipped with an additional subscript \(k\) . This principle is suitable in the sense that the coordinates with index \(s\) are recognized as the fiber coordinates for projections \(\rho _s,\,\, s=1,2,\,\ldots ,\,k\), i.e. the coordinates with index \(s\) disappear after the application of projection \(\rho _s\).

The coordinate form of the three projections \(\rho _s:T^{3}U\rightarrow T^2U,\,\,\, s=1,2,3,\) is given by the following diagram:

$$\begin{aligned} \begin{array}{c} (u^i, u_1^i,u^i_2,u^i_{12},u^i_3, u_{13}^i,u^i_{23},u^i_{123})\\ _{\rho _1}\swarrow \qquad \quad _{\rho _2}\downarrow \qquad \qquad \quad \searrow \,_{\rho _3}\\ (u^i, u^i_2,u^i_3,u^i_{23})\qquad (u^i, u^i_1,u^i_3,u^i_{13})\qquad (u^i, u^i_1,u^i_2,u^i_{12}).\\ \end{array} \end{aligned}$$

Remark 3.1

Let us note that the semiholonomity condition is connected to the notion of the osculating bundle, see [11], and can be defined as the equalizer of all possible projections, which corresponds to (1).

4 Connections

We start with the jet–like approach to connections. This rather structural description is quite suitable for determining natural operators on connections, for details see [5].

Definition 4.1

A general connection on the fibered manifold \(Y\rightarrow M\) is a section \(\varGamma :Y\rightarrow J^1Y\) of the first jet prolongation \(J^1Y\rightarrow Y.\)

Further generalization of this idea leads us to the definition of \(r\)-th order connection, which is a section of \(r\)-th order jet prolongation of a fibered manifold. According to the character of the target space we distinguish holonomic, semiholonomic and nonholonomic general connections. The coordinate form of a second order nonholonomic connection \(\varDelta :Y\rightarrow \widetilde{J}^2Y\) is given by

$$y^p_i=F^p_i(x,y),\quad y^p_{0i}=G^p_i(x,y), \quad y^p_{ij}=H^p_{ij}(x,y),$$

where \(F,G,H\) are arbitrary smooth functions. In case of linear connections all functions are linear in fiber coordinates.

Let us now recall the geometric concept of a connection and its extension to higher order connections. The following section is based on the paper [11].

Definition 4.2

A connection on bundle \(\pi :M_1\rightarrow M\) is defined by the structure \(\triangle _h\oplus \triangle _v\) on a manifold \(M_1\) where \(\triangle _v=\ker T\pi \) is vertical distribution tangent to the fibers and \(\triangle _h\) is horizontal distribution complementary to the distribution \(\triangle _v\) . The transport of the fibers along the path \(\gamma \subset M\) is realized by the horizontal lifts given by the distribution \(\triangle _h\) on the surface \(\pi ^{-1}(\gamma )\). If the bundle is a vector one and the transport of fibers along an arbitrary path is linear, then the connection is called linear.

We will assume that the base manifold \(M\) is of dimension \(n\) and the fibers are of dimension \(r\). Then

$$ \dim \triangle _h=n\,,\quad \dim \triangle _v=r\,. $$

On the neighborhood \(U\subset M_1,\) let us consider local base and fiber coordinates:

$$ (u^i, u^\alpha )\,,\, i=1,2,\,\ldots ,\, n\,;\,\alpha =n+1,\,\ldots ,\, n+r. $$

Base coordinates \((u^i) \,\,\) are determined by the projection \(\pi \) and the coordinates \((\bar{u}^i)\) on a neighborhood \(\bar{U}=\pi (U),\, u^i=\bar{u}^i\circ \pi \,.\)

Definition 4.3

On a neighborhood \(U\subset M_1\) we define a local \((\)adapted\()\) basis of the structure \(\triangle _h\oplus \triangle _v\,\),

$$\begin{aligned} (X_i \; X_\alpha )=\left( {\partial \over \partial u^j} \, {\partial \over \partial u^\beta } \right) \cdot \begin{pmatrix} \delta ^j_i &{} 0 \\ \varGamma ^\beta _i &{} \delta ^\beta _\alpha \\ \end{pmatrix},\quad \begin{pmatrix} \omega ^i \\ \omega ^\alpha \\ \end{pmatrix}= \begin{pmatrix} \delta ^i_j &{} 0\\ - \varGamma ^\alpha _j &{} \delta ^\alpha _\beta \end{pmatrix}\cdot \begin{pmatrix} du^{j}\\ du^{\beta } \end{pmatrix}. \end{aligned}$$

