Schur's lemma is a result in Riemannian geometry that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. It is essentially a degree of freedom counting argument.
Suppose
(
M
n
,
g
)
is a Riemannian manifold and
n
≥
3
. Then if
the sectional curvature is pointwise constant, that is, there exists some function
f
:
M
→
R
such that
s
e
c
t
(
Π
p
)
=
f
(
p
)
for all two-dimensional subspaces
Π
p
⊂
T
p
M
and all
p
∈
M
,
then
f
is constant, and the manifold has constant sectional curvature (also known as a space form when
M
is complete); alternatively
the Ricci curvature endomorphism is pointwise a multiple of the identity, that is, there exists some function
f
:
M
→
R
such that
R
i
c
(
X
p
)
=
f
(
p
)
X
p
for all
X
p
∈
T
p
M
and all
p
∈
M
,
then
f
is constant, and the manifold is Einstein.
The requirement that
n
≥
3
cannot be lifted. This result is far from true on two-dimensional surfaces. In two dimensions sectional curvature is always pointwise constant since there is only one two-dimensional subspace
Π
p
⊂
T
p
M
, namely
T
p
M
. Furthermore, in two dimensions the Ricci curvature endomorphism is always a multiple of the identity (scaled by Gauss curvature). On the other hand, certainly not all two-dimensional surfaces have constant sectional (or Ricci) curvature.
