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=Connection=
$\def\G{\varGamma}$

An infinitesimal structure on a (smooth) [[bundle]] which allows to define the [[parallel transport]] between [[Fibre space|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.
# Connections on the vector bundles, defined by an operator of covariant derivation;
# Affine connections on manifolds, defined on the tangent (and cotangent) bundle to a smooth manifold;
# Connections on $G$-bundles equipped with fibrewise action of a Lie group $G$;
# 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 [[Parallel transport#ehresmann|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<ref>E.g., if the fiber $F$, is compact, but also if it is a finite-dimensional vector- or $G$-space with $\G'$ linear or $G$-invariant.</ref> 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''<ref>The term ''linear connection'' today seems to be used as a complete synonym of the term ''affine connection''.</ref> 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 [[section]]s $\Gamma(E)=\{s:B\to E,\ \pi\circ s=\operatorname{id}_B\}$: 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<ref>I.e., a tuple of usual, "scalar" 1-forms, of cardinality equal to $\dim\mathfrak g$.</ref> 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 [[Bundle#G-cocycle|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<ref>The usual way to write the corresponding formulas explicitly involves canonical $\mathfrak g$-valued 1-form on the Lie group, associated with the right shift action of $G$ on itself.</ref>.  

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 $\theta_{jk,\alpha}$ 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]]<ref>This term has also other meanings in the theory of linear ordinary differential equations with analytic coefficients, see [[Stokes phenomenon]].</ref> in the given trivialization. 

The coefficients of the 1-forms $\theta_{jk}$ are called the [[Christoffel symbol]]s:
$$
\theta_{jk}=\sum_j\Gamma_{jk}^i(b)\,\rd b_i,\tag{C}
$$
where $(b_1,\dots,b_m)$ are local coordinates in an open set $U=U_\alpha$, and $\theta_{jk}$ 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<ref>Here $\rd M(\cdot)$ is the matrix 1-form on $U_{\alpha\beta}$ whose components are the differentials of the entries of the matrix function $M(\cdot)$.</ref> 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<ref>That is, $D_\gamma f(b)=\lim_{t\to 0}\frac1t (f(\gamma(t))-f(\gamma(0))$: note that scalar functions take values in the same space $\R$ over all points on the curve $\gamma$.</ref>:
$$
D_\gamma(s_1+s_2)=D_\gamma s_1+D_\gamma s_2,\qquad D_\gamma (f\cdot s)=(D_\gamma f)\cdot s+f\cdot D_\gamma s.
$$



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
$$
\nabla_{\frac\partial{\partial b_i}}\mathrm e_j=\sum_{k=1}^n\Gamma_{ij}^k(b)\mathrm e_k,\qquad\text{where}\quad\mathrm e_j(b)=(0,\dots,\underset{j}1,\dots,0),\quad j,k=1,\dots,n, \ i=1,\dots,m.
$$
These coefficients are the same Christoffel symbols (C).

====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 {{anchor|tensor-derivative}}====

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.

==Curvature of an affine connection==
''The main article is [[Curvature form]].''

The parallel transport between two different fibers of a vector bundle depends on the smooth (or piecewise-smooth) path connecting the base points. The curvature is a local differential expression of this dependence. More specifically, the curvature is a map which associates with any two vectors $u_a,w_a\in T_a B$ tangent to the base, the properly normalized linear operator of parallel transport along the perimeter of an infinitesimally small parallelogram, a closed path in the base, with sides parallel to $u_a$ and $w_a$. More precisely, let $u,w$ be two ''commuting'' vector fields on $B$ which extend the vectors $u_a=u(a)$ and $w_a=w(a)$. Then the curvature is the commutator of two differential operators<ref>For a general pair of vector fields $u,w\in\Gamma (TB)$, the curvature is defined as $[\nabla_u,\nabla_w]-\nabla_{[u,w]}$.</ref>,
$$
R_{u,w}=\nabla_u\nabla_w-\nabla_w\nabla_u:\Gamma(E)\to\Gamma(E).
$$
One can instantly verify that $R_{\varphi u,\psi w}=\varphi\psi R_{u,w}$ and is additive in each argument $u,w$ separately, thus $R$ depends in the bilinear (antisymmetric) way on the vectors defining the "infinitesimally small loop".

