Christoffel symbols, covariant derivative
edit
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
)
,
{\displaystyle \Gamma _{kij}={\frac {1}{2}}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,}
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
)
.
{\displaystyle {\begin{aligned}\Gamma ^{m}{}_{ij}&=g^{mk}\Gamma _{kij}\\&={\frac {1}{2}}\,g^{mk}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\,g^{mk}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,.\end{aligned}}}
Here
g
i
j
{\displaystyle g^{ij}}
is the inverse matrix to the metric tensor
g
i
j
{\displaystyle g_{ij}}
. In other words,
δ
i
j
=
g
i
k
g
k
j
{\displaystyle \delta ^{i}{}_{j}=g^{ik}g_{kj}}
and thus
n
=
δ
i
i
=
g
i
i
=
g
i
j
g
i
j
{\displaystyle n=\delta ^{i}{}_{i}=g^{i}{}_{i}=g^{ij}g_{ij}}
is the dimension of the manifold .
Christoffel symbols satisfy the symmetry relations
Γ
k
i
j
=
Γ
k
j
i
{\displaystyle \Gamma _{kij}=\Gamma _{kji}}
or, respectively,
Γ
i
j
k
=
Γ
i
k
j
{\displaystyle \Gamma ^{i}{}_{jk}=\Gamma ^{i}{}_{kj}}
,
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
{\displaystyle \Gamma ^{i}{}_{ki}={\frac {1}{2}}g^{im}{\frac {\partial g_{im}}{\partial x^{k}}}={\frac {1}{2g}}{\frac {\partial g}{\partial x^{k}}}={\frac {\partial \log {\sqrt {|g|}}}{\partial x^{k}}}}
and
g
k
ℓ
Γ
i
k
ℓ
=
−
1
|
g
|
∂
(
|
g
|
g
i
k
)
∂
x
k
{\displaystyle g^{k\ell }\Gamma ^{i}{}_{k\ell }={\frac {-1}{\sqrt {|g|}}}\;{\frac {\partial \left({\sqrt {|g|}}\,g^{ik}\right)}{\partial x^{k}}}}
where |g | is the absolute value of the determinant of the metric tensor
g
i
k
{\displaystyle g_{ik}}
. These are useful when dealing with divergences and Laplacians (see below).
The covariant derivative of a vector field with components
v
i
{\displaystyle v^{i}}
is given by:
v
i
;
j
=
(
∇
j
v
)
i
=
∂
v
i
∂
x
j
+
Γ
i
j
k
v
k
{\displaystyle v^{i}{}_{;j}=(\nabla _{j}v)^{i}={\frac {\partial v^{i}}{\partial x^{j}}}+\Gamma ^{i}{}_{jk}v^{k}}
and similarly the covariant derivative of a
(
0
,
1
)
{\displaystyle (0,1)}
-tensor field with components
v
i
{\displaystyle v_{i}}
is given by:
v
i
;
j
=
(
∇
j
v
)
i
=
∂
v
i
∂
x
j
−
Γ
k
i
j
v
k
{\displaystyle v_{i;j}=(\nabla _{j}v)_{i}={\frac {\partial v_{i}}{\partial x^{j}}}-\Gamma ^{k}{}_{ij}v_{k}}
For a
(
2
,
0
)
{\displaystyle (2,0)}
-tensor field with components
v
i
j
{\displaystyle v^{ij}}
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
{\displaystyle v^{ij}{}_{;k}=\nabla _{k}v^{ij}={\frac {\partial v^{ij}}{\partial x^{k}}}+\Gamma ^{i}{}_{k\ell }v^{\ell j}+\Gamma ^{j}{}_{k\ell }v^{i\ell }}
and likewise for tensors with more indices.
