Connections

Motivation

1. Local Trivializations and Transition Functions

Let $E\stackrel{\pi }{\to }M$ be a smooth vector bundle of rank $k$ over a manifold $M$. We can choose a trivializing open cover $\{U_{\alpha }\}$ of $M$ such that locally, $E{|}_{U_{\alpha }}\cong U_{\alpha }\times {\mathbb{R}}^k$. Over each $U_{\alpha }$, we can choose a local frame (a basis of smooth sections) denoted by the row vector: ${\vec{\sigma }}^{\alpha }=\{{\sigma }_1^{\alpha },\dots ,{\sigma }_k^{\alpha }\}$. On the overlap $U_{\alpha }\cap U_{\beta }$, the local frames are related by a smooth transition matrix $g_{\alpha \beta }^E:U_{\alpha }\cap U_{\beta }\to GL(k,\mathbb{R})$. Following the convention of right-multiplication for frames: ${\vec{\sigma }}^{\beta }={\vec{\sigma }}^{\alpha }\cdot g_{\alpha \beta }^E$ For any global section $s\in \Gamma (M,E)$, its local coordinate representations (column vectors) ${\phi }_{\alpha }$ and ${\phi }_{\beta }$ transform via the inverse relationship: ${\phi }_{\alpha }=g_{\alpha \beta }^E\cdot {\phi }_{\beta }$

2. The Failure of the Ordinary Derivative

If we naively apply a vector field $X\in \mathfrak{X}(M)$ to the local components ${\phi }_{\alpha }$ component-wise, the Leibniz (product) rule destroys the tensor transformation property on overlaps. Differentiating ${\phi }_{\alpha }=g_{\alpha \beta }^E\cdot {\phi }_{\beta }$ yields: $X({\phi }_{\alpha })=X(g_{\alpha \beta }^E)\cdot {\phi }_{\beta }+g_{\alpha \beta }^E\cdot X({\phi }_{\beta })$ The presence of the non-zero term $X(g_{\alpha \beta }^E)\cdot {\phi }_{\beta }$ implies that the ordinary directional derivative does not produce a globally well-defined section. To fix this, we must introduce a connection.

Connection

A connection on a vector bundle $E$ over smooth manifold $M$ is a map $\nabla :\mathfrak{X}(M)\times \Gamma (E)\to \Gamma (E)$, such that the following hold:

• $C^{\infty }(M)$-linearity in the first argument: ${\nabla }_{fX}s=f{\nabla }_{X}s$ for any $f\in C^{\infty }(M)$;
• $\mathbb{R}$-linearity in the second argument: ${\nabla }_X(as+bt)=a{\nabla }_{X}s+b{\nabla }_{X}t$;
• Leibniz rule: ${\nabla }_X(fs)=f{\nabla }_{X}s+(Xf)s$ for any $f\in C^{\infty }(M)$.

An important special case of a connection is the affine connection or covariant derivative:

Affine Connection

Let $M$ be a smooth manifold. An affine connection on $M$ is a connection on the tangent bundle. Explicitly, it is a map $\nabla :\mathfrak{X}(M)\times \mathfrak{X}(M)\to \mathfrak{X}(M)$, such that the following hold:

• $C^{\infty }(M)-$ linearity in the first argument: ${\nabla }_{aX+bY}Z=a{\nabla }_{X}Z+b{\nabla }_{Y}Z$;
• Leibniz rule: ${\nabla }_X(fY)=f{\nabla }_{X}Y+(Xf)Y$ for any $f\in C^{\infty }(M)$.

Affine connection is also called the covariant derivative.

Parallel Section

A section $s$ of the vector bundle $E$ is called parallel (with respect to a connection $\nabla$) if ${\nabla }_{X}s=0$ for all $X\in \mathfrak{X}(M)$.

