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 be a vector bundle with connection . Choose a neighborhood of that looks like with at the point . Since everything we do is local, we may just assume

This gives us a particularly nice frame for . Choose a basis for and then define sections of on at a point by parallel translation along the (euclidean!) line from to . Then is a global frame for and is, by construction, parallel along each straight line through the origin. One therefore has:

Using this choice of frame, we get an identification of with the trivial bundle . In particular, we can write the connection in the form

where is a matrix of 1-forms. We can write

where is a matrix-valued function. In this choice of coordinates, the curvature is an -valued 2-form, and it is given by

Our goal is to express the terms of the Taylor expansion of in terms of . First note that

which means that we get from the curvature. In particular, letting , we have for ,

But that isn’t enough to determine even the linear terms of the Taylor expansion of at .

In order to do this, consider the radial vector field

whose integral curves are the lines through the origin. The sections used to trivialize are parallel along the integral curves of , by assumption, which implies that or

where is a matrix-valued function. In coordinates, we get that

or, applying ,

This antisymmetry, combined with (2), gives

Proposition 2The linear terms are determined in terms of .

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

**2. The Riemannian case**

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

and can also be written as a 4-tensor

The 4-tensor 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 with its canonical metric.

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

where is an open subset of containing the origin. Restricting , we can assume that it is a diffeomorphism into , whose image is a neighborhood of .

Now is an open subset of a vector space and we can therefore choose coordinates which are linear on , and such that the vector fields forms an orthonormal basis for . In these coordinates, we have

and the connection coefficients . 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 3In exponential coordinates, one has:

where are the curvature values at (i.e. ).

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

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