View source for Non-linear connection

A [[Differential-geometric structure|differential-geometric structure]] defined for the category of smooth fibre spaces associated with a certain principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670901.png" />-bundle that determines the isomorphisms of the fibres (the parallel transfer) for the given non-linear connection along every piecewise-smooth curve in the base space of a bundle in the given category, which is compatible with the isomorphism of the corresponding fibres of the principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670902.png" />-bundle. Here it is assumed that the structure in question is not identical with the classical concept of a [[Linear connection|linear connection]], which is defined by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670903.png" />-invariant [[Horizontal distribution|horizontal distribution]] of one kind or another. A different meaning of the term non-linear connection [[#References|[5]]] consists in the fact that the transfer for the fibres of a vector bundle defined by a horizontal distribution ceases to have a linear character, that is, is not a linear isomorphism of these fibres.

The necessity of introducing and studying non-linear connections arose from the need to study various differential-geometric structures of higher orders (such as, for example, a [[Kawaguchi space|Kawaguchi space]]). The foundations of the general theory of non-linear connections are fairly well developed and applications of some special types (see [[#References|[2]]]–[[#References|[4]]]) have been investigated.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670904.png" /> be a smooth principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670905.png" />-bundle with structure Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670906.png" /> and canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670907.png" /> onto the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670908.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n0670909.png" /> be the category of all bundles associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709010.png" />. A bundle isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709011.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709013.png" />, is defined to be a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709014.png" /> that commutes with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709016.png" />. Any isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709017.png" /> can be described by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709020.png" />, hence is a diffeomorphism of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709022.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709023.png" /> of all isomorphisms between all possible fibres of a principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709024.png" /> is a smooth bundle with structure groupoid over the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709025.png" /> (a [[Groupoid|groupoid]] is a category with inverse elements). An isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709026.png" /> gives rise to a corresponding isomorphism of the fibres over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709027.png" /> of any associated bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709028.png" />, and the groupoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709029.png" /> serves for the whole category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709030.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709031.png" /> be the category of all piecewise-smooth curves in the base manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709032.png" />. A connection in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709033.png" /> of smooth bundles in the most general sense is any functor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709034.png" /></td> </tr></table>

that is the identity on the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709036.png" /> be the canonical projection of the groupoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709037.png" /> onto its base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709038.png" />, defined by the condition that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709041.png" />. In this way <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709042.png" /> is identified with the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709043.png" /> of all left and right units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709044.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709045.png" /> be the vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709046.png" /> formed by the fibres of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709048.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709049.png" /> be the fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709051.png" />-velocities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709052.png" /> (the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709053.png" /> are regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709054.png" />-jets of all possible smooth mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709055.png" /> with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709056.png" />). The bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709058.png" /> have canonical projections onto the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709059.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709060.png" /></td> </tr></table>

A connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709061.png" /> is called a non-linear connection of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709063.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709064.png" /> is the smallest number for which the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709065.png" /> determines a smooth mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709066.png" /></td> </tr></table>

such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709067.png" />. In turn, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709068.png" /> is determined by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709069.png" /> corresponding to it. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709070.png" /> and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709071.png" /> is fibrewise linear, the connection degenerates to a linear one on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709072.png" />. In the study of the properties of non-linear connections and in their classification a fundamental role is played by the structure equations of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709073.png" />. These can be written in the form of Pfaffian equations connecting the differentials of the relative coordinates of the geometric objects describing the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709075.png" />. In terms of the coefficients of the structure equations and by means of the operations of their differential prolongations and restrictions it has been established [[#References|[2]]] that a non-linear connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709076.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709077.png" /> gives rise to a linear connection of special structure in the smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709078.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709079.png" /> over the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067090/n06709080.png" /> and is completely characterized by this linear connection. The forms of these linear connections have been found and also their structure equations. A non-linear analogue has been found for the theorem on the holonomy group, and its statement involves not only the curvature, but also the linear hull of the distribution of horizontal cones, which replace in the non-linear case the subspace of the horizontal distribution of a linear connection.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Vagner,   "The theory of composite manifolds"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''8'''  (1950)  pp. 11–72  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.E. Evtushik,   "Non-linear connections of higher order"  ''Izv. Vuz. Mat.'' , '''2'''  (1969)  pp. 34–44  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.E. Evtushik,   "Holonomy of nonlinear connections"  ''Sib. Math. J.'' , '''14''' :  3  (1973)  pp. 370–379  ''Sibirsk. Mat. Zh.'' , '''14''' :  3  (1973)  pp. 536–548</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.E. Evtushik,   V.B. Tret'yakov,   "Structures that can be defined by a system of higher order differential equations"  ''Trudy Geom. Sem.'' , '''6'''  (1974)  pp. 243–255  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Kawaguchi,   "On the theory of non-linear connections I. Introduction to the theory of general non-linear connections"  ''Tensor, New Ser.'' , '''2'''  (1952)  pp. 123–142</TD></TR></table>