Connections and Curvature

Here's some motivation for the problem at hand. Suppose we have a tangent vector $\vec{v}$ at a point on the sphere $p \in S^2$ , and we want a way to consistently slide this vector along some path to get a "parallel" vector at $q$ . This is called parallel transport. We can do this by fixing some local coordinates and sliding the vector as we would in $\bbc^1$ , but the issue is that the vector we get might not agree with the vector we get by doing this with a different set of local coordinates.

To solve this, we can rotate $S^2$ to move $p$ along the sphere without rotating about the axis through $p$ . This is an example of a connection on the sphere, which is a special example since we use in some sense the Lie group structure of $S^2$ . We will see later that this generalizes, but that we can also define connections on other complex manifolds.

Another important motivaiton is that, given some vector bundle $\cale \to M$ over a complex manifold $M$ , suppose we want to differentiate a section $\sigma \colon M \to \cale$ at $p$ in the direction $v \in T_p M$ . Let's work naively and see where it goes wrong. Consider a path $\gamma \colon (-1,1) \to M$ with $\gamma(0) = p$ and $\gamma'(0) = v$ . Then we could try to define it as follows.

d\sigma(v)(p) = \lim_{t \to 0} \frac{\sigma(\gamma(t)) - \sigma(\gamma(0))}{t}

But we run into an issue since $\sigma(\gamma(t)) \in \cale_{\gamma(t)}$ and $\sigma(\gamma(0)) \in \cale_p$ are in distinct vector spaces, so subtracting them is meaningless. Thus we need a parallel transport of $\sigma(\gamma(t))$ to $\cale_p$ in order to define this difference, and hence the directional derivative of a section of the bundle.

Linear Connections

Let $\cale \to M$ be a smooth complex vector bundle over a smooth complex manifold $M$ . Let $\calo_{\cale} = \calo_{\cale}(M)$ denote the space of smooth (global) sections of $\cale \to M$ .

Definition A covariant derivative on $\cale \to M$ is a $\bbc$ -linear map $\nabla \colon \calo_{\cale} \to \Omega_M \otimes \calo_{\cale}$ such that the product rule

\nabla(f\sigma) = df \otimes \sigma + f \nabla \sigma

holds for all smooth functions $f$ on $M$ and all smooth sections $\sigma$ of $\cale$ .

Example Let $\cale_0 = M \times \bbc^k$ be the trivial rank $k$ vector bundle on $M$ . Then we can define a trivial connection $\nabla$ as follows. Take the standard basis $\{e_1,\ldots,e_k\}$ of $\bbc^k$ such that every section $\sigma$ of $\cale_0$ can be written as $\sigma = \sum_{i=1}^k \sigma^i e_i$ where $\sigma^i \colon M \to \bbc$ are smooth. Let

\nabla(\sigma) = \sum_{i=1}^k d\sigma^i \otimes e_i

We denote the trivial connection by $d\sigma$ .

We can use this example to describe a connection locally on a general vector bundle. Suppose $\cale$ is a rank $k$ bundle on $M$ , and $U$ be an open subset of $M$ over while $\cale$ is trivial. Then over $U$ , $\cale$ admits a smooth local frame of sections $\{e_1, \ldots, e_k\}$ where $e_i \colon U \to \cale_U$ . The frame defines a basis for the fiber $\cale_p$ for any $p \in U$ , so any local section $\sigma \colon U \to \cale_U$ can be written

\sigma = \sum_{i=1}^k \sigma^i e_i

where $\sigma^i \colon U \to \bbc$ are smooth. Then $\nabla(e_i) \in \Omega_M(U) \otimes \cale_U$ can be expanded in the local frame as follows.

\nabla(e_i) = \sum_{j=1}^k \omega_i^j \otimes e_j

where $\omega_i^j \in \Omega_M(U)$ are differential $1$ -forms on $U$ . We can put these together into a matrix

\omega = \begin{pmatrix} \omega_1^1 & \ldots & \omega_k^1 \\ \vdots & \ddots & \vdots \\ \omega_1^k & \ldots & \omega_k^k \end{pmatrix} \in \Omega_M(U) \otimes \End(\cale_U)

called the local connection form of $\nabla$ over $U$ , which is an endomorphism-valued 1-form. Then, using the product rule, we can see that

\begin{align*} \nabla(\sigma) = \nabla \left( \sum_{i=1}^k \sigma^i e_i\right) &= \sum_{i=1}^k \left(d\sigma^i \otimes e_i + \sigma^i \nabla e_i \right) \\ &= \sum_{i=1}^k \left(d\sigma^i \otimes e_i + \sigma^i \sum_{j=1}^k \omega_i^j \otimes e_j \right) \\ &= \sum_{i=1}^k \left(d\sigma^i \otimes e_i\right) + \sum_{i=1}^k\left(\sum_{j=1}^k \sigma^j \omega_j^i\right) \otimes e_i \\ &= \sum_{i=1}^k \left(d\sigma^i + \sum_{j=1}^k \sigma^j \omega_j^i\right) \otimes e_i \end{align*}

