In a smooth coordinate chart, the Christoffel symbols of the first kind are given by
Γ
k
i
j
=
1
2
(
∂
∂
x
j
g
k
i
+
∂
∂
x
i
g
k
j
−
∂
∂
x
k
g
i
j
)
=
1
2
(
g
k
i
,
j
+
g
k
j
,
i
−
g
i
j
,
k
)
,
and the Christoffel symbols of the second kind by
Γ
m
i
j
=
g
m
k
Γ
k
i
j
=
1
2
g
m
k
(
∂
∂
x
j
g
k
i
+
∂
∂
x
i
g
k
j
−
∂
∂
x
k
g
i
j
)
=
1
2
g
m
k
(
g
k
i
,
j
+
g
k
j
,
i
−
g
i
j
,
k
)
.
Here
g
i
j
is the inverse matrix to the metric tensor
g
i
j
. In other words,
δ
i
j
=
g
i
k
g
k
j
and thus
n
=
δ
i
i
=
g
i
i
=
g
i
j
g
i
j
is the dimension of the manifold.
Christoffel symbols satisfy the symmetry relations
Γ
k
i
j
=
Γ
k
j
i
or, respectively,
Γ
i
j
k
=
Γ
i
k
j
,
the second of which is equivalent to the torsion-freeness of the Levi-Civita connection.
The contracting relations on the Christoffel symbols are given by
Γ
i
k
i
=
1
2
g
i
m
∂
g
i
m
∂
x
k
=
1
2
g
∂
g
∂
x
k
=
∂
log
|
g
|
∂
x
k
and
g
k
ℓ
Γ
i
k
ℓ
=
−
1
|
g
|
∂
(
|
g
|
g
i
k
)
∂
x
k
where |g| is the absolute value of the determinant of the metric tensor
g
i
k
. These are useful when dealing with divergences and Laplacians (see below).
The covariant derivative of a vector field with components
v
i
is given by:
v
i
;
j
=
(
∇
j
v
)
i
=
∂
v
i
∂
x
j
+
Γ
i
j
k
v
k
and similarly the covariant derivative of a
(
0
,
1
)
-tensor field with components
v
i
is given by:
v
i
;
j
=
(
∇
j
v
)
i
=
∂
v
i
∂
x
j
−
Γ
k
i
j
v
k
For a
(
2
,
0
)
-tensor field with components
v
i
j
this becomes
v
i
j
;
k
=
∇
k
v
i
j
=
∂
v
i
j
∂
x
k
+
Γ
i
k
ℓ
v
ℓ
j
+
Γ
j
k
ℓ
v
i
ℓ
and likewise for tensors with more indices.
The covariant derivative of a function (scalar)
ϕ
is just its usual differential:
∇
i
ϕ
=
ϕ
;
i
=
ϕ
,
i
=
∂
ϕ
∂
x
i
Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,
(
∇
k
g
)
i
j
=
(
∇
k
g
)
i
j
=
0
as well as the covariant derivatives of the metric's determinant (and volume element)
∇
k
|
g
|
=
0
The geodesic
X
(
t
)
starting at the origin with initial speed
v
i
has Taylor expansion in the chart:
X
(
t
)
i
=
t
v
i
−
t
2
2
Γ
i
j
k
v
j
v
k
+
O
(
t
3
)
If one defines the curvature operator as
R
(
U
,
V
)
W
=
∇
U
∇
V
W
−
∇
V
∇
U
W
−
∇
[
U
,
V
]
W
and the coordinate components of the
(
1
,
3
)
-Riemann curvature tensor by
(
R
(
U
,
V
)
W
)
ℓ
=
R
ℓ
i
j
k
W
i
U
j
V
k
, then these components are given by:
R
ℓ
i
j
k
=
∂
∂
x
j
Γ
ℓ
i
k
−
∂
∂
x
k
Γ
ℓ
i
j
+
Γ
ℓ
j
s
Γ
i
k
s
−
Γ
ℓ
k
s
Γ
s
i
j
Lowering indices with
R
ℓ
i
j
k
=
g
ℓ
s
R
s
i
j
k
one gets
R
i
k
ℓ
m
=
1
2
(
∂
2
g
i
m
∂
x
k
∂
x
ℓ
+
∂
2
g
k
ℓ
∂
x
i
∂
x
m
−
∂
2
g
i
ℓ
∂
x
k
∂
x
m
−
∂
2
g
k
m
∂
x
i
∂
x
ℓ
)
+
g
n
p
(
Γ
n
k
ℓ
Γ
p
i
m
−
Γ
n
k
m
Γ
p
i
ℓ
)
.
The symmetries of the tensor are
R
i
k
ℓ
m
=
R
ℓ
m
i
k
and
R
i
k
ℓ
m
=
−
R
k
i
ℓ
m
=
−
R
i
k
m
ℓ
.
That is, it is symmetric in the exchange of the first and last pair of indices, and antisymmetric in the flipping of a pair.
The cyclic permutation sum (sometimes called first Bianchi identity) is
R
i
k
ℓ
m
+
R
i
m
k
ℓ
+
R
i
ℓ
m
k
=
0.
