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Connection

$\def\G{\varGamma}$

2020 Mathematics Subject Classification: Primary: 53.xx Secondary: 53Bxx53Cxx55Rxx [MSN][ZBL]

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 equivalent ways to describe the infinitesimal parallel transport.

1. Connections on abstract vector bundles, defined by an operator of covariant derivation on sections or the distribution of horizontal subspaces;
2. Connections on abstract $G$-bundles equipped with fibrewise action of a Lie group $G$ and invariant by this action;
3. Affine and linear connections on manifolds, defined on the tangent (and cotangent) bundle to a smooth manifold;
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, and the fibers have the natural structure of a homogeneous $G$-space.

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 $(G,e)\to (F,x)$, $g\mapsto x\cdot g$.

Linear and 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 linear connections. By definition, a connection is linear 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 $\pi$, $$ (s_1+s_2)(b)=s_1(b)+s_2(b),\qquad (f\cdot s_i)(b)=f(b)\cdot s_i(b),\qquad s_i(b)\in F_b=\pi^{-1}(b),\ f(b)\in \R. $$

Affine connection

If the fiber is a homogeneous $G$-space with $G=\R^n$ being the additive group of the vector $n$-space, then instead of the linearity one should require that the parallel transport is an affine map (composition of a linear map and a translation). Any affine connection on a bundle with $n$-dimensional fibers can be uniquely lifted to a suitable linear connection with $(n+1)$-dimensional leaves.

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 $\theta_{jk,\alpha}$ 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 in the given trivialization.

The coefficients of the 1-forms $\theta_{jk}$ are called the Christoffel symbols: $$ \theta_{jk}=\sum_i\Gamma_{jk}^i(b)\,\rd b_i,\tag{C} $$ 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 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 $$ \nabla_{\frac\partial{\partial b_i}}\mathrm e_j=\sum_{k=1}^n\Gamma_{ij}^k(b)\mathrm e_k,\qquad\text{where}\quad\mathrm e_j(b)=(0,\dots,\underset{j}1,\dots,0),\quad j,k=1,\dots,n, \ i=1,\dots,m. $$ 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

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, $$ R_{u,w}=\nabla_u\nabla_w-\nabla_w\nabla_u:\Gamma(E)\to\Gamma(E). $$ 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 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.

Connections on Riemannian manifolds

Main article: Levi-Civita connection.

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 derivation of tensors is identically zero.

A "miracle" [Be], also known as the Principal Lemma of Riemannian Geometry [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 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 $$ \left<\nabla_{\partial_i}\partial_j,\partial_k\right>,\quad \bigl<\nabla_{\partial_j}\partial_k,\partial_i\bigr>, \quad \left<\nabla_{\partial_k}\partial_i,\partial_j\right> $$ (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: $$ \Gamma_{ij}^\ell=\frac12\sum_{k=1}^m \biggl(\frac{\partial}{\partial b_i}g_{jk}+\frac{\partial}{\partial b_j}g_{ki}+\frac{\partial}{\partial b_k}g_{ij}\biggr)g^{k\ell}.\tag{LC} $$ 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.

Definition. The unique symmetric linear connection compatible with the Riemannian structure on a manifold, is called the Levi-Civita connection.

Example: smooth surfaces in $\R^3$ and Theorema Egregium

If $M$ is a 2-dimensional smooth surface in $\R^3$, then it naturally inherits the Riemannian metric from this embedding (vectors tangent to $M$ are also tangent to $\R^3$ with its flat standard Euclidean structure).

The ambient space also has the natural flat connection: in the canonical coordinates it has zero Cristoffel symbols and the covariant derivative $\nabla^\circ=\rd$ is the coordinate-wise differential: $\nabla^\circ_w u=\frac{\rd u}{\rd w}=(L_w u_1,L_w u_2,L_w u_3)$.

A vector field $u$ tangent to $M$ can be covariantly differentiated along any vector $w_a\in T_a M$ in the sense of the ambient flat connection, but the result in general will be not tangent to $M$ anymore: $\nabla^\circ_{w_a}u\in T_a\R^3\smallsetminus T_a M$. However, if we apply to $\nabla^\circ_w u$ the orthogonal projection $\operatorname{Proj}:T_a\R^3\to T_a M$ parallel to the unit normal vector field $\nu=\{\nu_a\}\perp T_a M$, the result will be a well-defined connection on $M$: the differential operators $$ \nabla=\operatorname{Proj}\circ \nabla^\circ,\qquad \nabla_w u=\frac{\rd u}{\rd w}-\bigl<\frac{\rd u}{\rd w},\nu\bigr>\cdot\nu, $$ will satisfy the Leibniz rule. One can verify by the direct computation, that this connection is compatible with the metric induced on $M$ by its embedding in $\R^3$.

