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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 $F$. An 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 generic fiber $F$ of a bundle $\pi:E\to B$ is a (finite-dimensional) vector space isomorphic to $\R^n$, then it is natural to consider affine connections. By definition, a connection is affine, if all the parallel transport operators are linear (invertible) maps between the corresponding fibers.

The linear structure on the generic fiber induces the structure of a module (over the ring $C^\infty(B)$ of smooth functions on the base) on the space of sections $\Gamma(E)=\{s:B\to E,\ \pi\circ s=\operatorname{id}_B\}$: 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 $\pi$, $$ (s_1+s_2)(b)=s_1(b)+s_2(b),\qquad (f\cdot s_i)(b)=f(b)\cdot s_i(b),\qquad s_i(b)\in F_b=\pi^{-1}(b),\ f(b)\in \R. $$

Connection 1-forms (Lie-algebra-valued and matrix-valued)

For each Ehresmann connection on the principal bundle the splitting of the tangent space $T_x E=V_x\oplus \G_x$ into the vertical and horizontal subspaces defines the linear projection of $T_x E$ into the vertical component $V_x$ parallel to the horizontal component $\G_x$. This projection can be interpreted as $\mathfrak g$-valued differential 1-form $\boldsymbol\omega$ on $TE$, which is "identical" on the vertical subspaces (recall that each vertical subspace $V_x$ can be identified with $\mathfrak g$) and invariant by the right action of $G$. The horizontal subspace $\G_x$ by construction is the common null space defined by the Pfaffian equations $\boldsymbol\omega=0$.

If the principal bundle $\pi$ is defined by local trivializations and the corresponding cocycle $\{g_{\alpha\beta}:U_{\alpha\beta}\to G\}$, this means that points from $G\times U_\alpha$ are identified with points from $G\times U_\beta$ over $U_{\alpha\beta}=U_\alpha\cap U_\beta$ by the gauge transformation rule $$ (v_\alpha,b)\simeq (v_\beta,b)\ \iff\ v_\alpha=g_{\alpha\beta}\cdot v_\beta,\ v_\beta=g_{\beta\alpha}\cdot v_\alpha.\tag{GT} $$

In the trivialization over $U_\alpha$ (resp., over $U_\beta$) the connection is defined the tuple of $\mathfrak g$-valued forms $\boldsymbol\omega_\alpha$ (resp., $\boldsymbol\omega_\beta$). To be self-consistent over the intersection $U_{\alpha\beta}$, one tuple should be transformed into the other by the gauge transformation: for any two sections $s_\alpha$ and $s_\beta=g_{\beta\alpha}\cdot s_\alpha$ over $U_{\alpha\beta}$, the linear operators $\boldsymbol\omega_\alpha\circ \rd s_\alpha: T_b B\to \mathfrak g$ and $\boldsymbol\omega_\beta\circ \rd s_\beta:T_b B\to\mathfrak g$ must coincide after computing the differentials.

For an Ehresmann connection on a vector bundle the tangent space to the linear fiber $F\simeq\R^n$ is naturally identified with itself, thus in a local trivialization $\R^n\times U_\alpha$ with the coordinates $(v_1,\dots,v_n,b_1,\dots,b_m)$ the connection form can be associated with the tuple of 1-forms $\omega_{i\alpha}=\rd v_i-\sum_{k=1}^m F_{ik\alpha}(v,b)\rd b_k$, where $m=\dim B$, and $i=1,\dots,n=\dim F$. If the connection is affine (see above), then the dependence of the coefficients $F_{ik\alpha}(v,b)$ on $v$ must be linear. This allows to write the (vector) connection form $\boldsymbol\omega_\alpha$ as $$ \boldsymbol\omega_\alpha=\rd v_\alpha-\Omega_\alpha\cdot v_\alpha,\qquad\Omega_\alpha=\begin{pmatrix} \theta_{11,\alpha}&\cdots&\theta_{1n,\alpha}\\ \vdots &\ddots&\vdots\\ \theta_{n1,\alpha}&\cdots&\theta_{nn,\alpha}\end{pmatrix},\quad v=\begin{pmatrix} v_{1,\alpha}\\ \vdots \\ v_{n,\alpha}\end{pmatrix}, \quad \rd v_\alpha=\begin{pmatrix} \rd v_{1,\alpha}\\ \vdots \\ \rd v_{n,\alpha}\end{pmatrix}, $$ with $\theta_{ij,\alpha}$ being 1-forms on the base $U_\alpha\subseteq B$, $v_\alpha=(v_{1,\alpha},\dots,v_{n,\alpha})$ the coordinates on the fiber and $\rd v_\alpha$ their respective differentials. The matrix 1-form $\Omega_\alpha$ is called the connection matrix in the given trivialization.