The horizontal distribution \(\triangle _h\,\) is the linear span of the vector fields \((X_i)\) and the annihilator of the forms \((\omega ^\alpha )\),

$$ X_i=\partial _i+\varGamma ^\beta _i\partial _\beta ,\quad \omega ^\alpha =du^\alpha -\varGamma ^\alpha _idu^i. $$

Definition 4.4

A classical affine connection on manifold \(M\) is seen as a linear connection on the bundle \(\pi _1:{\textit{TM}}\rightarrow M\). On the tangent bundle \({ TM}\rightarrow M\) one can define the structure \(\triangle _h\oplus \triangle _v\). The indices in the formulas are denoted by Latin letters all of them ranging from 1 to \(n\). The functions \(\varGamma ^\alpha _i\,, X_i\,, \omega ^\alpha \) are of the form \((\)in \(\varGamma ^\alpha _i\) the sign is changed to comply with the classical theory\()\) \(:\)

$$\begin{aligned} \varGamma ^\alpha _i\quad&\rightsquigarrow \quad -\varGamma ^i_{jk}u^k_1\,,\\ X_i=\partial _i+\varGamma ^\alpha _i\partial _\alpha \quad&\rightsquigarrow \quad X_i=\partial _i-\varGamma ^k_{ij}u^i_1\partial ^1_k\,,\\ \omega ^\alpha =du^\alpha -\varGamma ^\alpha _idu^i\quad&\rightsquigarrow \quad U^i_{12}=u^i_{12}+\varGamma ^i_{jk}u^k_1u^j_2\,. \end{aligned}$$

Definition 4.5

Higher order connections are defined as follows: on tangent bundle \({ TM}\) the structure \(\triangle \oplus \triangle _1\) is defined where \(\ker T\rho _1=\triangle _1\), on \(T({ TM})\) the structure \(\varDelta \oplus \varDelta _1\oplus \varDelta _2\oplus \varDelta _{12}\) is defined where \(\ker T\rho _s=\varDelta _s\oplus \varDelta _{12}\,, s=1,2,\) etc.

5 Connections on Two-Fold Fibered Manifolds

More generally, one can define a second order connection by means of a two-fold fibered manifold. Note that the Definition 4.5 is a special case of the following. A two-fold fibered manifold is a commutative diagram

where \(\rho _1,\rho _2\) and \(\pi _1,\pi _2\)—four fibered manifolds

$$\begin{aligned} \dim M=n,\,\,\,\dim \fancyscript{M}_1=n+r_1,\,\,\,\dim \fancyscript{M}_2=n+r_2,\,\,\,\dim \fancyscript{M}=n+r_1+r_2+r_{12}. \end{aligned}$$

The double projection

$$ \pi =\pi _1\circ \rho _2=\pi _2\circ \rho _1 : \fancyscript{M}\rightarrow M $$

divides a manifold \(\fancyscript{M}\) to \(n\)-parameter family of fibers of dimensions \((r_1+r_2+r_{12})\). Each fiber carries structure of another two fibers of dimensions \(r_1+r_{12}\)  and  \(r_2+r_{12}\)  and these two fibers have the common intersection of dimension \(r_{12}\).

A two-fold fibered manifold is called a vector bundle if both fibrations \(\pi _1,\) \( \pi _2, \rho _1\) and \(\rho _2\)—form vector bundles.

An example of a two-fold fibered manifold is the second order tangent bundle \(T^2M\) of a manifold \(M\). In this case \(n=r_1=r_2=r_{12}\).

Definition 5.1

A connection on a two-fold fibered manifold is defined by a structure on a manifold \(\fancyscript{M}\):

$$\begin{aligned} \varDelta \otimes \varDelta _1\otimes \varDelta _2\otimes \varDelta _{12}\,, \end{aligned}$$
(2)
$$ \dim \varDelta =n,\quad \dim \varDelta _1=r_1\,,\quad \dim \varDelta _2=r_2\,,\quad \dim \varDelta _{12}=r_{12}\,, $$
$$ \mathrm{Ker} T\rho _2=\varDelta _2\oplus \varDelta _{12}\,,\qquad \mathrm{Ker} T\rho _1=\varDelta _1\oplus \varDelta _{12} $$
$$ T\rho _2(\varDelta \oplus \varDelta _1)=T\fancyscript{M}_1,\quad T\rho _1(\varDelta \oplus \varDelta _2)=T\fancyscript{M}_2\,,$$
$$ T\pi \varDelta ={ TM}. $$

Remark 5.1

A connection on a two-fold vector fibered manifold is called linear if the structure (2) induces on the manifolds \(\pi _1, \pi _2, \rho _1\) and \(\rho _2\) linear connections.