Def Covariant Derivative of Real Functions Given a point $p\in M$ of the manifold $M$, a real function $f:M\to \mathbb{R}$ on the manifold and a tangent vector $v\in T_{p}M$, the covariant derivative of $f$ at $p$ along $v$, denoted ${\nabla }_{v}f$, is a scalar that represents the principal part of the change in the value of $f$ when the argument of $f$ is changed infinitesimally in the same direction as the displacement vector $v$. Formally, there is a differentiable curve $\gamma :[-1,1]\to M$ such that $\gamma (0)=p$ and ${\gamma }^{\prime }(0)=v$, and the covariant derivative of $f$ at $p$ is defined by ${\nabla }_{v}f=v(f)={\left(f\circ \gamma \right)}^{\prime }(0)={\lim }_{t\to 0}\frac{f(\gamma \left(t\right))-f(p)}{t}$When $X:M\to TM$ is a vector field on $M$, the covariate derivative ${\nabla }_{X}f:M\to \mathbb{R}$ is the function that assigns each point $p$ a scaler ${\nabla }_{X(p)}f$. Notice that this derivative exists without a definition of connection. So we simply write $Xf$ instead of ${\nabla }_{X}f$.

Local Connection and Connection Matrix

A connection (or covariant derivative) $\nabla$ is an operator that assigns to every vector field $X\in \mathfrak{X}(M)$ a differential operator ${\nabla }_X:\Gamma (E)\to \Gamma (E)$ satisfying the $C^{\infty }$-linearity in $X$ and the Leibniz rule for sections. Locally on $U_{\alpha }$, the action of ${\nabla }_X$ on the local frame ${\vec{\sigma }}^{\alpha }$ is completely determined by a $k\times k$ matrix of 1-forms, known as the connection matrix (or connection 1-form) ${\Gamma }^{\alpha }\in {\Omega }^1(U_{\alpha },{\mathfrak{gl}}_k(\mathbb{R}))$: ${\nabla }_X{\vec{\sigma }}^{\alpha }={\vec{\sigma }}^{\alpha }\cdot {\Gamma }^{\alpha }(X)$ Equivalently, in index notation: ${\nabla }_X{\sigma }_i^{\alpha }={\sum }_{j=1}^k{\Gamma }_i^{\alpha ,j}(X)\cdot {\sigma }_j^{\alpha }$

The Compatibility Condition

For the local connections to glue together into a valid global connection, the connection matrices ${\Gamma }^{\alpha }$ and ${\Gamma }^{\beta }$ must satisfy a specific transformation rule on $U_{\alpha }\cap U_{\beta }$. ${\Gamma }^{\alpha }(X)=g_{\alpha \beta }^E\cdot {\Gamma }^{\beta }(X)\cdot (g_{\alpha \beta }^E{)}^{-1}-X(g_{\alpha \beta }^E)\cdot (g_{\alpha \beta }^E{)}^{-1}$

Proof By applying ${\nabla }_X$ to the frame transformation equation ${\vec{\sigma }}^{\beta }={\vec{\sigma }}^{\alpha }\cdot g_{\alpha \beta }^E$, we obtain: ${\nabla }_X{\vec{\sigma }}^{\beta }={\nabla }_X({\vec{\sigma }}^{\alpha }\cdot g_{\alpha \beta }^E)$ Applying the Leibniz rule to the right side: ${\vec{\sigma }}^{\beta }\cdot {\Gamma }^{\beta }(X)=({\nabla }_X{\vec{\sigma }}^{\alpha })\cdot g_{\alpha \beta }^E+{\vec{\sigma }}^{\alpha }\cdot X(g_{\alpha \beta }^E)$ Substituting ${\nabla }_X{\vec{\sigma }}^{\alpha }={\vec{\sigma }}^{\alpha }\cdot {\Gamma }^{\alpha }(X)$ and ${\vec{\sigma }}^{\beta }={\vec{\sigma }}^{\alpha }\cdot g_{\alpha \beta }^E$: ${\vec{\sigma }}^{\alpha }\cdot g_{\alpha \beta }^E\cdot {\Gamma }^{\beta }(X)={\vec{\sigma }}^{\alpha }\cdot {\Gamma }^{\alpha }(X)\cdot g_{\alpha \beta }^E+{\vec{\sigma }}^{\alpha }\cdot X(g_{\alpha \beta }^E)$ Since the frame ${\vec{\sigma }}^{\alpha }$ consists of linearly independent basis vectors, we can equate the matrix coefficients: $g_{\alpha \beta }^E\cdot {\Gamma }^{\beta }(X)={\Gamma }^{\alpha }(X)\cdot g_{\alpha \beta }^E+X(g_{\alpha \beta }^E)$ Rearranging for ${\Gamma }^{\alpha }(X)$, we arrive at the compatibility condition: ${\Gamma }^{\alpha }(X)=g_{\alpha \beta }^E\cdot {\Gamma }^{\beta }(X)\cdot (g_{\alpha \beta }^E{)}^{-1}-X(g_{\alpha \beta }^E)\cdot (g_{\alpha \beta }^E{)}^{-1}$

Construction of Global Connection

A connection always exists on any vector bundle.

