I’d also like to describe some more quantitative versions of what curvature means. We saw in the previous post that curvature measures the sense in which a connection fails to be a local system, or in other words look locally like the standard connection on a trivial bundle. (This is perhaps part of the motivation of the use of curvature to construct de Rham representatives of the characteristic classes of a vector bundle, cf. Chern-Weil theory.)

1. Curvature as deviation from flatness

If you work in coordinates, it’s not immediately clear how to say to what extent a connection looks like a trivial connection, because the trivial connection can look very different if you change the frame. Curvature has the property of being tensorial and not depending on a given choice of frame.

But another way to say this is to try to express the connection in the best possible choice of coordinates, and see whether it looks like the standard one. Namely, let ${V \rightarrow M}$ be a vector bundle with connection ${\nabla}$. Choose a neighborhood of ${p \in M}$ that looks like ${\mathbb{R}^n}$ with ${p}$ at the point ${0}$. Since everything we do is local, we may just assume

$\displaystyle M = \mathbb{R}^n, p = 0.$

This gives us a particularly nice frame for ${V}$. Choose a basis ${e_1, \dots, e_m}$ for ${V_p}$ and then define sections ${E_1, \dots, E_m}$ of ${V}$ on ${\mathbb{R}^n}$ at a point ${q}$ by parallel translation along the (euclidean!) line from ${p=0}$ to ${q}$. Then ${E_1, \dots, E_m}$ is a global frame for ${V}$ and is, by construction, parallel along each straight line through the origin. One therefore has:

$\displaystyle \nabla E_i ( p) = 0, \quad 1 \leq i \leq m.$

Using this choice of frame, we get an identification of ${V}$ with the trivial bundle ${\mathbb{R}^m}$. In particular, we can write the connection ${\nabla: V \rightarrow \Omega^1(V)}$ in the form

$\displaystyle \nabla = d + \omega,$

where ${\omega \in \Omega^1(\mathrm{End}(\mathbb{R}^n))}$ is a matrix of 1-forms. We can write

$\displaystyle \omega = \sum \omega_i dx^i,$

where ${\omega_i}$ is a matrix-valued function. In this choice of coordinates, the curvature is an ${\mathrm{End}(\mathbb{R}^n)}$-valued 2-form, and it is given by

$\displaystyle \Omega = d \omega + \omega \wedge \omega. \ \ \ \ \ (1)$

Our goal is to express the terms of the Taylor expansion of ${\omega_i}$ in terms of ${\Omega(p)}$. First note that

$\displaystyle \Omega(p) = d \omega(p),$

which means that we get ${d \omega(p)}$ from the curvature. In particular, letting ${\Omega = \sum_{i < j} \Omega_{ij} dx^i \wedge dx^j}$, we have for ${i < j}$,

$\displaystyle \Omega_{ij}(p) = \partial_i \omega_j(p) - \partial_j \omega_i(p). \ \ \ \ \ (2)$

But that isn’t enough to determine even the linear terms of the Taylor expansion of ${\omega}$ at ${p}$.

In order to do this, consider the radial vector field

$\displaystyle \mathcal{R} = \sum x_i \frac{\partial}{\partial x_i},$

whose integral curves are the lines through the origin. The sections ${E_1, \dots, E_m}$ used to trivialize ${V}$ are parallel along the integral curves of ${\mathcal{R}}$, by assumption, which implies that ${\nabla_{\mathcal{R}} E_k = 0}$ or

$\displaystyle \iota_{\mathcal{R}} \omega_i = 0, \quad \forall i,$

where ${\iota_{\mathcal{R}} \omega_i}$ is a matrix-valued function. In coordinates, we get that

$\displaystyle \sum x_i \omega_i = 0,$

or, applying ${\partial_{i_1} \partial_{i_2}}$,

$\displaystyle \partial_{i_1} \omega_{i_2} + \partial_{i_2} \omega_{i_1} = 0.$

This antisymmetry, combined with (2), gives

$\displaystyle \Omega_{ij}(p) = 2 \partial_i \omega_j (p), \quad i < j.$

Proposition 2 The linear terms ${\partial_i \omega_j}$ are determined in terms of ${\Omega_{ij}(p)}$.

In other words, when you choose “good coordinates” for ${V}$, then you can make the connection 1-forms ${\omega}$ vanish at a given point ${p}$, but the curvature still shows up as the linear terms in the Taylor expansion of ${\omega}$.

2. The Riemannian case

Now suppose we’re working with a Riemannian manifold ${M}$ with metric ${g}$. Then the tangent bundle of ${M}$ comes equipped with a canonical connection ${\nabla}$, whose curvature will simply be called the curvature of ${M}$; that is, it is a tensor

$\displaystyle R(\cdot, \cdot, \cdot): TM \otimes TM \otimes TM \rightarrow TM,$

and can also be written as a 4-tensor

$\displaystyle R(X, Y, Z, W) = g( R(X, Y) Z, W).$

The 4-tensor ${R}$ satisfies a collection of symmetry and antisymmetry properties, and the Bianchi identity. In this case, one formulation of curvature is that it is the obstruction to flatness; that is, a Riemannian manifold has vanishing curvature tensor if and only if it is locally isometric to ${\mathbb{R}^n}$ with its canonical metric.

Once again, there’s a quantitative way of measuring this. For any ${M}$ and for any ${p \in M}$, there is a system of “especially good” coordinates around ${p}$ (unique up to an orthogonal linear transformation). There is an exponential map

$\displaystyle \exp_p: T_p M \supset U \rightarrow M,$

where ${U}$ is an open subset of ${T_p M}$ containing the origin. Restricting ${U}$, we can assume that it is a diffeomorphism into ${M}$, whose image is a neighborhood of ${p}$.

Now ${U}$ is an open subset of a vector space and we can therefore choose coordinates ${x_1, \dots, x_n}$ which are linear on ${U}$, and such that the vector fields ${\{\partial_i = \frac{\partial}{\partial x_i}\}}$ forms an orthonormal basis for ${T_p M}$. In these coordinates, we have

$\displaystyle g_{ij} = g(\partial_i, \partial_j) = \delta_{ij} + O(|x|^2),$

and the connection coefficients ${\Gamma_{ij}^k(p) = 0}$. In other words, one can choose local coordinates for a Riemannian manifold such that it looks “flat” to order two.

But the second-order terms are precisely what measure curvature. In fact, using similar but slightly more involved reasoning as above, one can prove:

Proposition 3 In exponential coordinates, one has:

$\displaystyle g_{ij}(\mathbf{x}) = \delta_{ij} - \frac{1}{3} \sum R_{iajb} x^a x^b + O(|x|^3),$

where ${R_{iajb} = R(\partial_j, \partial_b, \partial_a, \partial_i)(p)}$ are the curvature values at ${p}$ (i.e. ${\mathbf{x} = 0}$).

In the next post, I’d like to talk about the sign of the curvatures and what they mean geometrically.