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Connection

$\def\G{\varGamma}$

An infinitesimal structure on a (smooth) bundle which allows to define the parallel transport between fibres of the bundle.

Historical overview

Initially the notion of connection appeared in the works by Gregorio Ricci and Tullio Levi-Civita who constructed the infinitesimal parallel transport between tangent spaces of a Riemannian manifold. Later the construction was distilled from the Riemannian context by Éli Cartan who constructed connections on principal bundles (equipped by action of a Lie group) and introduced the notion of the connection form. Finally, Charles Ehresmann gave a general definition of connection via distribution of horizontal subspaces, which is the most general construction using only the smooth structure and local triviality of the bundle.

In practice when dealing with connections, one distinguishes several particular cases of bundles and several (interrelated) ways to describe the infinitesimal parallel transport.

1. Connections on the vector bundles, defined by an operator of covariant derivation;
2. Affine connections on manifolds, defined on the tangent (and cotangent) bundle to a smooth manifold;
3. Connections on $G$-bundles equipped with fibrewise action of a Lie group $G$;
4. Levi-Civita (metric) connections on Riemannian manifolds, defined on the tangent bundle and preserving the metric structure.

Ehresmann connection

Let $\pi:E\to B$ be a smooth bundle with a generic fiber $F$. The Ehresmann connection is a "horizontal" subbundle $\G\subset TE$ of the tangent bundle, complementary to the "vertical" subbundle $\operatorname{Ker}\rd \pi$. In other words, the tangent space $T_xE$ at any point $x\in E$ is split as a direct sum of the vertical subspace $V_x=T_x F_x$ tangent to the fiber $F_x=\pi^{-1}(\pi(x))$ through $x$, and a complementary subspace $\G_x\subset T_x$; by definition of complementarity, $\dim\G_x=\dim B$, and the restriction $\rd \pi(x)|_{\G_x}\to T_{\pi(x)}B$ is an isomorphism.

This splitting allows to define the parallel transport along any smooth curve $\gamma:[0,1]\to B$ between the fibers $F_0=\pi^{-1}(\gamma(0))$ and $F_1=\pi^{-1}(\gamma(1))$ as follows. Consider the induced bundle $\pi'=\gamma^*\pi$ over the new base $B'=[0,1]$: because of the triviality of the segment $[0,1]$, this bundle is trivial, with the total space $E'\simeq F\times [0,1]$, and the horizontal subbundle $\G$ induces a horizontal subbundle $\G'=\gamma^*\G$ on $E'$. But since the base is now one-dimensional, the distribution $\G'$ is integrable and defines the foliation on $E'$, everywhere transversal to the fibers. Under mild additional conditions the leaf (integral trajectory of the integrable distribution $\G'$) passing through the point $x\in F_0=F\times\{0\}$, crosses the fiber $F_1=F\times\{1\}$ at a unique point $\tau_0^1(x)\in F_1$. By the general theorems from ordinary differential equations, the map $\tau=\tau_0^1$ is a diffeomorphism (smooth and invertible self-map) of $F$.

$G$-invariant connection

Assume that $\pi:E\to B$ is a bundle with a Lie group $G$ acting freely and transitively on the fibers (and hence on the total space), say, by the right multiplication, $E\times G\to E$, $r:(x,g)\mapsto x\cdot g$, $\pi(x\cdot g)=\pi(x)$. Then is natural to consider the equivariant ($G$-invariant) connections for which the horizontal subspaces $\G_x$ and $\G_{x\cdot g}$ are related by the differential of the corresponding right shift $\rd r(\cdot,g)$.

Such invariance means that for any smooth curve $\gamma$ the corresponding parallel transport $\tau_\gamma:F_0\to F_1$ commutes with the action of $G$: $\tau(x\cdot g)=\tau(x)\cdot g$ for any $g\in G$. In other words, it is sufficient to construct the parallel transport for only one point on the fiber.

Note also that for principal $G$-bundles the tangent spaces $T_x F_b$ to the same fiber $\pi^{-1}(b)$ are canonically isomorphic to each other and to the tangent space $T_e G=\mathfrak g$ which is a Lie algebra of the group $G$. The isomorphism is defined by the differential of the map $(G,e)\to (F,x)$, $g\mapsto x\cdot g$.

Affine connection

If the fiber $F$ of a bundle $\pi:E\to B$ is a (finite-dimensional) vector space, then this linear structure induces the structure of a module (over the ring $C^\infty(B)$ of smooth functions on the base) on the space of sections: for any two sections $s_1,s_2:B\to E$ and any smooth function $f:B\to \R$ the sum $s_1+s_2:B\to E$ and the products $f\cdot s_i:B\to E$ are well-defined smooth sections of $\pi$.

