1Basics of Riemannian manifolds

III Riemannian Geometry

1 Basics of Riemannian manifolds
Before we do anything, we lay out our conventions. Given a choice of local
coordinates {x
k
}, the coefficients X
k
for a vector field X are defined by
X =
X
k
X
k
x
k
.
In general, for a tensor field X T M
q
T
M
p
, we write
X =
X
X
k
1
...k
q
`
1
...`
p
x
k
1
···
x
k
q
dx
`
1
··· dx
`
p
,
and we often leave out the signs.
For the sake of sanity, we will often use implicit summation convention, i.e.
whenever we write something of the form
X
ijk
Y
i`jk
,
we mean
X
i,j
X
ijk
Y
i`jk
.
We will use upper indices to denote contravariant components, and lower
indices for covariant components, as we have done above. Thus, we always sum
an upper index with a lower index, as this corresponds to applying a covector to
a vector.
We will index the basis elements oppositely, e.g. we write d
x
k
x
k
for a basis element of
T
M
, so that the indices in expressions of the form
A
k
d
x
k
seem to match up. Whenever we do not follow this convention, we will write out
summations explicitly.
We will also adopt the shorthands
k
=
x
k
,
k
=
k
.
With these conventions out of the way, we begin with a very brief summary
of some topics in the Michaelmas Differential Geometry course, starting from
the definition of a Riemannian metric.
Definition
(Riemannian metric)
.
Let
M
be a smooth manifold. A Riemannian
metric
g
on
M
is an inner product on the tangent bundle
T M
varying smoothly
with the fibers. Formally, this is a global section of
T
M T
M
that is fiberwise
symmetric and positive definite.
The pair (M, g) is called a Riemannian manifold.
On every coordinate neighbourhood with coordinates
x
= (
x
1
, ··· , x
n
), we
can write
g =
n
X
i,j=1
g
ij
(x) dx
i
dx
j
,
and we can find the coefficients g
ij
by
g
ij
= g
x
i
,
x
j
and are C
functions.
Example.
The manifold
R
k
has a canonical metric given by the Euclidean
metric. In the usual coordinates, g is given by g
ij
= δ
ij
.
Does every manifold admit a metric? Recall
Theorem
(Whitney embedding theorem)
.
Every smooth manifold
M
an embedding into
R
k
for some
k
. In other words,
M
is diffeomorphic to a
submanifold of R
k
. In fact, we can pick k such that k 2 dim M.
Using such an embedding, we can induce a Riemannian metric on
M
by
restricting the inner product from Euclidean space, since we have inclusions
T
p
M T
p
R
k
=
R
k
.
More generally,
Lemma.
Let (
N, h
) be a Riemannian manifold, and
F
:
M N
is an immersion,
then the pullback g = F
h defines a metric on M.
The condition of immersion is required for the pullback to be non-degenerate.
In Differential Geometry, if we do not have metrics, then we tend to consider
diffeomorphic spaces as being the same. With metrics, the natural notion of
isomorphism is
Definition
(Isometry)
.
Let (
M, g
) and (
N, h
) be Riemannian manifolds. We
say
f
:
M N
is an isometry if it is a diffeomorphism and
f
h
=
g
. In other
words, for any p M and u, v T
p
M, we need
h
(df)
p
u, (df)
p
v
= g(u, v).
Example.
Let
G
be a Lie group. Then for any
x
, we have translation maps
L
x
, R
x
: G G given by
L
x
(y) = xy
R
x
(y) = yx
These maps are in fact diffeomorphisms of G.
G
admits a Riemannian metric, but we might want
to ask something stronger does there exist a left-invariant metric? In other
words, is there a metric such that each L
x
is an isometry?
Recall the following definition:
Definition
(Left-invariant vector field)
.
Let
G
be a Lie group, and
X
a vector
field. Then X is left invariant if for any x G, we have d(L
x
)X = X.
We had a rather general technique for producing left-invariant vector fields.
Given a Lie group
G
, we can define the Lie algebra
g
=
T
e
G
. Then we can
produce left-invariant vector fields by picking some X
e
g, and then setting
X
a
= d(L
a
)X
e
.
The resulting vector field is indeed smooth, as shown in the differential geometry
course.
Similarly, to construct a left-invariant metric, we can just pick a metric at
the identity and the propagating it around using left-translation. More explicitly,
given any inner product on h·, ·i on T
e
G, we can define g by
g(u, v) = h(dL
x
1
)
x
u, (dL
x
1
)
x
vi
for all
x G
and
u, v T
x
G
. The argument for smoothness is similar to that
for vector fields.
Of course, everything works when we replace “left” with “right”. A Rie-
mannian metric is said to be bi-invariant if it is both left- and right-invariant.
These are harder to find, but it is a fact that every compact Lie group admits a
bi-invariant metric. The basic idea of the proof is to start from a left-invariant
metric, then integrate the metric along right translations of all group elements.
Here compactness is necessary for the result to be finite.
We will later see that we cannot drop the compactness condition. There are
non-compact Lie groups that do not admit bi-invariant metrics, such as
SL
(2
, R
).
Recall that in order to differentiate vectors, or even tensors on a manifold,
we needed a connection on the tangent bundle. There is a natural choice for the
connection when we are given a Riemannian metric.
Definition
(Levi-Civita connection)
.
Let (
M, g
) be a Riemannian manifold.
The Levi-Civita connection is the unique connection
: Ω
0
M
(
T M
)
1
M
(
T M
)
on M satisfying
(i) Compatibility with metric:
Zg(X, Y ) = g(
Z
X, Y ) + g(X,
Z
Y ),
(ii) Symmetry/torsion-free:
X
Y
Y
X = [X, Y ].
Definition
(Christoffel symbols)
.
In local coordaintes, the Christoffel symbols
are defined by
j
x
k
= Γ
i
jk
x
i
.
With a bit more imagination on what the symbols mean, we can write the
first property as
d(g(X, Y )) = g(X, Y ) + g(X, Y ),
while the second property can be expressed in coordinate representation by
Γ
i
jk
= Γ
i
kj
.
The connection was defined on
T M
, but in fact, the connection allows us to
differentiate many more things, and not just tangent vectors.
Firstly, the connection
induces a unique covariant derivative on
T
M
, also
denoted , defined uniquely by the relation
Xhα, Y i = h∇
X
α, Y i + hα,
X
Y i
for any X, Y Vect(M) and α
1
(M).
To extend this to a connection
on tensor bundles
T
q,p
(
T M
)
q
(T
M)
p
for any p, q 0, we note the following general construction:
In general, suppose we have vector bundles
E
and
F
, and
s
1
Γ(
E
) and
s
2
Γ(
F
). If we have connections
E
and
F
on
E
and
F
respectively, then
we can define
EF
(s
1
s
2
) = (
E
s
1
) s
2
+ s
1
(
F
s
2
).
Since we already have a connection on
T M
and
T
M
, this allows us to extend
the connection to all tensor bundles.
Given this machinery, recall that the Riemannian metric is formally a section
g
Γ(
T
M T
M
). Then the compatibility with the metric can be written in
the following even more compact form:
g = 0.