The (second) Bianchi identity is
∇
m
R
n
i
k
ℓ
+
∇
ℓ
R
n
i
m
k
+
∇
k
R
n
i
ℓ
m
=
0
,
that is,
R
n
i
k
ℓ
;
m
+
R
n
i
m
k
;
ℓ
+
R
n
i
ℓ
m
;
k
=
0
which amounts to a cyclic permutation sum of the last three indices, leaving the first two fixed.
Ricci and scalar curvatures
Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information.
The Ricci curvature tensor is essentially the unique nontrivial way of contracting the Riemann tensor:
R
i
j
=
R
ℓ
i
ℓ
j
=
g
ℓ
m
R
i
ℓ
j
m
=
g
ℓ
m
R
ℓ
i
m
j
=
∂
Γ
ℓ
i
j
∂
x
ℓ
−
∂
Γ
ℓ
i
ℓ
∂
x
j
+
Γ
m
i
j
Γ
ℓ
ℓ
m
−
Γ
m
i
ℓ
Γ
ℓ
j
m
.
The Ricci tensor
R
i
j
is symmetric.
By the contracting relations on the Christoffel symbols, we have
R
i
k
=
∂
Γ
ℓ
i
k
∂
x
ℓ
−
Γ
m
i
ℓ
Γ
ℓ
k
m
−
∇
k
(
∂
∂
x
i
(
log
|
g
|
)
)
.
The scalar curvature is the trace of the Ricci curvature,
R
=
g
i
j
R
i
j
=
g
i
j
g
ℓ
m
R
i
ℓ
j
m
.
The "gradient" of the scalar curvature follows from the Bianchi identity (proof):
∇
ℓ
R
ℓ
m
=
1
2
∇
m
R
,
that is,
R
ℓ
m
;
ℓ
=
1
2
R
;
m
.
The Einstein tensor Gab is defined in terms of the Ricci tensor Rab and the Ricci scalar R,
G
a
b
=
R
a
b
−
1
2
g
a
b
R
where g is the metric tensor.
The Einstein tensor is symmetric, with a vanishing divergence (proof) which is due to the Bianchi identity:
∇
a
G
a
b
=
G
a
b
;
a
=
0.
The Weyl tensor is given by
C
i
k
ℓ
m
=
R
i
k
ℓ
m
+
1
n
−
2
(
R
i
m
g
k
ℓ
−
R
i
ℓ
g
k
m
+
R
k
ℓ
g
i
m
−
R
k
m
g
i
ℓ
)
+
1
(
n
−
1
)
(
n
−
2
)
R
(
g
i
ℓ
g
k
m
−
g
i
m
g
k
ℓ
)
,
where
n
denotes the dimension of the Riemannian manifold.
The Weyl tensor satisfies the first (algebraic) Bianchi identity:
C
i
j
k
l
+
C
k
i
j
l
+
C
j
k
i
l
=
0.
The Weyl tensor is a symmetric product of alternating 2-forms,
C
i
j
k
l
=
−
C
j
i
k
l
C
i
j
k
l
=
C
k
l
i
j
,
just like the Riemann tensor. Moreover, taking the trace over any two indices gives zero,
C
i
j
k
i
=
0
The Weyl tensor vanishes (
C
=
0
) if and only if a manifold
M
of dimension
n
≥
4
is locally conformally flat. In other words,
M
can be covered by coordinate systems in which the metric
d
s
2
satisfies
d
s
2
=
f
2
(
d
x
1
2
+
d
x
2
2
+
…
d
x
n
2
)
This is essentially because
C
i
j
k
l
is invariant under conformal changes.
The gradient of a function
ϕ
is obtained by raising the index of the differential
∂
i
ϕ
d
x
i
, whose components are given by:
∇
i
ϕ
=
ϕ
;
i
=
g
i
k
ϕ
;
k
=
g
i
k
ϕ
,
k
=
g
i
k
∂
k
ϕ
=
g
i
k
∂
ϕ
∂
x
k
The divergence of a vector field with components
V
m
is
∇
m
V
m
=
∂
V
m
∂
x
m
+
V
k
∂
log
|
g
|
∂
x
k
=
1
|
g
|
∂
(
V
m
|
g
|
)
∂
x
m
.
The Laplace–Beltrami operator acting on a function
f
is given by the divergence of the gradient:
Δ
f
=
∇
i
∇
i
f
=
1
|
g
|
∂
∂
x
j
(
g
j
k
|
g
|
∂
f
∂
x
k
)
=
g
j
k
∂
2
f
∂
x
j
∂
x
k
+
∂
g
j
k
∂
x
j
∂
f
∂
x
k
+
1
2
g
j
k
g
i
l
∂
g
i
l
∂
x
j
∂
f
∂
x
k
=
g
j
k
∂
2
f
∂
x
j
∂
x
k
−
g
j
k
Γ
l
j
k
∂
f
∂
x
l
The divergence of an antisymmetric tensor field of type
(
2
,
0
)
simplifies to
∇
k
A
i
k
=
1
|
g
|
∂
(
A
i
k
|
g
|
)
∂
x
k
.