A connection is called ''flat'', if its curvature is zero, that is, the result of the parallel transport between two fibers does not depend on (small) variations of the path connecting the respective base points. The flatness is equivalent to vanishing of the curvature and commutation of the covariant derivatives along commuting vector fields. However, flat connections may have nontrivial parallel transport along non-contractible loops.

== 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 connection]]s.

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 in a ring|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<ref>{{Cite|Mi|Chapter II, Definition 8.5 and the footnote}}.</ref>. This is the condition generalizing the independence of the mixed derivatives on their order.

== Connection on Riemannian manifolds ==
For a connection $\nabla$ defined on the tangent bundle $TM$ of a [[Riemannian manifold]] $M$ (with the natural isomorphism between $TM$ and $T^*M$) it is natural to require certain compatibility between $\nabla$ and the metric structure. This condition can be formulated in several equivalent forms:
* The parallel transport along any curve is an isometric operator;
* The covariant derivative satisfies the Leibniz rule $\nabla_w\left< u,v \right>=\left< \nabla_w u,v \right> +\left< u,\nabla_w v \right>$, where $\nabla_w$ acts on the function $\left< u,w \right>$ as the usual Lie (directional) derivative;
* The covariant  derivative $\nabla g$ of the metric tensor $g$ (in the sense of the [[#tensor-derivative|derivation of tensors]] is identically zero.

A "miracle" {{cite|Be}}, also known as the Principal Lemma of Riemannian Geometry {{Cite|Mi|Lemma 8.6}} is the fact that for any Riemannian manifold there is a unique ''symmetric'' (torsion-less) connection compatible with the Riemannian metric. 

====Computational formulas====

The proof is given by a straightforward computation in any local coordinates: denoting by $g_{ij}=\left<\partial_i,\partial_j \right>$ the components of the metric tensor<ref>By definition, $\partial_i=\frac{\partial}{\partial b_i}$ are the coordinate vector fields on $M$, and $g_{ij}=g_{ji}$.</ref> and by $\Gamma_{ij}^k$ the Christoffel symbols, we obtain the  identities
$$
\nabla_{\partial_i}\left<\partial_j,\partial_k\right>=\left<\nabla_{\partial_i}\partial_j,\partial_k\right>+\left<\partial_j,\nabla_{\partial_i}\partial_k\right>,
$$
and two other similar identities obtained by cyclical permutation of the indices $i,j,k=1,\dots,m$, $m=\dim M$. Thus we have three linear equations with respect to only three unknowns
$$
\left<\nabla_{\partial_i}\partial_j,\partial_k\right>,\quad \left<\nabla_{\partial_j}\partial_k,\partial_i\right>, \quad \left<\nabla_{\partial_k}\partial_i,\partial_j\right>
$$
(the remaining terms coincide with one of these because of the symmetry of the scalar product $\left< \cdot,\cdot \right>$ and absence of the torsion, $\nabla_{\partial_i}\partial_j=\nabla_{\partial_j}\partial_i$).

This system of three equations is immediately solvable, yielding the identity
$$
\sum_{\ell}\Gamma_{ij}^\ell g_{\ell k}=\frac12\biggl(\frac{\partial}{\partial b_i}g_{jk}+\frac{\partial}{\partial b_j}g_{ki}+\frac{\partial}{\partial b_k}g_{ij}\biggr).
$$
Introducing the symmetric matrix function $g^{k\ell}$ inverse to the metric tensor, we can resolve the above system of linear equations with respect to the Christoffel symbols:
$$
\Gamma_{ij}^\ell=\frac12\sum_{k=1}^m \biggl(\frac{\partial}{\partial b_i}g_{jk}+\frac{\partial}{\partial b_j}g_{ki}+\frac{\partial}{\partial b_k}g_{ij}\biggr)g^{k\ell}.
$$
This proves the uniqueness of the connection: its Christoffel symbols are uniquely defined by the components of the metric tensor and their derivatives. One can verify that the connection defined by the above formulas, is indeed compatible with the metric.

