Lecture Notes for 120 - UCLA Department of Mathematics

8.2. RIEMANNIAN GEOMETRY 171
This expression shows how certain third order partials might not commute since
we have
2 3
@ 3 q
@u i @u j @u k
@ 3 q
@u j @u i @u k = ⇥
@q
@u 1 ···
@q
@u n
⇤ 6
4
But recall that since second order partials do commute we have
@ 3 q
@u i @u j @u k = @ 3 q
@u i @u k @u j
So we see that third order partials commute if and only if the Riemann curvature
vanishes. This can be used to establish the difficult existence part of the next result.
R 1 ijk
.
R n ijk
Theorem 8.2.1. [Riemann] The Riemann curvature vanishes if and only if
there are Cartesian coordinates around any point.
Proof. The easy direction is to assume that Cartesian coordinates exist. Certainly
this shows that the curvatures vanish when we use Cartesian coordinates, but
this does not guarantee that they also vanish in some arbitrary coordinate system.
For that we need to figure out how the curvature terms change when we change
coordinates. A long tedious calculation shows that if the new coordinates are called
v i and the curvature in these coordinates ˜R l ijk , then
˜R ijk l = @u↵ @u @u @v l
@v i @v j @v k @u R ↵ .
Thus we see that if the all curvatures vanish in one coordinate system, then they
vanish in all coordinate systems.
Conversely, to find Cartesian coordinates set up a system of differential equations
@q
@u i = U i
@ ⇥ ⇤
U1 ··· U
@u i n = ⇥ ⇤
U 1 ··· U n [ i]
whose integrability conditions are a consequence of having vanishing curvature. ⇤
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