The splitting of the tangent space $T_x E=V_x\oplus \G_x$ into the vertical and horizontal sections defines two commuting linear projections, the horizontal one $T_x E\overset {h_x}\longrightarrow\G_x\overset{\rd \pi}{\longrightarrow} T_{\pi(x)}B$ (parallel to the vertical direction $V_x$) and $v_x:T_x E\to V_x$(parallel to the horizontal direction $\G_x$).

If $s:B\to E$ is a smooth section and $w\in T_b B$ a vector tangent to the base at some point $b\in B$, $\rd s(b)\cdot w\in T_xE$ is a vector tangent to $E$ at $x=s(b)$. The "vertical component" of the this vector, $v_x\cdot \rd s(b)\cdot w\in T_x F_x$ is called the covariant derivative of the section $s$ in direction of $w$. The usual notation is $$ \nabla_w s(b)\in T_{s(b)}F_b. $$ Note that by construction the covariant derivative is linear with respect to $w$, as a composition of linear maps $\rd s(b):T_b B\to T_{s(b)}E$ and $v_x:T_x\to V_x$, $x=s(b)$. If $s_1,s_2$ are two connections, then the corresponding covariant derivative of their sum $s_1+s_2$ is tangent to the same fiber $\pi^{-1}(b)$ yet at the new point $s_1(b)+s_2(b)$, ditto for the section $f\cdot s$.

Definition. The Ehresmann connection on a vector bundle is called affine, if the covariant derivation operator is additive and satisfies the Leibniz rule: $$ \nabla_w (s_1+s_2)(b)=\nabla_w s_1(b)+\nabla_w s_2(b),\qquad \nabla_w (f\cdot s)(b)= (\rd f\cdot w)\cdot s(b)+f(b)\cdot \nabla_w s(b)\qquad\forall b\in B. $$

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Let be a smooth locally trivial fibration with typical fibre on which a Lie group acts effectively and smoothly. A connection on this fibre bundle is a mapping of the category of piecewise-smooth curves in the base into the category of diffeomorphisms of the fibres that associates with a curve (with initial point and end point ) a diffeomorphism satisfying the following axioms:

1) for , , , and one has

2) for an arbitrary trivializing diffeomorphism and for an , the diffeomorphism , where , is defined by the action of some element ;

3) for an arbitrary piecewise-smooth parametrization , the mapping , where is the image of under , defines a piecewise-smooth curve in that starts from the unit element ; moreover, with a common non-zero tangent vector define paths in with a common tangent vector that depends smoothly on and .

The diffeomorphism is called the parallel displacement along . The parallel displacements along all possible closed curves form the holonomy group of the connection at ; this group is isomorphic to a Lie subgroup of that does not depend on . A curve on is said to be horizontal for if for any and some piecewise-smooth parametrization of it. If and are given, then there always exists a unique horizontal curve , called the horizontal lift of the curve , such that ; it consists of the points . The set of horizontal lifts of all curves in determines the connection uniquely: maps the end points of all lifted curves of into the initial points.

A connection is called linear if depends linearly on for any and , or equivalently, if for any the tangent vectors of the horizontal curves beginning at form a vector subspace of , called the horizontal subspace. Here , where is the fibre through , that is, . The smooth distribution is called the horizontal distribution of the linear connection . It determines uniquely: its integral curves are the horizontal lifts.

A fibre bundle is called principal (respectively, a space of homogeneous type), and is denoted by (respectively, ), if acts simply transitively (respectively, transitively) on , that is, if for any there is exactly one (respectively, there is an) element that sends to . Suppose that acts on from the left; then a natural action from the right is defined on , where defines . Here is identified with the quotient manifold formed by the orbits , where is the stationary subgroup of a point from . More generally, can be identified with the quotient manifold of orbits relative to the action defined by .

A smooth distribution on is a horizontal distribution of some linear connection (which it determines uniquely) if and only if

for arbitrary and . All horizontal distributions on (respectively, ) are the images of such under the canonical projection (respectively, the natural lifts of such to under the canonical projection ). Often a linear connection is defined directly as a distribution with the properties mentioned above. It is known that on each , and so on every and , there is a linear connection.

A linear connection in is usually studied by using its connection form, which determines it uniquely and can be the basis for another definition. An important characteristic of a linear connection is the curvature form; this can be used to compute the Lie algebra of the holonomy group.