Remark 5.2

Similarly, one can define a connection on a \(k\)–fold fibered manifold. In such case the commutative diagram would be represented by a \(k\)–dimensional cube. These manifolds would correspond to the \(k\)–th tangent bundle \(T^kM\) of a manifold \(M\).

On the neighborhoods

$$ \fancyscript{U}\subset \,\fancyscript{M},\,\,\, \fancyscript{U}_1=\rho _2(\fancyscript{U})\subset \,\fancyscript{M}_1,\,\,\, \fancyscript{U}_2=\rho _1(\fancyscript{U})\subset \,\fancyscript{M}_2,\,\,\, U=\pi (\fancyscript{U})\subset \,M $$

we have the coordinate systems

\((u^i,\,\, u^{\alpha _1}, \,\, u^{\alpha _2},\,\, u^{\alpha _{12}}),\,\,\,(u^i,\,\, u^{\alpha _1}),\,\,\,(u^i,\,\, u^{\alpha _1}),\,\,\,(u^i).\)

The transformation of coordinates on the neighborhoods \(\fancyscript{U}\),

$$\begin{aligned} (u^i,\, u^{\alpha _1},\, u^{\alpha _2},\, u^{\alpha _{12}})\,\rightsquigarrow \,(\tilde{u}^i,\, \tilde{u}^{\alpha _1},\, \tilde{u}^{\alpha _2},\, \tilde{u}^{\alpha _{12}})=(a^i,\, a^{\alpha _1},\, a^{\alpha _2},\, a^{\alpha _{12}}), \end{aligned}$$

gives a Jacobi matrix:

$$\begin{aligned}\left( \begin{array}{cccc} a^i_j &{} 0 &{} 0 &{} 0 \\ a^{\alpha _1}_j &{} a^{\alpha _1}_{\beta _1} &{} 0 &{} 0 \\ a^{\alpha _2}_j &{}0&{} a^{\alpha _2}_{\beta _2} &{} 0 \\ a^{\alpha _{12}}_j &{} a^{\alpha _{12}}_{\beta _1}&{}a^{\alpha _{12}}_{\beta _2} &{} a^{\alpha _{12}}_{\beta _{12}} \\ \end{array} \right) . \end{aligned}$$

See [10, 12]. Let us mention that the local (adapted) basis of such decomposition is represented by a matrix of the form

$$\begin{aligned} \left( \begin{array}{cccc} \delta ^i_j &{} 0 &{} 0 &{} 0 \\ \varGamma ^{\alpha _1}_j &{} \delta ^{\alpha _1}_{\beta _1} &{} 0 &{} 0 \\ \varGamma ^{\alpha _2}_j &{}0&{} \delta ^{\alpha _2}_{\beta _2} &{} 0 \\ \varGamma ^{\alpha _{12}}_j &{} \varGamma ^{\alpha _{12}}_{\beta _1}&{}\varGamma ^{\alpha _{12}}_{\beta _2} &{} \delta ^{\alpha _{12}}_{\beta _{12}} \\ \end{array} \right) . \end{aligned}$$
(3)

The dual frame is given by the system of 1–forms:

$$\begin{aligned} \omega ^i&=du^i,\nonumber \\ \omega ^{\alpha _1}&=du^{\alpha _1}-\varGamma ^{\alpha _1}_idu^i,\nonumber \\ \omega ^{\alpha _2}&=du^{\alpha _2}-\varGamma ^{\alpha _2}_idu^i,\\ \omega ^{\alpha _{12}}&=du^{\alpha _{12}}-\varGamma ^{\alpha _{12}}_{\alpha _1} du^{\alpha _1}-\varGamma ^{\alpha _{12}}_{\alpha _2} du^{\alpha _2}-\bar{\varGamma }^{\alpha _{12}}_idu^i,\nonumber \\ \text {where}\quad&\varGamma ^{\alpha _{12}}_i-\bar{\varGamma }^{\alpha _{12}}_i=\varGamma ^{\alpha _{12}}_{\beta _1}\,\varGamma ^{\beta _1}_i +\varGamma ^{\alpha _{12}}_{\beta _2}\,\varGamma ^{\beta _2}_i. \end{aligned}$$

