Non-linear connection

A differential-geometric structure defined for the category of smooth fibre spaces associated with a certain principal $ G $-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 $ G $-bundle. Here it is assumed that the structure in question is not identical with the classical concept of a linear connection, which is defined by a $ G $-invariant horizontal distribution of one kind or another. A different meaning of the term non-linear connection [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). The foundations of the general theory of non-linear connections are fairly well developed and applications of some special types (see [2]–[4]) have been investigated.

Let $ \pi : X ( B , G ) \rightarrow B $ be a smooth principal $ G $-bundle with structure Lie group $ G $ and canonical projection $ \pi $ onto the base $ B $, and let $ K ( X ) $ be the category of all bundles associated with $ X $. A bundle isomorphism of $ G _ {x} = \pi ^ {-1} ( x ) $ onto $ G _ {y} = \pi ^ {-1} ( y ) $, $ x , y \in B $, is defined to be a mapping $ i : G _ {x} \rightarrow G _ {y} $ that commutes with the action of $ G $ on $ X $. Any isomorphism $ i $ can be described by $ i ( \xi _ {0} g ) = i ( \xi _ {0} ) g $, $ \xi _ {0} \in X $, $ g \in G $, hence is a diffeomorphism of the fibres $ G _ {x} $ and $ G _ {y} $. The set $ \Gamma ( X) $ of all isomorphisms between all possible fibres of a principal bundle $ X $ is a smooth bundle with structure groupoid over the base $ B \times B $ (a groupoid is a category with inverse elements). An isomorphism $ i \in \Gamma ( X) $ gives rise to a corresponding isomorphism of the fibres over $ x , y \in B $ of any associated bundle $ Y \in K ( X) $, and the groupoid $ \Gamma ( X) $ serves for the whole category $ K ( X) $.

Let $ \Lambda ( B) $ be the category of all piecewise-smooth curves in the base manifold $ B $. A connection in the category $ K ( X) $ of smooth bundles in the most general sense is any functor

$$ \gamma : \Lambda ( B) \rightarrow \Gamma ( X) $$

that is the identity on the base $ B \times B $. Let $ \alpha \times \beta : \Gamma ( X) \rightarrow B \times B $ be the canonical projection of the groupoid $ \Gamma ( X) $ onto its base $ B \times B $, defined by the condition that if $ \Gamma ( X) \ni i : G _ {x} \rightarrow G _ {y} $, then $ \alpha ( i) = x $, $ \beta ( i) = y $. In this way $ B $ is identified with the submanifold $ \widetilde{B} \subset \Gamma ( X) $ of all left and right units of $ \Gamma ( X) $. Let $ \Pi ( X) $ be the vector bundle over $ B \equiv \widetilde{B} $ formed by the fibres of the form $ T _ {e} [ \alpha ^ {-1} ( e) ] $, $ e \in B $, and let $ T ^ {p} ( B) $ be the fibre over $ B $ of $ p $-velocities of $ B $ (the elements of $ T ^ {p} ( B) $ are regular $ p $-jets of all possible smooth mappings $ \mathbf R \rightarrow B $ with source $ 0 \in \mathbf R $). The bundles $ \Pi ( X) $ and $ T ^ {p} ( B) $ have canonical projections onto the tangent bundle $ T ( B) $,

$$ \pi ^ \prime : \Pi ( X) \rightarrow T ( B) ,\ \ \pi ^ {p} : T ^ {p} ( B) \rightarrow T ( B) . $$

A connection $ \gamma $ is called a non-linear connection of order $ p = 1 , 2 \dots $ if $ p $ is the smallest number for which the functor $ \gamma $ determines a smooth mapping

$$ \gamma ^ {p} : T ^ {p} ( B) \rightarrow \Pi ( X) $$

such that $ \pi ^ \prime \circ \gamma ^ {p} = \pi ^ {p} $. In turn, $ \gamma $ is determined by the $ \gamma ^ {p} $ corresponding to it. When $ p = 1 $ and the mapping $ T ( B) \rightarrow \Pi ( X) $ is fibrewise linear, the connection degenerates to a linear one on $ K ( X) $. 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 $ \gamma ^ {p} : T ^ {p} ( B) \rightarrow \Pi ( X) $. These can be written in the form of Pfaffian equations connecting the differentials of the relative coordinates of the geometric objects describing the bundles $ T ^ {p} ( B) $ and $ \Pi ( X) $. 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 [2] that a non-linear connection $ \gamma ^ {p} $ in $ X ( B , G ) $ gives rise to a linear connection of special structure in the smooth $ G $-bundle $ X ( B , G) \otimes _ {B} T ^ {p} ( B) $ over the base $ T ^ {p} ( B) $ 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

[1]	V.V. Vagner, "The theory of composite manifolds" Trudy Sem. Vektor. Tenzor. Anal. , 8 (1950) pp. 11–72 (In Russian)
[2]	L.E. Evtushik, "Non-linear connections of higher order" Izv. Vuz. Mat. , 2 (1969) pp. 34–44 (In Russian)
[3]	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
[4]	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)
[5]	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