The coefficients of the 1-forms $\theta_{ik}$ are called the Christoffel symbols: $$ \theta_{ik}=\sum_j\Gamma_{ik}^j(b)\,\rd b_j,\tag{C} $$ where $(b_1,\dots,b_m)$ are local coordinates in an open set $U=U_\alpha$, and $\theta_{ik}$ the 1-forms constituting the connection matrix form over $U$. The Christoffel symbols are smooth functions on the base of a vector bundle.

Passing from one trivialization to another (over $U_\beta$) means replacing the coordinate vector $v_\alpha$ by the new coordinate vector $v_\beta=M_{\beta\alpha}\cdot v_\alpha$ with $M_{\alpha\beta}(\cdot)=M^{-1}_{\beta\alpha}(\cdot)$ being a smooth matrix-valued function (cocycle) on $U_{\alpha\beta}$. Applying this gauge transformation, we conclude by the Leibniz rule that $$ \rd v_\beta=\rd (M_{\beta\alpha}\cdot v_\alpha)=\rd M_{\beta\alpha}\cdot v_\alpha+M_{\beta\alpha}\cdot \rd v_\alpha=(\rd M_{\beta\alpha}+M_{\beta\alpha}\Omega_\alpha)\cdot M_{\alpha\beta}\cdot v_\beta, $$ that is, $$ \Omega_\beta=\rd M_{\beta\alpha}\cdot M_{\alpha\beta}+M_{\beta\alpha}\cdot\Omega_\alpha \cdot M_{\alpha\beta} \tag{GC} $$ (note the mnemonic order of indices $\alpha,\beta$). The transformation law (GC), sometimes called the gauge transformation of the connection matrux form, involves the "matrix logarithmic derivative" term $\rd M\cdot M^{-1}$ and hence the matrices $\Omega_\alpha$ "do not form a tensor".

Example. Let $\gamma:[0,1]\to B$ a smooth path, $t\mapsto \gamma(t)$, entirely belonging to one chart $U=U_\alpha$, with the corresponding matrix connection form $\Omega=\Omega_\alpha$. Then the induced connection on $\R^n\times[0,1]$ is defined by system of linear ordinary differential equations with variable coefficients of the form $$ \frac{\rd v}{\rd t}=A(t)\cdot v,\qquad A(t)=i_{\dot\gamma(t)}\,\Omega(\gamma(t)),\tag{LS} $$ with the matrix of coefficients $A$ obtained by evaluation of the matrix 1-form $\Omega$ on the velocity vector $\dot\gamma(t)$. The result of the parallel transport along $\gamma$ is the value at $t=1$ of the fundamental matrix solution $V(t)$ to the system (LS) with the initial condition $V(0)=\operatorname{id}$.

Covariant derivative

Any smooth section $s\in\Gamma(E)$ of a vector bundle $\pi:E\to B$ equipped with an affine connection, can be differentiated along any smooth curve $\gamma:(\R^1,0)\to (B,b)$ in the base. By definition, the result, called the "absolute", or covariant derivative at the initial moment $t=0$, i.e., at the point $b=\gamma(0)$, is the limit $$ D_\gamma s(b)=\lim_{t\to 0}\tfrac1t(\tau_t^0 \bigl(s(t)\bigr)-s(0))\in \pi^{-1}(b), $$ where $\tau_t^0=(\tau_0^t)^{-1}:\pi^{-1}\bigl(\gamma(t)\bigr)\to\pi^{-1}\bigl(\gamma(0)\bigr)$ is the parallel transport between two close fibers over the points $\gamma(t)$ and $\gamma(0)$ on the curve.