==...==
<references/>
----

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1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c0251409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514012.png" /> one has

<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/c/c025/c025140/c02514013.png" /></td> </tr></table>

2) for an arbitrary trivializing diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514014.png" /> and for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514015.png" />, the diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514017.png" />, is defined by the action of some element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514018.png" />;

3) for an arbitrary piecewise-smooth parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514019.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514021.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514022.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514023.png" />, defines a piecewise-smooth curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514024.png" /> that starts from the unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514025.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514026.png" /> with a common non-zero tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514027.png" /> define paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514028.png" /> with a common tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514029.png" /> that depends smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514031.png" />.

The diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514032.png" /> is called the [[Parallel displacement(2)|parallel displacement]] along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514033.png" />. The parallel displacements along all possible closed curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514034.png" /> form the [[Holonomy group|holonomy group]] of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514035.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514036.png" />; this group is isomorphic to a Lie subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514037.png" /> that does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514038.png" />. A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514040.png" /> is said to be horizontal for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514041.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514042.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514043.png" /> and some piecewise-smooth parametrization of it. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514045.png" /> are given, then there always exists a unique horizontal curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514046.png" />, called the horizontal lift of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514047.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514048.png" />; it consists of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514049.png" />. The set of horizontal lifts of all curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514051.png" /> determines the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514052.png" /> uniquely: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514053.png" /> maps the end points of all lifted curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514054.png" /> into the initial points.

A connection is called linear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514055.png" /> depends linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514056.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514058.png" />, or equivalently, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514059.png" /> the tangent vectors of the horizontal curves beginning at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514060.png" /> form a vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514062.png" />, called the horizontal subspace. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514064.png" /> is the fibre through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514065.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514066.png" />. The smooth distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514067.png" /> is called the [[Horizontal distribution|horizontal distribution]] of the linear connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514068.png" />. It determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514069.png" /> uniquely: its integral curves are the horizontal lifts.

A fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514070.png" /> is called principal (respectively, a space of homogeneous type), and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514071.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514072.png" />), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514073.png" /> acts simply transitively (respectively, transitively) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514074.png" />, that is, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514075.png" /> there is exactly one (respectively, there is an) element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514076.png" /> that sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514077.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514078.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514079.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514080.png" /> from the left; then a natural action from the right is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514081.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514082.png" /> defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514083.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514084.png" /> is identified with the quotient manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514085.png" /> formed by the orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514086.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514087.png" /> is the stationary subgroup of a point from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514088.png" />. More generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514089.png" /> can be identified with the quotient manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514090.png" /> of orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514091.png" /> relative to the action defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514092.png" />.

A smooth distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514093.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514094.png" /> is a horizontal distribution of some linear connection (which it determines uniquely) if and only if

<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/c/c025/c025140/c02514095.png" /></td> </tr></table>

for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514097.png" />. All horizontal distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514098.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c02514099.png" />) are the images of such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140100.png" /> under the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140101.png" /> (respectively, the natural lifts of such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140102.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140103.png" /> under the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140104.png" />). Often a linear connection is defined directly as a distribution with the properties mentioned above. It is known that on each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140105.png" />, and so on every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140107.png" />, there is a linear connection.