In case of linear connection the elements of the matrix (3) are of the form

$$\begin{aligned} \varGamma ^{\alpha _1}_j=\varGamma ^{\alpha _1}_{j\beta _1}u^{\beta _1}&,\quad \varGamma ^{\alpha _2}_j= \varGamma ^{\alpha _2}_{j\beta _2}u^{\beta _2},\\ \varGamma _{\beta _1}^{\alpha _{12}}=\varGamma ^{\alpha _{12}}_{{\beta _1}{\beta _2}}u^{\beta _2},&\quad \,\,\,\varGamma _{\beta _2}^{\alpha _{12}}=\varGamma ^{\alpha _{12}}_{{\beta _2}{\beta _1}}u^{\beta _1},\\ \varGamma ^{\alpha _{12}}_j=\varGamma ^{\alpha _{12}}_{j{\beta _1}{\beta _2}}u^{\beta _1} u^{\beta _2}+\varGamma ^{\alpha _{12}}_{j{\beta _{12}}}u^{\beta _{12}}&,\quad \bar{\varGamma }^{\alpha _{12}}_j=\bar{\varGamma }^{\alpha _{12}}_{j{\beta _1}{\beta _2}}u^{\beta _1} u^{\beta _2}+\bar{\varGamma }^{\alpha _{12}}_{j{\beta _{12}}}u^{\beta _{12}},\end{aligned}$$
$$ \varGamma ^{\alpha _{12}}_{j{\beta _1}{\beta _2}}-\bar{\varGamma }^{\alpha _{12}}_{j{\beta _1}{\beta _2}}= \varGamma ^{\alpha _{12}}_{{\gamma _2}{\beta _1}}\varGamma ^{\gamma _2}_{j\beta _2}, $$

where the coefficients depend on the base coordinates \(u^i\) only.

6 Ehresmann Prolongation

First, let us now recall a concept of a product of two connections.

Given two higher order connections \(\varGamma :Y\rightarrow \widetilde{J}^{r}Y\) and \(\overline{\varGamma }:Y\rightarrow \widetilde{J}^{s}Y,\) the product of \(\varGamma \) and \(\overline{\varGamma }\) is the \((r+s)\)-th order connection \(\varGamma *\overline{\varGamma }:Y\rightarrow \widetilde{J}^{r+s}Y\) defined by

$$ \varGamma *\overline{\varGamma }=\widetilde{J}^{s}\varGamma \circ \overline{\varGamma }. $$

Particularly, if both \(\varGamma \) and \(\overline{\varGamma }\) are of the first order, then \(\varGamma *\overline{\varGamma }:Y\rightarrow \widetilde{J}^{2}Y\) is semiholonomic if and only if \(\varGamma =\overline{\varGamma }\) and \(\varGamma *\overline{\varGamma }\) is holonomic if and only if \(\varGamma \) is curvature-free, [9, 13].

As an example we show the coordinate expression of an arbitrary nonholonomic second order connection and of the product of two first order connections. The coordinate form of \(\varDelta :Y\rightarrow \widetilde{J}^2Y\) is

$$ y^p_i=F^p_i(x,y),\quad y^p_{0i}=G^p_i(x,y), \quad y^p_{ij}=H^p_{ij}(x,y), $$

where \(F,G,H\) are arbitrary smooth functions. Further, if the coordinate expressions of two first order connections \(\varGamma ,\overline{\varGamma }:Y\rightarrow J^1Y\) are

$$\begin{aligned} \varGamma :\quad y^p_i=F^p_i(x,y),\qquad \overline{\varGamma }:\quad y^p_i=G^p_i(x,y), \end{aligned}$$
(4)

then the second order connection \(\varGamma *\overline{\varGamma }:Y\rightarrow \widetilde{J}^2Y\) has equations

$$\begin{aligned} y^p_i = F^p_i, \quad y^p_{0i} = G^p_i,\quad y^p_{ij} = \frac{\partial F^p_i}{\partial x^j}+\frac{\partial F^p_i}{\partial y^q}G^q_j. \end{aligned}$$

For linear connections, the coordinate form would be obtained by substitution

$$\begin{aligned} F^p_i&=F^p_{iq}y^q,\\ G^p_i&=G^p_{iq}y^q \end{aligned}$$

in the Eq. (4), where \(F^p_{iq}\) and \(G^p_{iq}\) are functions of the base manifold coordinates \(x_i.\) For order three see [8].