From this definition and the linearity of $\tau$ it follows that $D_\gamma$ is the additive operation satisfying the Leibniz rule, if we extend it to smooth functions by the natural way as the Lie derivation: $$ D_\gamma(s_1+s_2)=D_\gamma s_1+D_\gamma s_2,\qquad D_\gamma (f\cdot s)=(D_\gamma f)\cdot s+f\cdot D_\gamma s. $$

In each trivializing chart $U_\alpha$ a smooth section $s(\cdot)$ can be identified with a vector-function $s_\alpha:U_\alpha\to\R^n$ and the covariant derivative can be expressed through the matrix 1-form of the connection $\Omega_\alpha$ as follows: $$ D_\gamma s(b)=i_{\dot \gamma(0)}\bigl(\rd s_\alpha-\Omega_\alpha\cdot s_\alpha\bigr)=\left.\frac{\rd s_\alpha(\gamma(t))}{\rd t}\right|_{t=0}-A(b)\cdot s_\alpha(b),\qquad A(b)=(i_{\dot\gamma(0)}\Omega_\alpha)(b). $$ This computation shows that the covariant derivative in fact depends only on the velocity vector $w=\dot\gamma(0)$ of the curve $\gamma$, and does this in a linear way. Thus the covariant derive becomes a differential operator (usually denoted by $\nabla_w$) which generalizes the directional derivative $\nabla_w f=i_w\rd f=L_w f$: $$ \nabla_w:\Gamma(E)\to\Gamma(E),\quad \forall f\in C^\infty(B),\qquad \nabla_{fw}=f\nabla_w,\quad \nabla_w(fs)=f\,(\nabla_w s)+(\nabla_w f)\, s. $$

Covariant derivative and the Christoffel symbols

To describe completely a linear connection in any trivializing chart $F\times U\simeq \{(v,b):v=(v_1,\dots,v_n)\in\R^n,\ b=(b_1,\dots,b_m)\in V\subseteq\R^m\}$, it is sufficient to specify the coefficients of the expansion of the covariant derivatives along all coordinate axes of $n$ linear independent sections $\mathrm e_1,\dots\mathrm e_n\in\Gamma(E)$, e.g., the coefficients $\Gamma_{ij}^k=\Gamma_{ij}^k(b)$ of the expansions $$ \nabla_{\frac\partial{\partial b_i}}\mathrm e_j=\sum_{k=1}^n\Gamma_{ij}^k(b)\mathrm e_k,\qquad\text{where}\quad\mathrm e_j(b)=(0,\dots,\underset{j}1,\dots,0),\quad j,k=1,\dots,n, \ i=1,\dots,m. $$

Covariant derivative and parallel transport

A section $s\in\Gamma(E)$ is said to be parallel along $\gamma$, if $D_\gamma s\equiv0$. If $s$ is such a section, it defines the result of the parallel transport of the vector $s(\gamma(0))\in F_{\gamma(0)}$ to be the vector $s(\gamma(t))\in F_{\gamma(t)}$ for all points $\gamma(t)$ on the curve.

For any vector $v\in F_{a}$ and any curve $\gamma:(\R^1,0)\to (B,a)$ one can construct a unique section $s$ defined along $\gamma$ and parallel along it with the initial condition $s(a)=v$. This section is constructed as a solution to a system of linear ordinary differential equations $$ \frac{\rd }{\rd t}s(\gamma(t))=i_{\dot\gamma(t)}\Omega_\alpha(\gamma(t))\cdot s(\gamma(t)),\qquad s:(\R^1,0)\to\R^n $$ in any trivializing chart.

Parallel transport is independent of the parametrization of the curve

If $s$ is a section parallel along a curve $\gamma$, then this fact is not changed by another choice of the parametrization of $\gamma$. Indeed, this choice would replace the derivative $D_\gamma$ by a (non-constant) multiple, without changing solutions of the equation $D s=0$.

Dual of the covariant derivative

The construction of the covariant derivative via the parallel transport allows to derivate also sections of the dual bundle. For a parametrized curve $\gamma:(\R^1,0)\to (B,b)$ he family of linear operators $\{\tau_t^s:F_t\to F_s,\ t,s\in(\R^1,0)\}$ defines the parallel translation between fibers $F_t=\pi^{-1}(t)$ and $F_s=\pi^{-1}(s)$ along $\gamma$. The dual bundle has fibers $F_t^*$ dual to $F_t$, and the parallel transport is realized by the adjoint operators $(\tau_t^s)^*={\tau^*}_s^t$ in the opposite direction. However, this allows to define the covariant directional derivative of sections of the dual bundle $E^*$ using the same construction. If $s\in\Gamma(E)$ and $s^*\in\Gamma(E^*)$ are two sections of the dual bundles, then the parallel transport agrees with the pairing: $$ \nabla_w\left< s, s^* \right>=\left< \nabla_w s,s^* \right> +\left< s,\nabla^*_w s^*\right> $$ (the left hand side is the Lie derivative of the scalar function). This identity can be used to define the covariant derivative $\nabla^*_w$ of the section $s^*$ and the corresponding 1-form $\nabla^*$. If in a trivializing chart $U_\alpha$ the covariant derivation takes the form $\nabla=\rd -\Omega_\alpha$, then $\nabla^*=\rd +\Omega_\alpha^*$, where $\Omega^*_\alpha$ is the transpose (in the real case) of the matrix 1-form $\Omega_\alpha$.