The covariant derivative of a function (scalar)
ϕ
{\displaystyle \phi }
is just its usual differential:
∇
i
ϕ
=
ϕ
;
i
=
ϕ
,
i
=
∂
ϕ
∂
x
i
{\displaystyle \nabla _{i}\phi =\phi _{;i}=\phi _{,i}={\frac {\partial \phi }{\partial x^{i}}}}
Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,
(
∇
k
g
)
i
j
=
0
,
(
∇
k
g
)
i
j
=
0
{\displaystyle (\nabla _{k}g)_{ij}=0,\quad (\nabla _{k}g)^{ij}=0}
as well as the covariant derivatives of the metric's determinant (and volume element)
∇
k
|
g
|
=
0
{\displaystyle \nabla _{k}{\sqrt {|g|}}=0}
The geodesic
X
(
t
)
{\displaystyle X(t)}
starting at the origin with initial speed
v
i
{\displaystyle 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
)
{\displaystyle X(t)^{i}=tv^{i}-{\frac {t^{2}}{2}}\Gamma ^{i}{}_{jk}v^{j}v^{k}+O(t^{3})}
Curvature tensors
edit
Definitions
edit
R
i
j
k
l
=
∂
Γ
i
k
l
∂
x
j
−
∂
Γ
j
k
l
∂
x
i
+
(
Γ
i
k
p
Γ
j
p
l
−
Γ
j
k
p
Γ
i
p
l
)
{\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}-{\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}+{\big (}\Gamma _{ik}^{p}\Gamma _{jp}^{l}-\Gamma _{jk}^{p}\Gamma _{ip}^{l}{\big )}}
R
(
u
,
v
)
w
=
∇
u
∇
v
w
−
∇
v
∇
u
w
−
∇
[
u
,
v
]
w
{\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w}
R
j
k
l
i
=
∂
Γ
l
j
i
∂
x
k
−
∂
Γ
k
j
i
∂
x
l
+
(
Γ
k
p
i
Γ
l
j
p
−
Γ
l
p
i
Γ
k
j
p
)
{\displaystyle {R_{jkl}^{i}}={\frac {\partial \Gamma _{lj}^{i}}{\partial x^{k}}}-{\frac {\partial \Gamma _{kj}^{i}}{\partial x^{l}}}+{\big (}\Gamma _{kp}^{i}\Gamma _{lj}^{p}-\Gamma _{lp}^{i}\Gamma _{kj}^{p}{\big )}}
R
i
k
=
R
i
j
k
j
{\displaystyle R_{ik}={R_{ijk}}^{j}}
Ric
(
v
,
w
)
=
tr
(
u
↦
R
(
u
,
v
)
w
)
{\displaystyle \operatorname {Ric} (v,w)=\operatorname {tr} (u\mapsto R(u,v)w)}
R
=
g
i
k
R
i
k
{\displaystyle R=g^{ik}R_{ik}}
R
=
tr
g
Ric
{\displaystyle R=\operatorname {tr} _{g}\operatorname {Ric} }
Traceless Ricci tensor
edit
Q
i
k
=
R
i
k
−
1
n
R
g
i
k
{\displaystyle Q_{ik}=R_{ik}-{\frac {1}{n}}Rg_{ik}}
Q
(
u
,
v
)
=
Ric
(
u
,
v
)
−
1
n
R
g
(
u
,
v
)
{\displaystyle Q(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{n}}Rg(u,v)}
(4,0) Riemann curvature tensor
edit
R
i
j
k
l
=
R
i
j
k
p
g
p
l
{\displaystyle R_{ijkl}={R_{ijk}}^{p}g_{pl}}
Rm
(
u
,
v
,
w
,
x
)
=
g
(
R
(
u
,
v
)
w
,
x
)
{\displaystyle \operatorname {Rm} (u,v,w,x)=g{\big (}R(u,v)w,x{\big )}}
W
i
j
k
l
=
R
i
j
k
l
−
1
n
(
n
−
1
)
R
(
g
i
k
g
j
l
−
g
i
l
g
j
k
)
−
1
n
−
2
(
Q
i
k
g
j
l
−
Q
j
k
g
i
l
−
Q
i
l
g
j
k
+
Q
j
l
g
i
k
)
{\displaystyle W_{ijkl}=R_{ijkl}-{\frac {1}{n(n-1)}}R{\big (}g_{ik}g_{jl}-g_{il}g_{jk}{\big )}-{\frac {1}{n-2}}{\big (}Q_{ik}g_{jl}-Q_{jk}g_{il}-Q_{il}g_{jk}+Q_{jl}g_{ik}{\big )}}
W
(
u
,
v
,
w
,
x
)
=
Rm
(
u
,
v
,
w
,
