--- date: 2022-01-15 21:49 modification date: Wednesday 9th February 2022 10:11:15 title: connection aliases: [connection, "Levi-Cevita connection", "affine connection", "integrable connection", vertical space, horizontal space, Riemannian geometry] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #geomtop/differential-geometry - Refs: - Course notes: #resources/course-notes - Links: - [ASD](Unsorted/anti%20self%20dual.md) - [flat connection](Unsorted/flat%20connection.md) - [Milnor fiber](Milnor%20fiber) - [Algebraic de Rham](Unsorted/algebraic%20de%20Rham%20cohomology.md) - [Local systems biject with monodromy representations](Unsorted/Local%20systems%20biject%20with%20monodromy%20representations.md) - [flat connection](Unsorted/flat%20connection.md) - [First fundamental form](First fundamental form) - [Second fundamental form](Second fundamental form) - [mean curvature](mean%20curvature) - [Laplace-Beltrami operator](Laplace-Beltrami operator.md) - Types of curvature: - [Gaussian curvature](Gaussian curvature) - [Ricci curvature](Ricci curvature.md) - [Sectional curvature](Sectional curvature) - [Holonomy Classification](Holonomy Classification.md) --- # connection ![](attachments/Pasted%20image%2020220502135750.png) # Motivations - Provides lifts of curves in $M$ to curves in $\Frame(M)$. - Connects nearby tangent spaces, tangent vector fields can be differentiated as if they were functions $f \in C^\infty(M; V)$ for a fixed vector space $V$, - The main invariants of an affine connection are its [curvature](Unsorted/curvature.md). - If both vanish, $\Gamma(TM)$ is almost a [Lie algebra](Lie%20algebra.md). - Can define a [covariant derivative](covariant%20derivative.md) - Sometimes flat connections are referred to as *integrable connections*. # Differential geometry definitions - **Vertical subspace**: for $\pi: P\to B$, the **vertical subspace** at $p$ is $V_p P \da \ker (d\pi)$. - Intuition: $V_p P \leq \T_p P$ is the subspace parallel to the fiber. Uniquely defined by $\pi$. - **Horizontal subspace**: $H_p P \da (V_p P)^\perp$, the orthogonal complement. - Subspace orthogonal to fibers. Not uniquely defined, need a connection. - **Connections on fiber bundles**: - For $\pi: P\to B$ a fiber bundle a **connection** is a $C^\infty$ assignment $p\to H_p P$ of points in $P$ to horizontal subspaces such that $R_g$ preserves the horizontal subspaces in the sense of $$H_{R_g(p)} P = (dR_g)_p H_p P \qquad \forall g\in G.$$ - Works for any fiber bundle: define a connection as a smooth horizontal subspace distribution; for principal $G\dash$bundles require $G\dash$equivariance. - Alternative definitions of connections on fiber bundles: - For a fixed $p\in P$, every $H_p P$ induces a projection $\T_p P\to \lieg \da V_p P$, which is thus a $\lieg\dash$valued 1-form on $P$ callied the connection 1-form $\omega$ of $\nabla$. Satisfies: - $\omega$ vanishes on horizontal vectors. - $\omega(X^*) = X$ for all $X\in \lieg$. - $\omega(dR_g X) = \Ad_{g\inv} \omega(X)$. - **Connections on manifolds**: - Defined as $$\begin{align} \nabla: \Gamma_{C^\infty}(\T^{2, 0} M) &\to \Gamma_{C^\infty}(\T M) \\ X\tensor Y &\mapsto \nabla_{X} Y \end{align}$$where $\Gamma_{C^\infty}$ denotes taking smooth global sections, such that for all $f\in C^\infty(M; \RR)$, - $C^\infty(M; \RR)\dash$linear in the first variable: $$\nabla_{f \mathrm{X}} \mathrm{Y}=f \nabla_{\mathrm{X}} \mathrm{Y}$$ - Leibniz rule in the