If $\sigma is written in column vector notation using the$ \sigma^i$'s given by the local frame, then

\nabla(\sigma) = d\sigma + \omega \sigma

We see that $d\sigma$ is the trivial connection on the trivial bundle $\cale_U$ . This is a general fact about connections.

Remark Any connection differs from another by an endomorphism-valued 1-form.

In the case above, we see that any local connection $\nabla \mid_U$ on $U$ differs from the trivial connection $d$ by an endomorphism-valued 1-form $\omega$ .

For any $p \in M$ we can canonically identify $\Omega_{M,p} \otimes \cale_p$ with the vector space of linear maps $T_p M \to \cale_p$ , and so the above gives for every section $\sigma$ and $p \in M$ a smooth assignment (in terms of $p$ ) to a $\bbc$ -linear map $\nabla_p \sigma \colon T_p M \to \cale_p$ . We say that $\sigma$ is parallel in the direction of $v \in T_p M$ if $(\nabla_{p} \sigma) (v) = 0$ . We will use the shorthand $\nabla_v \sigma = 0$ .

This also makes sense if, say, $X$ is a vector field on $M$ , i.e. a section of the tangent bundle $X \colon M \to TM$ , in which case we use the notation $\nabla_X \sigma \colon M \to \cale$ to denote the section of $\cale$ given by $(\nabla_X \sigma)(p) = (\nabla_p \sigma)(X(p))$ .

Unimportant Remark If $\sigma$ is a section such that $\nabla_v \sigma = 0$ for all $v \in T_p M$ , we can define the horizontal subspace of the connection at $\sigma(p) = e \in \cale$ to be

H_e = (d\sigma)_p(T_p M) \subseteq T_e \cale

The following are true.

$H_e$ depends smoothly on $e$ .
The $H_e$ glue to a vector subbundle $H$ of the tangent bundle $T\cale$ (called the horizontal subbundle).
There is another subbundle $V$ of $T\cale$ (called the vertical subbundle) such that $T \cale = V \oplus H$ .
For every $e \in E$ , $H_e \cap V_e = 0$ . In fact, we can take this to be the definition of our connection. This is called the Ehresmann connection, and it allows us to generalize the notion to smooth fiber bundles like principal $G$ -bundles.

Curvature

The curvature of a connection $\nabla$ measures the failure of the connection to be flat, i.e., the failure of parallel transport to be path-independent. Mathematically, it is defined as follows.

Definition The curvature of a connection $\nabla$ is the $\bbc$ -linear map

F_{\nabla} \colon \calo_{\cale} \to \Omega^2_M \otimes \calo_{\cale}

given by $F_{\nabla}(\sigma)(X,Y) = \nabla_X \nabla_Y \sigma - \nabla_Y \nabla_X \sigma - \nabla_{[X,Y]} \sigma$ for vector fields $X,Y$ on $M$ .

In terms of a local frame as above, if $\omega$ is the connection form, then the curvature is given by

F_{\nabla} = d\omega + \omega \wedge \omega

This is sometimes called the structure equation.

Holomorphic Structures

When $M$ is a complex manifold, we can ask for our connection to be compatible with the complex structure. This leads to the notion of a holomorphic structure.

Definition A connection $\nabla$ on a complex vector bundle $\cale \to M$ over a complex manifold is called compatible with the complex structure if

\nabla = \nabla^{1,0} + \nabla^{0,1}

where $\nabla^{1,0}$ and $\nabla^{0,1}$ are the $(1,0)$ and $(0,1)$ components respectively.

The key relationship between connections and holomorphic structures is given by the following theorem:

Theorem Let $\cale \to M$ be a complex vector bundle over a complex manifold. Then $\cale$ admits a holomorphic structure if and only if there exists a connection $\nabla$ such that $F_{\nabla}^{0,2} = 0$ .

This is part of the Newlander-Nirenberg theorem and shows that the existence of holomorphic structures is fundamentally a differential-geometric question.

Finally, we note that on a Kähler manifold, the Levi-Civita connection provides a canonical way to study the geometry through parallel transport, and its curvature gives rise to important invariants like the Ricci curvature.