The Principal Lemma of Riemannian Geometry implies that the curvature of this connection (defined the explicit embedding of $M$ into $\R^3$) depends in fact only on the "intrinsic" (metric) geometry and is not changed by isometric bending of the surface $M$. This fact was discovered by Gauss and called by him Theorema Egregium, "the remarkable theorem".

Curvature of a metric

The curvature of the Levi-Civita connection is referred simply as the curvature of the metric. As was mentioned, the curvature defines (and can in turn be characterized) by a family of isometric operators (parallel transport) between two tangent spaces $T_a M$ and $T_b M$ at two different points of $M$, which depends on a (piecewise) smooth path connecting these points.

In the simplest nontrivial case of 2-surfaces, an isometry corresponding to the parallel transport along a small closed loop $\gamma$ beginning and ending at the specified point $a\in M$, is a rotation by a small angle depending on the loop (including the orientation of the latter). From the tensorial properties of the curvature it follows that the rotational angle is proportional to the area encircled by the loop. The proportionality coefficient is referred to as the Gauss curvature of a 2-dimensional metric surface at the specified point $a$.

Note that the Gauss curvature has a sign (positive or negative). In fact, the sign of the curvature (if it is constant for all points of $M$) is probably the most important local characteristic which affects global properties of the surface, see [Gr].

For Riemannian manifolds of higher dimensions the curvature tensor has several independent components which can be combined in different ways for different purposes, see Ricci curvature.

Other types of connections

As follows from this summary, many physical and geometric constructions can be formalized as connections (or parallel transport) in appropriate bundles (vector or principal). We mention here only the most important particular cases, referring to the Encyclopaedia pages when possible.

• Weyl connection. A connection $\nabla$ on a Riemannian manifold with the metric tensor $g$, which preserves the conformal structure, i.e., $\nabla_w g=\theta g$ for some positive smooth funcion $\theta:M\to\R_+$ on $M$.
• Yang-Mills connection. A connection on a vector bundle over a (pseudo-)Riemannian manifold, whose curvature form is harmonic in the sense of the Hodge-Laplace operator. Plays a key role in the modern physical field theory.
• Symplectic connection. A connection on the tangent space of an even-dimensional symplectic manifold, which preserves the symplectic structure. Unlike the metric connection, is unrelated to any local geometry of the symplectic manifolds (which is trivial by the Darboux theorem) and is never unique.
• Gauss-Manin connection. If $\pi:E\to B$ is a topological bundle with, say, a compact generic fiber $F$, the fibers $F_b=\pi^{-1}(b)\subset E$ are homeomorphic to each other, although in a non-canonical way. However, for any two sufficiently close points $b_1,b_2\in B$ the fibers $F_i=F_{b_i}$ are homeomorhic and the conjugating homeomorphism is in some sense close to identity. This allows to identify the homology groups $H_k(F_1,G)$ and $H_k(F_2,G)$ in a canonical way for any coefficients group $G$ and all dimensions $k=0,1,\dots,\dim F$. This identification defines a locally flat connection on the bundle with the same base $B$ and the fibers $H_k(\pi^{-1}(b),G)$. The dual connection is defined on the bundle with the dual fibers $H^k(\pi^{-1}(b),G)$ and is also locally flat. For instance, if $G=\R$ and the cohomology is inderstood in the de Rham sense, a $k$-form $\omega\in\varLambda^k(E)$ is locally parallel along a path $\gamma$ in the base $B$, if the restriction of this form on each fiber $\pi^{-1}(b)$ is closed and all periods of this restriction are constant along $\gamma$. The explicit form of the covariant derivative associated with a Gauss-Manin connection is usually written in the algebraic category.
• Hermitian connection. Defined on bundles whose fibers carry a natural structure of complex Hermitian spaces, in particular, on Hermitian manifolds.

This list should be continued.

Related geometric notions

There are many other geometric notions intimately related to connections. For more details refer to the respective EOM articles.

• Holonomy group. For a given base point $a\in B$, the holonomy group is the set of all self-maps of the fiber, obtained as parallel transport along closed loops beginning and ending at $a$. For non-simply connected base $B$ the holonomy group may be nontrivial even for (locally) flat connections.
• Geodesic line (sometimes geodesic curve or just geodesic). A smooth curve on a Riemannian manifold, whose velocity vector is parallel to itself along the curve. Geodesics are locally shortest paths (provide the minimum of the arc length integral among all curves connecting two sufficiently close points on the geodesic).
• Geodesic curvature, sectional curvature, Ricci curvature, mean curvature, principal curvature. Various combinations of components of the general metric curvature tensor. One has to distinguish between the intrinsic invariants which depend only on the metric (e.g., induced on an embedded submanifold of a Riemannian manifold) and extrinsic, depending on the embedding itself.

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