second variable: $$\nabla_{\mathrm{X}}(f \mathrm{Y})=\partial_{X} f \mathrm{Y}+f \nabla_{\mathrm{X}} \mathrm{Y}$$where $\partial_X$ is the directional derivative. - Equivalently, a principal $\GL_n(\RR)$ connection on the frame bundle $\Frame(M)$ - **Integrable connections**: - Definition: a $k\dash$dimensional distribution $\delta$ on $M$ is **integrable** iff for all $p\in M$ there exist submanifolds $Y_p \leq M$ with $p\in Y_p$ and $Y_p$ everywhere tangent to $\delta$. - Idea: let $z_1,\cdots, z_n$ be local coordinates at $p$, get $\dz_1,\cdots, \dz_n$ a basis of $\Omega^1_X$ in $U\ni p$ with dual basis $\dd{}{z_1},\cdots, \dd{}{z_n}$ for $(\Omega^1_X)\dual \cong \T_X$. - *Contract* $\nabla$ with the vector field $\dd{}{z_i}$ to get a map $V\to V$ satisfying the Leibniz rule wrt $\dd{}{z_i}$. - Think of this as $\Phi_i: \dd{}{z_i}\actson H^0(U; V)$; then $\nabla$ is integrable iff all actions $\Phi_i$ commute. - Idea: allows moving "horizontally" between fibers in a bundle, but is path-dependent iff nonzero curvature. - If zero curvature, on $U\ni p$, can write a basis of $V$ consisting of horizontal sections, and the connection in this basis is $$\nabla\qty{\sum f_i s_i} = \sum (s_i \tensor df_i)$$ - Parallel transport yields a well-defined monodromy representation $\rho: \pi_1(X, x)\to \GL(V_x)$. - Differential geometers refer to integrable connections as flat connections; in AG flat connection has a different meaning. - **Parallel transport**: - Take $v\in \T_b B$ and $p\in P_b$, lift to $\tilde v\in H_p P \leq \T_p P$. Liftable since $\pi: H_p P \iso \T_b B$. - For $a,b\in B$, pick $\gamma: a\mapsto b$ a curve, lift $\gamma'\in \T B$ to $\tilde \gamma'\in HP \leq \T P$ with initial points $\tilde a\in P_a$. - Solve the ODE to yield a diffeomorphism $\Phi_{\nabla, \gamma}: P_a\iso P_b$. Call this map the **parallel transport** along $\gamma$. - **Riemannian metric**: $(M, g)$ where $M$ is a manifold, $g\in \T^{2, 0} M$ with $g_p\in \Aut \T_p M$ positive definite for all $p\in M$. - **Levi-Cevita connection**: $(M, g, \nabla)$ where - Torsion-free: $$\nabla_X Y - \nabla_Y X = [X, Y].$$ - Affine (invariant under parallel transport): $$\nabla_X . g = 0\qquad \forall X.$$ Equivalently, $X.g(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$. - This uniquely determines a connection $\nabla_{\LC}$. - The unique affine torsion-free connection for which [parallel transport](parallel%20transport) is an isometry. - Get an equation: $$2 g\left(X, \nabla_Z Y\right)=Z \cdot g(X, Y)+g(Z,[X, Y])+Y \cdot g(X, Y)+g(Y,[X, Y])-X \cdot g(Y, Z)-g(X,[Y, Z])$$ - **Riemannian manifold**: $(M, g, \nabla_{\LC})$. - For $\omega$ a connection 1-form and $\Omega$ its curvature 2-form ($F_A \da dA + A\wedge A$), one has a [Maurer-Cartan](Unsorted/Maurer-Cartan.md) equation $$\Omega = d\omega + {1\over 2}[\omega\,\omega], \qquad [\omega \wedge \omega](X, Y)=[\omega(X), \omega(Y)]-[\omega(Y), \omega(X)]=2[\omega(X), \omega(Y)]$$ - **Distributions**: - For $M\in \smooth\Mfd$, a $k\dash$dimensional distribution $\delta$ is the assignment of $\T_p M\to V_p$ where $V_p\leq \T_p M$ is a $k\dash$dimensional subspace for each $p\in M$. - So a map $M\to \Gr_k(\TM)$. - Can define a connection as a horizontal distribution on the total space of a principal bundle. - Distributions are integral $\iff$ $[\delta\, \delta] \subseteq \delta$, i.e. if $X,Y\in H^0(\T M)$ with $X,Y\subset D$ then $[X\, Y]\subset D$. - Connections