The Hessian of a map
ϕ
:
M
→
N
is given by
(
∇
(
d
ϕ
)
)
i
j
γ
=
∂
2
ϕ
γ
∂
x
i
∂
x
j
−
M
Γ
k
i
j
∂
ϕ
γ
∂
x
k
+
N
Γ
γ
α
β
∂
ϕ
α
∂
x
i
∂
ϕ
β
∂
x
j
.
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let
h
and
k
be symmetric covariant 2-tensors. In coordinates,
h
i
j
=
h
j
i
k
i
j
=
k
j
i
Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted
h
∧
◯
k
. The defining formula is
(
h
∧
◯
k
)
i
j
k
l
=
h
i
k
k
j
l
+
h
j
l
k
i
k
−
h
i
l
k
j
k
−
h
j
k
k
i
l
Clearly, the product satisfies
h
∧
◯
k
=
k
∧
◯
h
An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations
g
i
j
=
δ
i
j
and
Γ
i
j
k
=
0
(but these may not hold at other points in the frame). These coordinates are also called normal coordinates. In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only.
R
i
k
ℓ
m
=
1
2
(
∂
2
g
i
m
∂
x
k
∂
x
ℓ
+
∂
2
g
k
ℓ
∂
x
i
∂
x
m
−
∂
2
g
i
ℓ
∂
x
k
∂
x
m
−
∂
2
g
k
m
∂
x
i
∂
x
ℓ
)
R
ℓ
i
j
k
=
∂
∂
x
j
Γ
ℓ
i
k
−
∂
∂
x
k
Γ
ℓ
i
j
Let
g
be a Riemannian metric on a smooth manifold
M
, and
φ
a smooth real-valued function on
M
. Then
g
~
=
e
2
φ
g
is also a Riemannian metric on
M
. We say that
g
~
is conformal to
g
. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with
g
~
, while those unmarked with such will be associated with
g
.)
g
~
i
j
=
e
2
φ
g
i
j
Γ
~
k
i
j
=
Γ
k
i
j
+
δ
i
k
∂
j
φ
+
δ
j
k
∂
i
φ
−
g
i
j
∇
k
φ
Note that the difference between the Christoffel symbols of two different metrics always form the components of a tensor.
We can also write this in a coordinate-free manner:
∇
~
F
∗
X
F
∗
Y
=
F
∗
(
∇
X
Y
+
X
(
φ
)
Y
+
Y
(
φ
)
X
−
g
(
X
,
Y
)
grad
φ
)
,
(where
F
:
M
→
N
is the conformal map, i.e.:
F
∗
g
~
=
e
2
φ
g
, and
X
,
Y
are vector fields.)
d
V
~
=
e
n
φ
d
V
Here
d
V
is the Riemannian volume element.
R
~
i
j
k
l
=
e
2
φ
(
R
i
j
k
l
−
[
g
∧
◯
(
∇
∂
φ
−
∂
φ
∂
φ
+
1
2
∥
∇
φ
∥
2
g
)
]
i
j
k
l
)
Here
∧
◯
is the Kulkarni–Nomizu product defined earlier in this article. The symbol
∂
k
denotes partial derivative, while
∇
k
denotes covariant derivative.
R
~
i
j
=
R
i
j
−
(
n
−
2
)
[
∇
i
∂
j
φ
−
(
∂
i
φ
)
(
∂
j
φ
)
]
+
(
△
φ
−
(
n
−
2
)
∥
∇
φ
∥
2
)
g
i
j
Beware that here the Laplacian
△
is minus the trace of the Hessian on functions,
△
f
=
−
∇
i
∂
i
f
Thus the operator
−
△
is elliptic because the metric
g
is Riemannian.
△
~
f
=
e
−
2
φ
(
△
f
−
(
n
−
2
)
∇
k
φ
∇
k
f
)
R
~
=
e
−
2
φ
(
R
+
2
(
n
−
1
)
△
φ
−
(
n
−
2
)
(
n
−
1
)
∥
∇
φ
∥
2
)
If the dimension
n
>
2
, then this simplifies to
R
~
=
e
−
2
φ
[
R
+
4
(
n
−
1
)
(
n
−
2
)
e
−
(
n
−
2
)
φ
/
2
△
(
e
(
n
−
2
)
φ
/
2
)
]
C
~
i
j
k
l
=
C
i
j
k
l
We see that the (3,1) Weyl tensor is invariant under conformal changes.
Let
ω
be a differential
p
-form. Let
∗
be the Hodge star, and
δ
the codifferential. Under a conformal change, these satisfy
∗
~
=
e
(
n
−
2
p
)
φ
∗
[
δ
~
ω
]
(
v
1
,
v
2
,
…
,
v
p
−
1
)
=
e
−
2
φ
[
δ
ω
−
(
n
−
2
p
)
ω
(
∇
φ
,
v
1
,
v
2
,
…
,
v
p
−
1
)
]