The idea of a connection first arose in 1917 in the work of T. Levi-Civita [1] on parallel displacement of a vector in Riemannian geometry. The notion of an affine connection was introduced by H. Weyl in 1918. In the 1920s E. Cartan (see [3]–[5]) investigated projective and conformal connections (cf. Projective connection; Conformal connection). In 1926 he gave the general concept of a "non-holonomic space with a fundamental group" (see Connections on a manifold), and identified these spaces from the point of view of the general theory of connections. In the 1940s V.V. Vagner developed an even more general concept that is close in spirit (but not in terms of the method) to the modern idea of a connection. 1950 was a decisive year; in it there appeared the survey by Vagner [6], the first notes of G.F. Laptev, which disclosed new approaches, especially analytic ones, and the work of C. Ehresmann [7] that laid the foundation of the modern global theory of connections. See also Weyl connection; Linear connection; Riemannian connection; Symplectic connection; Hermitian connection.

References

[1]	T. Levi-Civita, "Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana" Rend. Cir. Mat. Palermo , 42 (1917) pp. 173–205
[2]	H. Weyl, "Raum, Zeit, Materie" , Springer (1923)
[3]	E. Cartan, "Les espaces à connexion conforme" Ann. Soc. Polon. Math. , 2 (1924) pp. 171–221
[4]	E. Cartan, "Sur les variétés à connexion projective" Bull. Soc. Math. France , 52 (1924) pp. 205–241
[5]	E. Cartan, "Les groupes d'holonomie des espaces généralisés" Acta Math. , 48 (1926) pp. 1–42
[6]	V.V. Vagner, "Theory of a composite manifold" Trudy Sem. Vektor i Tenzor Anal. , 8 (1950) pp. 11–72 (In Russian)
[7]	Ch. Ehresmann, "Les connexions infinitésimal dans une espace fibré différentiable" , Colloq. de Topologie Bruxelles, 1950 , G. Thone & Masson (1951) pp. 29–55
[8]	G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[9]	K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956)
[10]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1955) (Translated from French)
[11]	Ü.G. Lumiste, "Connection theory in bundle spaces" J. Soviet Math. , 1 (1973) pp. 363–390 Itogi Nauk. Ser. Algebra. Topol. Geom. 1969 , 21 (1971) pp. 123–168

Comments

Consider a smooth locally trivial fibre bundle . A smooth section is a smooth mapping such that . This concept generalizes that of a function (where is the fibre of ), which is the same as a section of the trivial fibre bundle . In several areas of mathematics it is important to consider sections instead of just functions. E.g. in gauge field theory. But then one would also like to have something like the partial derivatives of a section available, i.e. the quantity that describes to first order how changes as varies (infinitesimally). This requires comparing the fibres of at neighbouring points, but there is nothing in the concept of a fibre bundle as it stands that allows one to do this. For this some extra structure is needed, and that is provided by the idea of a connection.

It would be simplest if for every two points one could prescribe an isomorphism in a consistent way, i.e. such that for all triples . Here , the fibre of over , is of course . This, however, would make the bundle trivial, and this is in general not possible. The next best thing would be to have for every smooth path from to an isomorphism (which may depend on the path ) from the fibre at the initial point of the path to the fibre at the final point, subject to certain natural restrictions. This is precisely what a connection is.

There are — at least — three intuitively natural ways of describing a connection.

i) Provide for every smooth path from to an isomorphism subject to the three conditions 1), 2), 3).

ii) For each let be the kernel of the tangent mapping at . The subspace of the tangent space to at is called the vertical tangent subspace to at . Now for each define a complementary subspace at , called the horizontal tangent subspace at . Thus, and induces an isomorphism . The are required to vary smoothly with . In the case of linear connections, cf. above, this is the infinitesimal version of i).

iii) Let be a vector bundle. Then a linear connection can also be specified by giving so to speak the partial derivatives of a section directly (covariant differentiation). This leads to the specification of a bilinear mapping , where is the space of vector fields on and is the space of sections of , with certain properties; cf. Linear connection for these properties in the case . One consequence of these properties is that , , , depends only on at . If is a smooth path starting in with tangent vector at , then

where is parallel displacement defined by .

An elegant and convenient way to describe a linear connection in the case that is a vector bundle is as follows. Let

be a local chart of and a trivialization of . Then above one has the following local trivialization of :

where the right-hand arrow is projection in the first two factors. A linear connection on is now given by a bundle mapping (i.e. the diagram

is commutative, and is linear in the fibres), such that locally the mapping looks like

The are the Christoffel symbols (relative to the trivialization ; in case , can be taken equal to so that the Christoffel symbols depend only on the chart ).

Given the connection , the horizontal subspace is defined by

and the covariant derivative of a section along a vector field is the section .

In the case of infinite-dimensional manifolds and bundles this last notion of a linear connection appears to be the appropriate replacement of the more traditional covariant derivative , cf. [a2], Sect. 1.1.

References

[a1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. 4
[a2]	W. Klingenberg, "Lectures on closed geodesics" , Springer (1979)