x
)
−
1
n
(
n
−
1
)
R
(
g
(
u
,
w
)
g
(
v
,
x
)
−
g
(
u
,
x
)
g
(
v
,
w
)
)
−
1
n
−
2
(
Q
(
u
,
w
)
g
(
v
,
x
)
−
Q
(
v
,
w
)
g
(
u
,
x
)
−
Q
(
u
,
x
)
g
(
v
,
w
)
+
Q
(
v
,
x
)
g
(
u
,
w
)
)
{\displaystyle W(u,v,w,x)=\operatorname {Rm} (u,v,w,x)-{\frac {1}{n(n-1)}}R{\big (}g(u,w)g(v,x)-g(u,x)g(v,w){\big )}-{\frac {1}{n-2}}{\big (}Q(u,w)g(v,x)-Q(v,w)g(u,x)-Q(u,x)g(v,w)+Q(v,x)g(u,w){\big )}}
G
i
k
=
R
i
k
−
1
2
R
g
i
k
{\displaystyle G_{ik}=R_{ik}-{\frac {1}{2}}Rg_{ik}}
G
(
u
,
v
)
=
Ric
(
u
,
v
)
−
1
2
R
g
(
u
,
v
)
{\displaystyle G(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{2}}Rg(u,v)}
Identities
edit
Basic symmetries
edit
R
i
j
k
l
=
−
R
j
i
k
l
{\displaystyle {R_{ijk}}^{l}=-{R_{jik}}^{l}}
R
i
j
k
l
=
−
R
j
i
k
l
=
−
R
i
j
l
k
=
R
k
l
i
j
{\displaystyle R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij}}
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
W
i
j
k
l
=
−
W
j
i
k
l
=
−
W
i
j
l
k
=
W
k
l
i
j
{\displaystyle W_{ijkl}=-W_{jikl}=-W_{ijlk}=W_{klij}}
g
i
l
W
i
j
k
l
=
0
{\displaystyle g^{il}W_{ijkl}=0}
The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
R
j
k
=
R
k
j
{\displaystyle R_{jk}=R_{kj}}
G
j
k
=
G
k
j
{\displaystyle G_{jk}=G_{kj}}
Q
j
k
=
Q
k
j
{\displaystyle Q_{jk}=Q_{kj}}
First Bianchi identity
edit
R
i
j
k
l
+
R
j
k
i
l
+
R
k
i
j
l
=
0
{\displaystyle R_{ijkl}+R_{jkil}+R_{kijl}=0}
W
i
j
k
l
+
W
j
k
i
l
+
W
k
i
j
l
=
0
{\displaystyle W_{ijkl}+W_{jkil}+W_{kijl}=0}
Second Bianchi identity
edit
∇
p
R
i
j
k
l
+
∇
i
R
j
p
k
l
+
∇
j
R
p
i
k
l
=
0
{\displaystyle \nabla _{p}R_{ijkl}+\nabla _{i}R_{jpkl}+\nabla _{j}R_{pikl}=0}
(
∇
u
Rm
)
(
v
,
w
,
x
,
y
)
+
(
∇
v
Rm
)
(
w
,
u
,
x
,
y
)
+
(
∇
w
Rm
)
(
u
,
v
,
x
,
y
)
=
0
{\displaystyle (\nabla _{u}\operatorname {Rm} )(v,w,x,y)+(\nabla _{v}\operatorname {Rm} )(w,u,x,y)+(\nabla _{w}\operatorname {Rm} )(u,v,x,y)=0}
Contracted second Bianchi identity
edit
∇
j
R
p
k
−
∇
p
R
j
k
=
−
∇
l
R
j
p
k
l
{\displaystyle \nabla _{j}R_{pk}-\nabla _{p}R_{jk}=-\nabla ^{l}R_{jpkl}}
(
∇
u
Ric
)
(
v
,
w
)
−
(
∇
v
Ric
)
(
u
,
w
)
=
−
tr
g
(
(
x
,
y
)
↦
(
∇
x
Rm
)
(
u
,
v
,
w
,
y
)
)
{\displaystyle (\nabla _{u}\operatorname {Ric} )(v,w)-(\nabla _{v}\operatorname {Ric} )(u,w)=-\operatorname {tr} _{g}{\big (}(x,y)\mapsto (\nabla _{x}\operatorname {Rm} )(u,v,w,y){\big )}}
Twice-contracted second Bianchi identity
edit
g
p
q
∇
p
R
q
k
=
1
2
∇
k
R
{\displaystyle g^{pq}\nabla _{p}R_{qk}={\frac {1}{2}}\nabla _{k}R}
div
g
Ric
=
1
2
d
R
{\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR}
Equivalently:
g
p
q
∇
p
G
q
k
=
0
{\displaystyle g^{pq}\nabla _{p}G_{qk}=0}
div
g
G
=
0
{\displaystyle \operatorname {div} _{g}G=0}
Ricci identity
edit
If