Proof

1. Choose a trivializing cover: Let $\{U_{\alpha }\}$ be an open cover of $M$ over which $E$ is trivial.
2. Define trivial connections: On each $U_{\alpha }$, define the trivial connection ${\nabla }^{\text{triv},\alpha }$ which simply differentiates the local components of a section in that chart.
3. Choose a PoU: Let $\{{\rho }_{\alpha }\}$ be a smooth partition of unity subordinate to the cover $\{U_{\alpha }\}$ (so $\text{supp}({\rho }_{\alpha })\subset U_{\alpha }$ and ${\sum }_{\alpha }{\rho }_{\alpha }=1$).
4. Glue the connections: For any global section $s\in \Gamma (E)$, define the global connection $\nabla$ by:${\nabla }_{X}s:={\sum }_{\alpha }{\nabla }_X^{\text{triv},\alpha }({\rho }_{\alpha }s)$Because ${\rho }_{\alpha }s$ has compact support strictly inside $U_{\alpha }$, it extends smoothly by zero to all of $M$. The sum is locally finite, and one can directly verify that this glued operator satisfies the Leibniz rule globally.

Theorem

The space of all connections $\nabla :\mathfrak{X}(M)\times \Gamma (E)\to \Gamma (E)$ is an affine space $\Gamma (M,T^{\ast }M\otimes {\mathfrak{gl}}_k(\mathbb{R}))$

Connection Coefficients

Christoffel Symbols of The Second Kind

Suppose we are working in a chart $\phi :U\to V\subset {\mathbb{R}}^n$ for $M$, with corresponding coordinate tangent vectors ${\partial }_1,\dots ,{\partial }_n$, and write ${\nabla }_j$ for ${\nabla }_{{\partial }_j}$. The Christoffel symbols (of the second kind) or connection coefficients of a connection $\nabla$ with respect to this chart are defined by ${\nabla }_i{\partial }_j={\Gamma }_{ij}^k{\partial }_k$

Proposition

The connection is determined on $U$ by the connection coefficients.

Proof Write $X=X^i{\partial }_i$. Then we have for any vector $v=v^k{\partial }_k$, ${\nabla }_{v}X={\nabla }_v\left(X^i{\partial }_i\right)=v(X^i){\partial }_i+X^i{\nabla }_v{\partial }_i=v(X^i){\partial }_i+v^k{X}^i{{\Gamma }_{ki}}^j{\partial }_j$

Parallel Transport

Parallel

Let $M$ be a manifold with an affine connection $\nabla$ Then a vector field $X$ is said to be parallel if for any vector field $Y$, ${\nabla }_{Y}X=0$.

Remark

Intuitively, parallel vector fields have all their derivatives equal to zero and are therefore in some sense constant.

Parallel Transport

If $\gamma :I\to M$ is a smooth map parameterized by an interval $I=[a,b]$ and $\xi \in T_{\gamma (a)}M$, then a vector field $X$ along $\gamma$ is called the parallel transport of $\xi$ along $\gamma$ if it satisfies the following conditions:

• ${\nabla }_{{\gamma }^{\prime }(t)}X=0$ for all $t\in [a,b]$;
• $X{|}_{\gamma (a)}=\xi$.

Geodesic

A curve $\gamma :[0,1]\to M$ on a smooth manifold with an affine connection $\nabla$ is a geodesic if ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0.$

Natural Dual Connection

Dual Connection

Given a connection $\nabla$ on a vector bundle $E$ over manifold $M$, the dual connection ${\nabla }^{\ast }$ on the dual bundle is naturally defined by the following equation: $({\nabla }_X^{\ast }\phi )(v)=X(\phi (v))+\phi ({\nabla }_{X}v)$for all $X\in \mathcal{X}(M)$.