Over the past couple of days I have been brushing up on introductory differential geometry. I’ve blogged about this subject a fair bit in the past, but I’ve never really had a good feel for it. I’d therefore like to make this post, and the next, a “big picture” one, rather than focusing on the technical details.

1. Curvature of a connection

Let ${M }$ be a manifold, and let ${V \rightarrow M}$ be a vector bundle. Suppose given a connection ${\nabla}$ on ${V}$. This determines, and is equivalent to, the data of parallel transport along each (smooth) curve ${\gamma: [0, 1] \rightarrow M}$. In other words, for each such ${\gamma}$, one gets an isomorphism of vector spaces

$\displaystyle T_{\gamma}: V_{\gamma(0)} \simeq V_{\gamma(1)}$

with certain nice properties: for example, given a concatenation of two smooth curves, the parallel transport behaves transitively. Moreover, a homotopy of curves induces a homotopy of the parallel transport operators.

In particular, if we fix a point ${p \in M}$, we get a map

$\displaystyle \Omega_p M \rightarrow \mathrm{GL}( V_p)$

that sends a loop at ${p}$ to the induced automorphism of ${V_p}$ given by parallel transport along it. (Here we’ll want to take ${\Omega_p M}$ to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.)

There’s a homotopy theoretical analog of this. Given a fibration ${E \rightarrow B}$, and given any path ${\gamma: [0, 1] \rightarrow B}$, we get an identification between the fiber

$\displaystyle E_{\gamma(0)} \simeq E_{\gamma(1)} ,$

which is well-defined up to coherent homotopy. We can think of this as the homotopy-theoretical analog of parallel transport. Another way to say it is that there is a coherently associative action of the loop space ${\Omega B}$ on a given fiber ${E_{b}}$. In the theory of ${(\infty, 1)}$-categories, a very powerful way to express these ideas is given by the Grothendieck construction, which states that to give a fibration over ${B}$ is equivalent to giving a functor

$\displaystyle \Pi_\infty B \rightarrow \mathrm{Spaces}$

where ${\Pi_\infty B}$ is the “fundamental ${\infty}$-groupoid of ${B}$.” In other words, a fibration over ${B}$ is determined by a space (the fiber) and homotopy coherent “parallel transport.”

What does this all have to do with curvature? The curvature of the vector bundle ${V \rightarrow M}$ with connection ${\nabla}$ is defined to be the operator ${R}$ (a ${\mathrm{End}(V)}$-valued 2-form) with

$\displaystyle R(X, Y, s) = \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X, Y]}s ,$

for vector fields ${X, Y}$ and sections ${s}$ of ${V}$. It measures the failure of the differential operator ${\nabla}$ to be symmetric: that is, for mixed partials to be equal.

If mixed partials are equal, that is if ${R \equiv 0}$, then the connection is said to be flat and a consequence is that the infinitesimal holonomy is trivial. In other words, if the connection is flat and if ${U}$ is a contractible neighborhood of ${p}$, then the composite

$\displaystyle \Omega_p U \rightarrow \Omega_p M \rightarrow \mathrm{GL}(V_p)$

is trivial. In other words, if mixed partials are equal, then parallel transport between two points doesn’t depend on the choice of curve connecting them, as long as they’re close enough. Equivalently — the connection is flat precisely when two homotopic paths lead to equal parallel transports.

2. Flat connections and local systems

Let’s now say what flatness means homotopically. Given a vector bundle ${V \rightarrow M}$, we can think of that as a functor of ${(\infty, 1)}$-categories

$\displaystyle \Pi_\infty M \rightarrow \mathrm{Vect}_{\mathbb{R}}^{\simeq}$

where ${\Pi_\infty M}$ is, again, ${M}$ treated as an ${\infty}$-groupoid and ${\mathrm{Vect} _{\mathbb{R}}^{\simeq}}$ is the topological category of finite-dimensional real vector spaces and isomorphisms between them. This assertion is a restatement of the fact that ${\bigsqcup BO(n)}$ is the classifying space for real vector bundles.

We can also say this in the following manner. To give a vector bundle ${V \rightarrow M}$ is equivalent to giving:

• For each ${p \in M}$, a choice of finite-dimensional real vector space ${V_p}$.
• For each path ${\gamma}$ from ${p}$ to ${q}$, an isomorphism ${T_\gamma: V_p \simeq V_q}$.
• For each homotopy of paths ${\gamma, \widetilde{\gamma}}$ between ${p,q}$, a homotopy

$\displaystyle T_{\gamma} \simeq T_{\widetilde{\gamma}} \in \mathrm{Iso}(V_p, V_q).$

• For each “homotopy of homotopies” of paths a “homotopy of homotopies” between the associated operators, and
• Coherence data for all this.

The choice of a connection — and it is not really a choice, since the space of connections is contractible — is one way to express the above data.

When a vector bundle comes with a flat connection, then any two homotopic paths lead to the same parallel transport data. That means that a flat connection determines a choice of factorization

$\displaystyle \Pi_\infty M \rightarrow \Pi_{\leq 1} M \rightarrow \mathrm{Vect}_{\mathbb{R}}^{\simeq},$

where ${\Pi_{\leq 1} M}$ is the (ordinary!) groupoid of points on ${M}$ and homotopy classes of paths between them. This is also known as a local system on ${M}$. In fact, one may state:

Theorem 1 There is an equivalence of categories between local systems of ${\mathbb{R}}$-vector spaces on ${M}$ and pairs ${(V, \nabla)}$ where ${V}$ is a vector bundle on ${M}$ and ${\nabla}$ is a flat connection on ${V}$.

There is another more explicit way of describing the correspondence. Given a vector bundle ${V \rightarrow M}$ with a flat connection ${\nabla}$, we can obtain a local system ${\mathcal{F}}$ (considered here as a locally constant sheaf on ${M}$) on ${M}$ as follows. Given an open subset ${U \subset M}$, let

$\displaystyle \mathcal{F}(U) = \left\{\mathrm{horizontal \ sections \ of \ } V(U)\right\}$

where a section ${s}$ is horizontal if ${\nabla s = 0}$.

In other words, curvature measures the fact that the vector bundle really depends on higher-order homotopical data, rather than simply the fundamental groupoid. There are more precise versions of this statement, for connections on principal ${G}$-bundles, for instance the Ambrose-Singer theorem that states that the image of curvature is exactly the Lie algebra of the infinitesimal holonomy group.

In the next post, I’d like to describe some of the more quantitative versions of this statement, and specialize to Riemannian manifolds, where there are many additional things to say.