X
{\displaystyle X}
is a vector field then
∇
i
∇
j
X
k
−
∇
j
∇
i
X
k
=
−
R
i
j
p
k
X
p
,
{\displaystyle \nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},}
which is just the definition of the Riemann tensor. If
ω
{\displaystyle \omega }
is a one-form then
∇
i
∇
j
ω
k
−
∇
j
∇
i
ω
k
=
R
i
j
k
p
ω
p
.
{\displaystyle \nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.}
More generally, if
T
{\displaystyle T}
is a (0,k)-tensor field then
∇
i
∇
j
T
l
1
⋯
l
k
−
∇
j
∇
i
T
l
1
⋯
l
k
=
R
i
j
l
1
p
T
p
l
2
⋯
l
k
+
⋯
+
R
i
j
l
k
p
T
l
1
⋯
l
k
−
1
p
.
{\displaystyle \nabla _{i}\nabla _{j}T_{l_{1}\cdots l_{k}}-\nabla _{j}\nabla _{i}T_{l_{1}\cdots l_{k}}={R_{ijl_{1}}}^{p}T_{pl_{2}\cdots l_{k}}+\cdots +{R_{ijl_{k}}}^{p}T_{l_{1}\cdots l_{k-1}p}.}
edit
A classical result says that
W
=
0
{\displaystyle W=0}
if and only if
(
M
,
g
)
{\displaystyle (M,g)}
is locally conformally flat, i.e. if and only if
M
{\displaystyle M}
can be covered by smooth coordinate charts relative to which the metric tensor is of the form
g
i
j
=
e
φ
δ
i
j
{\displaystyle g_{ij}=e^{\varphi }\delta _{ij}}
for some function
φ
{\displaystyle \varphi }
on the chart.
Gradient, divergence, Laplace–Beltrami operator
edit
Kulkarni–Nomizu product
edit
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let
A
{\displaystyle A}
and
B
{\displaystyle B}
be symmetric covariant 2-tensors. In coordinates,
A
i
j
=
A
j
i
B
i
j
=
B
j
i
{\displaystyle A_{ij}=A_{ji}\qquad \qquad B_{ij}=B_{ji}}
Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted
A
∧
◯
B
{\displaystyle A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B}
. The defining formula is
(
A
∧
◯
B
)
i
j
k
l
=
A
i
k
B
j
l
+
A
j
l
B
i
k
−
A
i
l
B
j
k
−
A
j
k
B
i
l
{\displaystyle \left(A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B\right)_{ijkl}=A_{ik}B_{jl}+A_{jl}B_{ik}-A_{il}B_{jk}-A_{jk}B_{il}}
Clearly, the product satisfies
A
∧
◯
B
=
B
∧
◯
A
{\displaystyle A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B=B{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}A}
In an inertial frame
edit
Conformal change
edit
Let
g
{\displaystyle g}
be a Riemannian or pseudo-Riemanniann metric on a smooth manifold
M
{\displaystyle M}
, and
φ
{\displaystyle \varphi }
a smooth real-valued function on
M
{\displaystyle M}
. Then
g
~
=
e
2
φ
g
{\displaystyle {\tilde {g}}=e^{2\varphi }g}
is also a Riemannian metric on
M
{\displaystyle M}
. We say that
g
~
{\displaystyle {\tilde {g}}}
is (pointwise) conformal to
g
{\displaystyle 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
~
{\displaystyle {\tilde {g}}}
, while those unmarked with such will be associated with
g
{\displaystyle g}
.)