A linear connection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140108.png" /> is usually studied by using its [[Connection form|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|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 [[#References|[1]]] on parallel displacement of a vector in Riemannian geometry. The notion of an [[Affine connection|affine connection]] was introduced by H. Weyl in 1918. In the 1920s E. Cartan (see [[#References|[3]]]–[[#References|[5]]]) investigated projective and conformal connections (cf. [[Projective connection|Projective connection]]; [[Conformal connection|Conformal connection]]). In 1926 he gave the general concept of a  "non-holonomic space with a fundamental group"  (see [[Connections on a manifold|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 [[#References|[6]]], the first notes of G.F. Laptev, which disclosed new approaches, especially analytic ones, and the work of C. Ehresmann [[#References|[7]]] that laid the foundation of the modern global theory of connections. See also [[Weyl connection|Weyl connection]]; [[Linear connection|Linear connection]]; [[Riemannian connection|Riemannian connection]]; [[Symplectic connection|Symplectic connection]]; [[Hermitian connection|Hermitian connection]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,   "Raum, Zeit, Materie" , Springer  (1923)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Cartan,   "Les espaces à connexion conforme"  ''Ann. Soc. Polon. Math.'' , '''2'''  (1924)  pp. 171–221</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Cartan,   "Sur les variétés à connexion projective"  ''Bull. Soc. Math. France'' , '''52'''  (1924)  pp. 205–241</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Cartan,   "Les groupes d'holonomie des espaces généralisés"  ''Acta Math.'' , '''48'''  (1926)  pp. 1–42</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.V. Vagner,   "Theory of a composite manifold"  ''Trudy Sem. Vektor i Tenzor Anal.'' , '''8'''  (1950)  pp. 11–72  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Ch. Ehresmann,   "Les connexions infinitésimal dans une espace fibré différentiable" , ''Colloq. de Topologie Bruxelles, 1950'' , G. Thone &amp; Masson  (1951)  pp. 29–55</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  K. Nomizu,   "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
Consider a smooth locally trivial fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140109.png" />. A smooth section is a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140110.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140111.png" />. This concept generalizes that of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140112.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140113.png" /> is the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140114.png" />), which is the same as a section of the trivial fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140115.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140116.png" /> changes as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140117.png" /> varies (infinitesimally). This requires comparing the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140118.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140119.png" /> one could prescribe an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140120.png" /> in a consistent way, i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140121.png" /> for all triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140122.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140123.png" />, the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140124.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140125.png" />, is of course <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140126.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140127.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140128.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140129.png" /> an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140130.png" /> (which may depend on the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140131.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140132.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140133.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140134.png" /> an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140135.png" /> subject to the three conditions 1), 2), 3).

ii) For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140136.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140137.png" /> be the kernel of the tangent mapping at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140138.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140139.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140140.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140141.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140142.png" /> is called the vertical tangent subspace to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140143.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140144.png" />. Now for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140145.png" /> define a complementary subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140146.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140147.png" />, called the horizontal tangent subspace at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140148.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140150.png" /> induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140151.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140152.png" /> are required to vary smoothly with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140153.png" />. In the case of linear connections, cf. above, this is the infinitesimal version of i).

iii) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140154.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140155.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140156.png" /> is the space of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140157.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140158.png" /> is the space of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140159.png" />, with certain properties; cf. [[Linear connection|Linear connection]] for these properties in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140160.png" />. One consequence of these properties is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140163.png" />, depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140164.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140165.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140166.png" /> is a smooth path starting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140167.png" /> with tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140168.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140169.png" />, then

<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/c/c025/c025140/c025140170.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140171.png" /> is parallel displacement defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140172.png" />.

An elegant and convenient way to describe a linear connection in the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140173.png" /> is a vector bundle is as follows. Let

<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/c/c025/c025140/c025140174.png" /></td> </tr></table>

be a local chart of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140175.png" /> and a trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140176.png" />. Then above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140177.png" /> one has the following local trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140178.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/c/c025/c025140/c025140179.png" /></td> </tr></table>

where the right-hand arrow is projection in the first two factors. A linear connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140180.png" /> is now given by a bundle mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140181.png" /> (i.e. the diagram

<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/c/c025/c025140/c025140182.png" /></td> </tr></table>

is commutative, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140183.png" /> is linear in the fibres), such that locally the mapping looks like

<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/c/c025/c025140/c025140184.png" /></td> </tr></table>

<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/c/c025/c025140/c025140185.png" /></td> </tr></table>

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140186.png" /> are the Christoffel symbols (relative to the trivialization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140187.png" />; in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140189.png" /> can be taken equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140190.png" /> so that the Christoffel symbols depend only on the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140191.png" />).

Given the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140192.png" />, the horizontal subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140193.png" /> is defined by

<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/c/c025/c025140/c025140194.png" /></td> </tr></table>

and the covariant derivative of a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140195.png" /> along a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140196.png" /> is the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140197.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025140/c025140198.png" />, cf. [[#References|[a2]]], Sect. 1.1.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,   "Lectures on closed geodesics" , Springer  (1979)</TD></TR></table>