In practice, however, the dual covariant derivative $\nabla^*$ is denoted by the same symbol $\nabla$.

Covariant derivation of tensor and exterior products

The calculus of covariant derivations on a vector bundle $E$ and its dual $E^*$ extends naturally on bundles which fiberwise are tensor or exterior products of $E$ and $E^*$. The corresponding formulas always have the form of suitable Leibniz identities, e.g., for a section $s_1\otimes s_2$ of the tensor product bundle $E\otimes E$ its covariant derivative is computed as follows, $$ \nabla(s_1\otimes s_2)=(\nabla s_1)\otimes s_2+s_1\otimes(\nabla s_2). $$ Derivative of a wedge product of two sections $\xi_1\land\xi_2$ of the bundle $E^*\land E^*$ follows the rule $$ \nabla(\xi_1\land \xi_2)=(\nabla \xi_1)\land \xi_2+(-1)^n\xi_1\land (\nabla \xi_2),\qquad n=\dim F, $$ etc. An important case is the maximal exterior power $\bigwedge ^n E$ of the $n$-dimensional vector bundle $E$, a line (1-dimensional) bundle called the determinant bundle. By Liouville--Ostrogradskii formula, the corresponding covariant derivative is the trace $\operatorname{tr}\nabla$ of the connection $\nabla$. In a trivializing chart in which $\nabla=\rd-\Omega_\alpha$, the trace takes the form $\operatorname{tr}\nabla=\rd-\operatorname{tr}\Omega_\alpha$, where the trace of a matrix 1-form $\Omega_\alpha$ is the sum of (scalar) diagonal 1-forms.

Connections on tangent/cotangent bundles of a smooth manifold

A very important particular case of linear connections are connections on the tangent bundle $TM$, which by duality descend on the cotangent bundle $T^*M$ and their tensor/wedge products. For those connections one can compare the covariant derivatives $\nabla_ w u$ and $\nabla_u w$ for two vector fields $u,w\in\Gamma(TM)$, sections of the same tangent bundle. This allows to introduce a subclass of linear connections, called symmetric connections.

A connection $\nabla$ is symmetric, if for any two vector fields $u,w$ $$ \nabla_w u-\nabla_u w-[w,u]=0,\tag{T} $$ with the commutator of two vector fields appearing in the right hand side.

Torsion tensor

The left hand side of the expression (T) depends on two vector fields $u,w$, but is a tensor field, not a derivation: if we replace these fields by $\varphi u$ and $\psi w$, with $\varphi,\psi$ being two (scalar) smooth functions, the result will be multiplied by the product $\varphi\psi$: $$ \nabla_{\psi w}u-\nabla_u (\psi w)-[\psi w,u]=\psi\,\nabla_w u-(\nabla_u\psi)\, w-\psi\,\nabla_u w +(L_u \psi)w-\psi\,[w,u]=\psi\,(\nabla_w u-\nabla_u w-[w,u]), $$ where $L_u\psi=\nabla_u \psi=i_u\rd \psi$ is the Lie derivative of $\psi$ along $u$, and the same regarding $\varphi$.

This is called the torsion tensor of the connection $\nabla$. Its vanishing means that the "second covariant derivative" in two different directions is symmetric. To be more precise, note that the expression $$ \sigma_{u,w}=\nabla_u\circ \nabla_w-\nabla_u w $$ can be considered as a second order differential operator on smooth functions, $f\mapsto\sigma_{u,w}f$. The value $(\sigma_{u,w}f)(a)$ at each point $a$ depends only on the vectors $u(a)$ and $w(a)$ (the computation is the same as above). The connection is symmetric, if $\sigma_{u,w}=\sigma_{w,u}$ for any two vector fields. This is the condition generalizing the independence of the mixed derivatives on their order.

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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)