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 2 The 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 3 In 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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