---- > [!theorem] Theorem. ([[three views on connections]]) > The notion of **connection** on a [[vector bundle]] $E \xrightarrow{\pi}B$ can arise in three equivalent ways: > > 1. As (smooth) [[field of horizontal subspaces]] in $TE$ depending linearly on the fiber coordinates, [[connection on a vector bundle|(this is our initial definition)]] ; > 2. As a system of matrix-valued $1$-[[differential form|forms]] $(A_{j}^{i})_{i,j=1}^{m}$, amounting to a system of $nm^{2}$ smooth **connection coefficients** $(\Gamma_{jk}^{i})_{i,j=1,\dots,m}^{k=1,\dots,n}$, assigned to every [[vector bundle|local trivialization]] $U$ of $E$ and obeying the [[transformation law]] $A^{\psi \Phi}=\psi A^{\Phi}\psi ^{-1} + \psi d \psi ^{-1} = \psi A ^{\Phi}\psi ^{-1}- (d \psi) \psi ^{-1}$ > on overlaps; > 3. As a [[covariant derivative on a vector bundle|covariant derivative]] $\nabla^{E}$ on the [[section|sections]] of $E$ and, more generally, on the [[differential form with values in a vector bundle|differential forms with values in ]] $E$. Here are two more views that one can work in: 4. As a [[partial covariant derivative]] $\nabla^{E}:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)$, i.e. an operator $\nabla^{E}(X,s)= \nabla^{E}_{X}s$ that is $C^{\infty}$-[[linear map|linear]] in $X$, additive in $s_{2}$, and Leibniz in $s$, meaning $\nabla^{E}_{X}(fs_{})=df \otimes s +(Xf)s$. 5. characterization in terms of parallel transport. Moreover, parallel transport gives an additional view on torsion-free connections (they are precisely those for which parallel transport is an isometry). > [!proof]- Proof. ([[three views on connections]]) > ~ It is nicest to show $1 \iff 2$ and $1 \iff 3$. **$1 \iff 2$.** This amounts to the derivation of the transformation law in [[connection on a vector bundle]]. **$1 \iff 3$.** That any connection $A$ gives rise to an operator $d_{A}$ that is a covariant derivative is shown in the [[covariant derivative on a vector bundle|covariant derivative]] note. Conversely, suppose we're given a covariant derivative $\nabla^{E}:\Gamma(E) \to \Gamma(T^{*}B \otimes E)$. [[covariant derivative on a vector bundle|We know]] $\nabla^{E}$ is a local operator, meaning it will be enough to define the prospective [[connection on a vector bundle|connection]] $A$ as a matrix of $1$-forms $A^{i}_{j}=\Gamma^{i}_{jk} \ dx^{k}$ in a local trivialization $U$ of $E$, then show it satisfies the required [[transformation law]]. To this end, let $s$ be a local section of $E$ over $U$; view $s$ as a vector-valued function $s:U \to \mathbb{R}^{n}$. With $\boldsymbol e_{i}:U \to \mathbb{R}^{n}$ the standard basis, $s=s ^{i} \boldsymbol e_{i}$. Now take $\Gamma^{i}_{jk}:= i\text{th component of } (\nabla^{E} \boldsymbol e_{j}) \left( \frac{ \partial }{ \partial x^{k} } \right): U \to \mathbb{R}^{n}.$ We have $\begin{align} \nabla^{E} s &= \nabla^{E}(s ^{i} \boldsymbol e_{i}) \\ &= ds ^{i} \otimes e_{i} + s ^{j} \Gamma^{i}_{jk} \ dx^{k} \otimes e_{i} \\ & = (ds ^{i} + s ^{j} \Gamma^{i}_{jk} \ dx^{k} ) \otimes e_{i} \\ &= d_{A} s, \end{align}$ where $A$ is given in the local trivialization as $A=\Gamma^{i}_{jk} \ dx^{k}$ a matrix of $1$-forms. The same check as in [[covariant derivative on a vector bundle#^47f224|here]] (showing $d_{A}$ satisfies) goes through to show $A$ has the needed [[transformation law]]. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```