In the above process, if \(\varGamma =\overline{\varGamma }\), the connection \(\varGamma *\varGamma \) is called the Ehresmann prolongation of \(\varGamma \), iteratively we obtain the \(r\)–th Ehresmann prolongation of \(\varGamma \). We show that Ehresmann prolongation plays an important role in determining all natural operators transforming first order connections into higher order connections. Let us note that also natural transformations of semiholonomic jet prolongation functor \(\overline{J}^r\) are involved. To find the details about this topic we refer to [5–7]. For our purposes, it is enough to consider \(r=2.\) We use the notation of [5], where the map \(e:\overline{J}^2Y\rightarrow \overline{J}^2Y\) is obtained from the natural exchange map \(e_{\Lambda }:J^1J^1Y\rightarrow J^1J^1Y\) as a restriction to the subbundle \(\overline{J}^2Y\subset J^1J^1Y\). Note that while \(e_{\Lambda }\) depends on the linear connection \(\Lambda \) on \(M\), its restriction \(e\) is independent of any auxiliary connections. We remark, that originally the map \(e_{\Lambda }\) was introduced by M. Modugno. We also recall that J. Pradines introduced a natural map \(\overline{J}^2Y\rightarrow \overline{J}^2Y\) with the same coordinate expression.

Now we are ready to recall the following assertion, see [7] for the proof.

Proposition 6.1

All natural operators transforming first order connection \(\varGamma :Y\rightarrow J^1 Y\) into second order semiholonomic connection \(Y\rightarrow \overline{J}^2 Y\) form a one parameter family

$$\begin{aligned} \varGamma \mapsto k\cdot (\varGamma *\varGamma )+(1-k)\cdot e(\varGamma *\varGamma ),\qquad k\in \mathbb {R}. \end{aligned}$$

This shows the importance of Ehresmann prolongation in the theory of prolongations of connections.

7 Tangent Functor and Ehresmann Prolongation

If we apply the tangent functor \(T\) two times on a projection \(\pi :E\rightarrow M\) and a section \(\sigma : M\rightarrow E\) we obtain

$$ T\pi :{ TE}\rightarrow { TM}\,,\,\,T^2\pi :T^2E\rightarrow T^2M, $$
$$ T\sigma : { TM}\rightarrow { TE},\,T^2\sigma : T^2M\rightarrow T^2E, $$

respectively. The mappings \(\sigma ,\ T\sigma \) and \(T^2\sigma \) define the sections of fibered manifolds \(\pi ,\ T\pi \) and \(T^2\pi .\)

Let us consider local coordinates on the following manifolds in the form

$$\text {on}\quad M,\, { TM},\, T^2M\,:\,\,\,(x^i),\,\,(x^i, x^i_1),\,\, (x^i, x^i_1, x^i_2, x^i_{12}),$$
$$\text {and on}\quad E,\, { TE},\, T^2E\,:\,\,\,(y^p),\,\, (y^p, y^p_1),\,\, (y^p, y^p_1, y^p_2, y^p_{12}).$$

Let us also consider for a function \(f\) defined on a manifold \(M\), its following differentials on \(T^2M\) in local coordinate form:

$$ f_1\doteq f_ix^i_1,\,\,\,f_2\doteq f_ix^i_2,\,\,f_{12}\doteq f_{ij}x^i_1x^j_2+f_ix^i_{12},\,\,\text {where}\,\, f_i=\frac{\partial f}{\partial x^i}\,,\,f_{ij}=\frac{\partial ^2 f}{\partial x^i\partial x^j}. $$

Furthermore, \(f_1=df\circ \rho _1\,,\,\,f_2=df\circ \rho _2\,,\,\,f_{12}=d\,^2f\). We use these notations in the formulae bellow.