Levi-Civita connection
edit
Γ
~
i
j
k
=
Γ
i
j
k
+
∂
φ
∂
x
i
δ
j
k
+
∂
φ
∂
x
j
δ
i
k
−
∂
φ
∂
x
l
g
l
k
g
i
j
{\displaystyle {\widetilde {\Gamma }}_{ij}^{k}=\Gamma _{ij}^{k}+{\frac {\partial \varphi }{\partial x^{i}}}\delta _{j}^{k}+{\frac {\partial \varphi }{\partial x^{j}}}\delta _{i}^{k}-{\frac {\partial \varphi }{\partial x^{l}}}g^{lk}g_{ij}}
∇
~
X
Y
=
∇
X
Y
+
d
φ
(
X
)
Y
+
d
φ
(
Y
)
X
−
g
(
X
,
Y
)
∇
φ
{\displaystyle {\widetilde {\nabla }}_{X}Y=\nabla _{X}Y+d\varphi (X)Y+d\varphi (Y)X-g(X,Y)\nabla \varphi }
(4,0) Riemann curvature tensor
edit
R
~
i
j
k
l
=
e
2
φ
R
i
j
k
l
−
e
2
φ
(
g
i
k
T
j
l
+
g
j
l
T
i
k
−
g
i
l
T
j
k
−
g
j
k
T
i
l
)
{\displaystyle {\widetilde {R}}_{ijkl}=e^{2\varphi }R_{ijkl}-e^{2\varphi }{\big (}g_{ik}T_{jl}+g_{jl}T_{ik}-g_{il}T_{jk}-g_{jk}T_{il}{\big )}}
where
T
i
j
=
∇
i
∇
j
φ
−
∇
i
φ
∇
j
φ
+
1
2
|
d
φ
|
2
g
i
j
{\displaystyle T_{ij}=\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi +{\frac {1}{2}}|d\varphi |^{2}g_{ij}}
Using the Kulkarni–Nomizu product :
Rm
~
=
e
2
φ
Rm
−
e
2
φ
g
∧
◯
(
Hess
φ
−
d
φ
⊗
d
φ
+
1
2
|
d
φ
|
2
g
)
{\displaystyle {\widetilde {\operatorname {Rm} }}=e^{2\varphi }\operatorname {Rm} -e^{2\varphi }g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}\left(\operatorname {Hess} \varphi -d\varphi \otimes d\varphi +{\frac {1}{2}}|d\varphi |^{2}g\right)}
Ricci tensor
edit
R
~
i
j
=
R
i
j
−
(
n
−
2
)
(
∇
i
∇
j
φ
−
∇
i
φ
∇
j
φ
)
−
(
Δ
φ
+
(
n
−
2
)
|
d
φ
|
2
)
g
i
j
{\displaystyle {\widetilde {R}}_{ij}=R_{ij}-(n-2){\big (}\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi {\big )}-{\big (}\Delta \varphi +(n-2)|d\varphi |^{2}{\big )}g_{ij}}
Ric
~
=
Ric
−
(
n
−
2
)
(
Hess
φ
−
d
φ
⊗
d
φ
)
−
(
Δ
φ
+
(
n
−
2
)
|
d
φ
|
2
)
g
{\displaystyle {\widetilde {\operatorname {Ric} }}=\operatorname {Ric} -(n-2){\big (}\operatorname {Hess} \varphi -d\varphi \otimes d\varphi {\big )}-{\big (}\Delta \varphi +(n-2)|d\varphi |^{2}{\big )}g}
Scalar curvature
edit
R
~
=
e
−
2
φ
R
−
2
(
n
−
1
)
e
−
2
φ
Δ
φ
−
(
n
−
2
)
(
n
−
1
)
e
−
2
φ
|
d
φ
|
2
{\displaystyle {\widetilde {R}}=e^{-2\varphi }R-2(n-1)e^{-2\varphi }\Delta \varphi -(n-2)(n-1)e^{-2\varphi }|d\varphi |^{2}}