are integral $\iff$ flat. Follows from the Frobenius theorem and a calculation: $$\begin{aligned}\Omega(X, Y) & =d \omega(h X, h Y) \\& =h X(\omega(h Y))-h Y(\omega(h X))-\omega([h X, h Y]) \\& =-\omega([h X, h Y])\end{aligned}$$ - Measuring path lengths: $$L(\gamma):=\int_\gamma \sqrt{g_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t))} d t$$ - Measuring distances: $$d(p, q):=\inf \ts{L(\gamma) \st \gamma:a\mapsto b}$$ - **Torsion and curvature**: $$T(X, Y):=\nabla_X Y-\nabla_Y X-[X, Y] ; \quad R(X, Y):=\nabla_X \nabla_Y-\nabla_Y \nabla_X-\nabla_{[X, Y]}$$ - **Sectional curvature**: - Let $M$ be a Riemannian manifold and $p \in M$ a point. Let $S$ be a two-dimensional subspace of $T_p M$. The sectional curvature is defined by $$K(S)=-\frac{g_p\left(R_p(Y, Z) Y, Z\right)}{|Y \wedge Z|^2},$$(c.f. [He78, Thm. I.12.2]) where $Y$ and $Z$ are linearly independent vectors in $S$ and $|Y \wedge Z|$ denotes the area of the parallelogram spanned by $Y$ and $Z$. # AG definitions - **Connection**: for $\Omega_X$ the sheaf of differentials and $V\in \Coh(X)$ locally free, a connection on $V$ is an additive bundle map $\nabla: V\to V\tensor \Omega_X$ satisfying $$\nabla(f.s) = f.\nabla(s) + s\tensor df\qquad f\in H^0(U; \OO_U),\,\,s\in H^0(U; V)$$ - Extending this map: $$\begin{align*}\nabla_1: V\tensor \Omega_X &\to V\tensor \Omega^2_X \\ s\tensor \omega &\mapsto \nabla(s) \wedge \omega + s\tensor d\omega \\ \\\wedge: (V\tensor \Omega^1_X)\tensor \Omega^1_X &\to V\tensor \Omega^2_X \\ v\tensor \omega_1 \tensor \omega_2 &\mapsto v\tensor (\omega_1 \wedge \omega_2)\end{align*}$$ - **Horizontal section**: $\nabla(s) = 0$. - **Curvature**: $\nabla^2 \da \nabla_1 \circ \nabla: V\to V\tensor \Omega^2_X$. - **Integrable connections**: vanishing curvature, so $\nabla^2 = 0$. - Always true if $\dim X = 1$. - **Parallel transport**: - For $p\in M, \ker \nabla$ is a local system and $\exists U\ni p$ such that $\ro{\ker \nabla}U = \ul{E_p}\slice{U}$. - Since $\ul{E_p}\slice{U} \tensor_\CC\OO_U \iso \ro E U$, for any $q\in U$ taking fibers induces a parallel transport isomorphism: $$\Phi_{p, q}: E(p) \to E(q)$$ # Unsorted ![attachments/Pasted image 20210613122858.png](attachments/Pasted%20image%2020210613122858.png) ![attachments/Pasted image 20210613122923.png](attachments/Pasted%20image%2020210613122923.png) - [covariant derivative](covariant%20derivative.md) with respect to two variables do not need to commute. The failure of the commutativity of partial covariant derivatives is closely related to the notion of [curvature](curvature.md). ![attachments/Pasted image 20210613124535.png](attachments/Pasted%20image%2020210613124535.png) ![attachments/Pasted image 20210613124542.png](attachments/Pasted%20image%2020210613124542.png) [Ricci curvature](Ricci%20curvature.md) : ![attachments/Pasted image 20210613124635.png](attachments/Pasted%20image%2020210613124635.png) ![](attachments/Pasted%20image%2020220408005504.png) ![](attachments/Pasted%20image%2020220408005512.png) ![](attachments/Pasted%20image%2020220408005531.png) ![](attachments/Pasted%20image%2020220408005540.png) # Residues ![](attachments/Pasted%20image%2020220209100826.png) Semistability, involves [SNC](Unsorted/simple%20normal%20crossings.md) divisors: ![](attachments/Pasted%20image%2020220209100838.png) ![](attachments/Pasted%20image%2020220209100929.png) # Meromorphic connection ![](attachments/Pasted%20image%2020220411000854.png)