---- > [!definition] Definition. ([[connection on a manifold]]) > Let $M$ be a [[smooth manifold]]. A **connection on $M$** is a [[connection on a vector bundle|connection]] on the [[tangent bundle]] $TM$. Its coefficients $\Gamma^{i}_{jk}$ are called **Christoffel symbols**. ^definition > [!proposition] The transformation law for Christoffel symbols. > The [[connection on a vector bundle|transformation law for an arbitrary connection]] on a [[vector bundle]] $E$ specializes as follows in the present case $E=TM$. > Recall the general transformation law for $A^{i}_{j}= \Gamma^{i}_{jk} \ dx^{k}$: $\Gamma^{i}_{jk}=\Gamma^{i'}_{j'k'} \ \psi^{i}_{i'} \psi^{j}_{j'} \frac{ \partial x^{k'} }{ \partial x^{k} } + \psi_{i'}^{i} \frac{ \partial \psi^{i'}_{j} }{ \partial x^{k} } .$ For the [[tangent bundle]] $TM$, the [[transition functions for a vector bundle over a smooth manifold|transition]] matrices $\psi=(\psi^{i}_{i'})$ are just the Jacobians $(\frac{ \partial x^{i} }{ \partial x^{i'} })$ of the [[transition map|transition maps]] for $M$. Substituting: $\Gamma^{i}_{jk}=\Gamma^{i'}_{j'k'} \frac{ \partial x^{i} }{ \partial x^{i'} } \frac{ \partial x^{j'} }{ \partial x^{j} } \frac{ \partial x^{k'} }{ \partial x^{k} } + \frac{ \partial x^{i} }{ \partial x^{i'} } \frac{ \partial^{2} x^{i'} }{ \partial x^{j} \partial x^{k} } .$ ^proposition > [!definition] Definition. (Torsion of a connection on $M$) > Observe, cf. the [[transformation law]] above, that $\widetilde{\Gamma}^{i}_{jk}:=\Gamma^{i}_{kj}$ is a valid connection $\widetilde{\Gamma}$ on $M$, in general different from $\Gamma$. The difference (working in coordinates, so that we can indeed subtract connections) $\Gamma^{i}_{jk}-\Gamma_{kj}^{i}$is called the **torsion** of a connection $D=(\Gamma^{i}_{jk})$ on $M$. A **symmetric connection** is one whose torsion is zero. With notation as in [[partial covariant derivative]], the computation below shows that a connection $D$ on $M$ is symmetric if and only if $D_{X}Y-D_{Y}X=[X,Y]$ for all [[vector field|vector fields]] $X,Y \in \mathscr{V}(M)$. ^definition Recall the local formula for $[X,Y]$ proven in [[vector field|Lie bracket of vector fields]]:[^1] $[X,Y]^{i}= X^{k}\frac{ \partial Y^{i} }{ \partial x^{k} } - Y^{k} \frac{ \partial X^{i} }{ \partial x^{k} } .$ And recall the local formulae for $D_{X}Y$, $D_{Y}X$: $\begin{align} (D_{X}Y)^{i}&=\frac{ \partial Y^{i} }{ \partial x^{k} }X^{k} + \Gamma^{i}_{jk} Y^{j}X^{k} \\ (D_{Y}X)^{i} &= \frac{ \partial X^{i} }{ \partial x^{k} } Y^{k} + \Gamma^{i}_{jk} X^{j} Y^{k}. \end{align} $ Since we're summing over both $j,k$ in the second term of $(D_{Y}X)^{i}$, it is safe to swap them: $(D_{Y}X)^{i}=\frac{ \partial X^{i} }{ \partial x^{k} } Y^{k} + \Gamma^{i}_{kj}X^{k}Y^{j}.$ Now $(D_{X}Y-D_{Y}X-[X,Y])^{i}=(\Gamma^{i}_{jk}-\Gamma_{kj}^{i})X^{k}Y^{j}$ which vanishes iff $\Gamma^{i}_{jk}=\Gamma^{i}_{kj}$, i.e., iff the connection is symmetric. ---- #### [^1]: Warning: the indices are different in that note, specifically $i$ there is now $k$ here and $j$ there is now $i$ here. ---- #### 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 > ``` ---- > [!definition] Definition. ([[Christoffel symbols]]) > Let $S$ be a [[orientable manifold|oriented]] [[surface|regular surface]] with local [[coordinate patch|coordinates]] $X$. Analogous to the [[Frenet frame|Frenet trihedron]], the triple $\{ X_{u}, X_{v}, N \}$ can be taken as a [[frame]] of $\mathbb{R}^{3}$ varying [[smooth|smoothly]] on $S$. We have a [[system of ODEs|system]] for such the derivatives of such [[frame|frames]] similar to the [[Frenet frame|Frenet formulas]] for [[parameterized curve|curves]]: $\begin{cases} X_{uu}= & \Gamma_{11}^{1}X_{u} + \Gamma_{11}^{2}X_{v} + L\b N \\ X_{uv}= & \Gamma_{12}^{1} X_{u}+ \Gamma_{12}^{2}X_{v} + M\b N \\ X_{vu}= & \Gamma_{21}^{1}X_{u} + \Gamma_{21}^{2}X_{v} + M\b N \\ X_{vv} = & \Gamma_{22}^{1}X_{u} + \Gamma_{22}^{2}X_{v} + N\b N \\ \b N_{u}= & a_{11}X_{u} + a_{21}X_{v} \\ \b N_{v}= & a_{12}X_{u}+a_{22}X_{v}. \end{cases}$ In the above, $\begin{bmatrix}L & M \\ M & N\end{bmatrix}$ is the [[matrix]] of the [[second fundamental form]] in the [[coordinate patch|patch]] $X$, and $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}=-\begin{bmatrix} E & F \\ F & G \end{bmatrix} \begin{bmatrix} L & M \\ M & N \end{bmatrix}=-\frac{1}{EG-F^{2}} \begin{bmatrix} E & -F \\ -F & E \end{bmatrix} \begin{bmatrix} L & M \\ M & N \end{bmatrix}$ (see [[principal curvature#^e0b64a|here]]). The coefficients $\Gamma_{ij}^{k}$, $i,j,k=1,2$ are called the **Christoffel symbols of $S$ in the patch $X$**. Note that [[equality of mixed partials for C2 functions|because]] $X_{uv}=X_{vu}$, the symbols are symmetric in the lower indices. So there are six $\Gamma$ to compute in total. This note is incomplete. See Do Carmo 4.3. ---- #### ---- #### 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 > ``` #reformatrevisebatch04