if
n
≠
2
{\displaystyle n\neq 2}
this can be written
R
~
=
e
−
2
φ
[
R
−
4
(
n
−
1
)
(
n
−
2
)
e
−
(
n
−
2
)
φ
/
2
Δ
(
e
(
n
−
2
)
φ
/
2
)
]
{\displaystyle {\tilde {R}}=e^{-2\varphi }\left[R-{\frac {4(n-1)}{(n-2)}}e^{-(n-2)\varphi /2}\Delta \left(e^{(n-2)\varphi /2}\right)\right]}
Traceless Ricci tensor
edit
R
~
i
j
−
1
n
R
~
g
~
i
j
=
R
i
j
−
1
n
R
g
i
j
−
(
n
−
2
)
(
∇
i
∇
j
φ
−
∇
i
φ
∇
j
φ
)
+
(
n
−
2
)
n
(
Δ
φ
−
|
d
φ
|
2
)
g
i
j
{\displaystyle {\widetilde {R}}_{ij}-{\frac {1}{n}}{\widetilde {R}}{\widetilde {g}}_{ij}=R_{ij}-{\frac {1}{n}}Rg_{ij}-(n-2){\big (}\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi {\big )}+{\frac {(n-2)}{n}}{\big (}\Delta \varphi -|d\varphi |^{2}{\big )}g_{ij}}
Ric
~
−
1
n
R
~
g
~
=
Ric
−
1
n
R
g
−
(
n
−
2
)
(
Hess
φ
−
d
φ
⊗
d
φ
)
+
(
n
−
2
)
n
(
Δ
φ
−
|
d
φ
|
2
)
g
{\displaystyle {\widetilde {\operatorname {Ric} }}-{\frac {1}{n}}{\widetilde {R}}{\widetilde {g}}=\operatorname {Ric} -{\frac {1}{n}}Rg-(n-2){\big (}\operatorname {Hess} \varphi -d\varphi \otimes d\varphi {\big )}+{\frac {(n-2)}{n}}{\big (}\Delta \varphi -|d\varphi |^{2}{\big )}g}
(3,1) Weyl curvature
edit
W
~
i
j
k
l
=
W
i
j
k
l
{\displaystyle {{\widetilde {W}}_{ijk}}^{l}={W_{ijk}}^{l}}
W
~
(
X
,
Y
,
Z
)
=
W
(
X
,
Y
,
Z
)
{\displaystyle {\widetilde {W}}(X,Y,Z)=W(X,Y,Z)}
for any vector fields
X
,
Y
,
Z
{\displaystyle X,Y,Z}
Volume form
edit
det
g
~
=
e
n
φ
det
g
{\displaystyle {\sqrt {\det {\widetilde {g}}}}=e^{n\varphi }{\sqrt {\det g}}}
d
μ
g
~
=
e
n
φ
d
μ
g
{\displaystyle d\mu _{\widetilde {g}}=e^{n\varphi }\,d\mu _{g}}
Hodge operator on p-forms
edit
∗
~
i
1
⋯
i
n
−
p
j
1
⋯
j
p
=
e
(
n
−
2
p
)
φ
∗
i
1
⋯
i
n
−
p
j
1
⋯
j
p
{\displaystyle {\widetilde {\ast }}_{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}=e^{(n-2p)\varphi }\ast _{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}}
∗
~
=
e
(
n
−
2
p
)
φ
∗
{\displaystyle {\widetilde {\ast }}=e^{(n-2p)\varphi }\ast }
Codifferential on p-forms
edit
d
∗
~
j
1
⋯
j
p
−
1
i
1
⋯
i
p
=
e
−
2
φ
(
d
∗
)
j
1
⋯
j
p
−
1
i
1
⋯
i
p
−
(
n
−
2
p
)
e
−
2
φ
∇
i
1
φ
δ
j
1
i
2
⋯
δ
j
p
−
1
i
p