If the section \(\sigma \) is defined by local functions \(\varGamma ^p\), then the sections \(T\sigma \) and \(T^2\sigma \) are defined by its differentials \(\varGamma _1^p\) ,  \(\varGamma _2^p\)   and   \(\varGamma _{12}^p,\)

$$\begin{aligned}&\sigma :\,\,x^i\rightsquigarrow y^p=\varGamma ^p,\nonumber \\&T\sigma :\,\, (x^i,x^i_1)\rightsquigarrow (y^p,y^p_1)=(\varGamma ^p,\varGamma ^p_1),\nonumber \\&T^2\sigma :\,\,(x^i, x^i_1, x^i_2, x^i_{12})\rightsquigarrow (y^p, y^p_1, y^p_2, y^p_{12})=(\varGamma ^p, \varGamma ^p_1, \varGamma ^p_2, \varGamma ^p_{12}),\nonumber \\&\text {where}\quad \varGamma ^p_1=\varGamma ^p_ix^i_1,\quad \varGamma ^p_2=\varGamma ^p_ix^i_2, \quad \varGamma ^p_{12}=\varGamma ^p_{ij}x^i_1x^j_2+\varGamma ^p_ix^i_{12}. \end{aligned}$$
(5)

The case when the coefficients \(\varGamma ^p_i,\,\,\varGamma ^p_{ij}\) in (5) are arbitrary functions, corresponds to a nonholonomic connection on the fibered manifold \(\pi \).

The case when \(\varGamma ^p_{ij}=\displaystyle {\frac{\partial \varGamma ^p_i}{\partial x^j}}\),  where   \(\varGamma ^p_i\) are arbitrary functions corresponds to a semiholonomic connection on the fibered manifold \(\pi \).

The case when \(\varGamma ^p_1=d\varGamma ^p\circ \rho _1\,,\,\,\varGamma ^p_2=d\varGamma ^p\circ \rho _2\,,\,\,\varGamma ^p_{12}=d\,^2\varGamma ^p\,,\,\,\) corresponds to a holonomic connection on the fibered manifold \(\pi \).

The functions \(\varGamma ^p_i,\,\varGamma ^p_{ij}\) define nonholonomic, semiholonomic or holonomic Ehresmann prolongation of a connection, respectively.

Remark 7.1

Nonholonomic prolongation induces a connection on a double fibered manifold

$$ J\,\rightarrow \, E\,\rightarrow \, M\,:\,\,\,y^p_i\,\rightsquigarrow \,y^p\,\rightsquigarrow \, x^i. $$

On the fibered manifold \(E\,\rightarrow \,M\) the fiber transformations are given by the Pfaff system

$$ \omega ^p\equiv dy^p - \varGamma ^p_idx^i=0, $$

more precisely, along a curve \(x^i(t)\) – by the system of first order ODEs

$$\begin{aligned} \dot{y}^p=\varGamma ^p_i\,\dot{x}^i. \end{aligned}$$
(6)

In case \((\varGamma ^p_{12}, x^i_1, x^j_2, x^i_{12})\,\rightsquigarrow \,(\ddot{y}^p\,,\,\dot{x}^i\,,\,\dot{x}^j\,,\,\ddot{x}^i)\) we obtain the system of second order ODEs:

$$ \varGamma ^p_{12}=\varGamma ^p_{ij}x^i_1x^j_2+\varGamma ^p_ix^i_{12}\,\,\rightsquigarrow \,\, \ddot{y}^p=\varGamma ^p_{ij}\dot{x}^i\dot{x}^j+\varGamma ^p_i\ddot{x}^i. $$

Considering the system (6), we obtain for fiber coordinates \(y^\alpha , y^\alpha _i\) system of first order ODEs

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{y}^p &{}=\,\varGamma ^p_i\,\dot{x}^i,\\ \dot{y}^p_i &{}=\,\varGamma ^p_{ij}\,\dot{x}^j. \end{array} \right. \end{aligned}$$

The sections of fibers along a curve \(x^i(t)\) are given.

The horizontal distribution \(\triangle _h\) is \(n\)-dimensional and described by the vector field

$$ X_i=\partial _i+\varGamma ^p_i\,\partial _p+\varGamma ^p_{ij}\,\partial ^j_p\,,\quad \text {where}\quad \partial _i=\frac{\partial }{\partial x^i},\,\, \partial _p=\frac{\partial }{\partial y^p},\,\,\partial ^j_p=\frac{\partial }{\partial y^p_j}\,. $$