{\displaystyle {\widetilde {d^{\ast }}}_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}=e^{-2\varphi }(d^{\ast })_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}-(n-2p)e^{-2\varphi }\nabla ^{i_{1}}\varphi \delta _{j_{1}}^{i_{2}}\cdots \delta _{j_{p-1}}^{i_{p}}}
d
∗
~
=
e
−
2
φ
d
∗
−
(
n
−
2
p
)
e
−
2
φ
ι
∇
φ
{\displaystyle {\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }}
Laplacian on functions
edit
Δ
~
Φ
=
e
−
2
φ
(
Δ
Φ
+
(
n
−
2
)
g
(
d
φ
,
d
Φ
)
)
{\displaystyle {\widetilde {\Delta }}\Phi =e^{-2\varphi }{\Big (}\Delta \Phi +(n-2)g(d\varphi ,d\Phi ){\Big )}}
Hodge Laplacian on p-forms
edit
Δ
d
~
ω
=
e
−
2
φ
(
Δ
d
ω
−
(
n
−
2
p
)
d
∘
ι
∇
φ
ω
−
(
n
−
2
p
−
2
)
ι
∇
φ
∘
d
ω
+
2
(
n
−
2
p
)
d
φ
∧
ι
∇
φ
ω
−
2
d
φ
∧
d
∗
ω
)
{\displaystyle {\widetilde {\Delta ^{d}}}\omega =e^{-2\varphi }{\Big (}\Delta ^{d}\omega -(n-2p)d\circ \iota _{\nabla \varphi }\omega -(n-2p-2)\iota _{\nabla \varphi }\circ d\omega +2(n-2p)d\varphi \wedge \iota _{\nabla \varphi }\omega -2d\varphi \wedge d^{\ast }\omega {\Big )}}
The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian.
Second fundamental form of an immersion
edit
Suppose
(
M
,
g
)
{\displaystyle (M,g)}
is Riemannian and
F
:
Σ
→
(
M
,
g
)
{\displaystyle F:\Sigma \to (M,g)}
is a twice-differentiable immersion. Recall that the second fundamental form is, for each
p
∈
M
,
{\displaystyle p\in M,}
a symmetric bilinear map
h
p
:
T
p
Σ
×
T
p
Σ
→
T
F
(
p
)
M
,
{\displaystyle h_{p}:T_{p}\Sigma \times T_{p}\Sigma \to T_{F(p)}M,}
which is valued in the
g
F
(
p
)
{\displaystyle g_{F(p)}}
-orthogonal linear subspace to
d
F
p
(
T
p
Σ
)
⊂
T
F
(
p
)
M
.
{\displaystyle dF_{p}(T_{p}\Sigma )\subset T_{F(p)}M.}
Then
h
~
(
u
,
v
)
=
h
(
u
,
v
)
−
(
∇
φ
)
⊥
g
(
u
,
v
)
{\displaystyle {\widetilde {h}}(u,v)=h(u,v)-(\nabla \varphi )^{\perp }g(u,v)}
for all
u
,
v
∈
T
p
M
{\displaystyle u,v\in T_{p}M}
Here
(
∇
φ
)
⊥
{\displaystyle (\nabla \varphi )^{\perp }}
denotes the
g
F
(
p
)
{\displaystyle g_{F(p)}}
-orthogonal projection of
∇
φ
∈
T
F
(
p
)
M
{\displaystyle \nabla \varphi \in T_{F(p)}M}
onto the
g
F
(
p
)
{\displaystyle g_{F(p)}}
-orthogonal linear subspace to
d
F
p
(
T
p
Σ
)
⊂
T
F
(
p
)
M
.
{\displaystyle dF_{p}(T_{p}\Sigma )\subset T_{F(p)}M.}
Mean curvature of an immersion
edit
In the same setting as above (and suppose
Σ
{\displaystyle \Sigma }
has dimension
n
{\displaystyle n}
), recall that the mean curvature vector is for each
p
∈
Σ
{\displaystyle p\in \Sigma }
an element
H
p
∈
T
F
(
p
)
M
{\displaystyle {\textbf {H}}_{p}\in T_{F(p)}M}
defined as the
g
{\displaystyle g}
-trace of the second fundamental form. Then
H
~
=
e
−
2
φ
(
H
−
n
(
∇
φ
)
⊥
)
.
{\displaystyle {\widetilde {\textbf {H}}}=e^{-2\varphi }({\textbf {H}}-n(\nabla \varphi )^{\perp }).}
Note that this transformation formula is for the mean curvature vector , and the formula for the mean curvature
H
{\displaystyle H}
in the hypersurface case is
H
~
=
e
−
φ
(
H
−
n
⟨
∇
φ
,
η
⟩
)
{\displaystyle {\widetilde {H}}=e^{-\varphi }(H-n\langle \nabla \varphi ,\eta \rangle )}
where
η
{\displaystyle \eta }
is a (local) normal vector field.
Variation formulas
edit
Principal symbol
edit
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.
The principal symbol of the map
g
↦
Rm
g
{\displaystyle g\mapsto \operatorname {Rm} ^{g}}
assigns to each
ξ
∈
T
p
∗
M
{\displaystyle \xi \in T_{p}^{\ast }M}
a map from the space of symmetric (0,2)-tensors on
T
p
M
{\displaystyle T_{p}M}
to the space of (0,4)-tensors on
T
p
M
,
{\displaystyle T_{p}M,}
given by
v
↦
ξ
j
ξ
k
v
i
l
+
ξ
i
ξ
l
v
j
k
−
ξ
i
ξ
k
v
j
l
−
ξ
j
ξ
l
v
i
k
2
=
−
1
2
(
ξ
⊗
ξ
)
∧
◯
v
.
{\displaystyle v\mapsto {\frac {\xi _{j}\xi _{k}v_{il}+\xi _{i}\xi _{l}v_{jk}-\xi _{i}\xi _{k}v_{jl}-\xi _{j}\xi _{l}v_{ik}}{2}}=-{\frac {1}{2}}(\xi \otimes \xi ){~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}v.}
The principal symbol of the map
g
↦
Ric
g
{\displaystyle g\mapsto \operatorname {Ric} ^{g}}
assigns to each
ξ
∈
T
p
∗
M
{\displaystyle \xi \in T_{p}^{\ast }M}
an endomorphism of the space of symmetric 2-tensors on
T
p
M
{\displaystyle T_{p}M}
given by
v
↦
v
(
ξ
♯
,
⋅
)
⊗
ξ
+
ξ
⊗
v
(
ξ
♯
,
⋅
)
−
(
tr
g
p
v
)
ξ
⊗
ξ
−
|
ξ
|
g
2
v
.
{\displaystyle v\mapsto v(\xi ^{\sharp },\cdot )\otimes \xi +\xi \otimes v(\xi ^{\sharp },\cdot )-(\operatorname {tr} _{g_{p}}v)\xi \otimes \xi -|\xi |_{g}^{2}v.}
The principal symbol of the map
g
↦
R
g
{\displaystyle g\mapsto R^{g}}
assigns to each
ξ
∈
T
p
∗
M
{\displaystyle \xi \in T_{p}^{\ast }M}
an element of the dual space to the vector space of symmetric 2-tensors on
T
p
M
{\displaystyle T_{p}M}
by
v
↦
|
ξ
|
g
2
tr
g
v
+
v
(
ξ
♯
,
ξ
♯
)
.
{\displaystyle v\mapsto |\xi |_{g}^{2}\operatorname {tr} _{g}v+v(\xi ^{\sharp },\xi ^{\sharp }).}
See also
edit
Notes
edit
References
edit
Arthur L. Besse. "Einstein manifolds." Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. Springer-Verlag, Berlin, 1987. xii+510 pp. ISBN 3-540-15279-2