Useful Formulae in Riemannian Geometry - Perimeter Institute

1. Curvature tensors
Conventions, Definitions, Identities, and Other Useful Formulae
Consider a d + 1 dimensional manifold M with metric g µν . The covariant derivative on M that is metriccompatible
with g µν is ∇ µ .
Christoffel
Riemann Tensor
Ricci Tensor
Schouten Tensor
Weyl Tensor
Γ λ µν = 1 2 gλρ (∂ µ g ρν + ∂ ν g µρ − ∂ ρ g µν ) (1)
R λ µσν = ∂ σ Γ λ µν − ∂ ν Γ λ µσ + Γ κ µνΓ λ κσ − Γ κ µσΓ λ κν (2)
S µν = 1
d − 1
R µν = δ σ λ Rλ µσν (3)
(
R µν − 1 )
2 d g µνR
∇ ν S µν = ∇ µ S ν ν (5)
C λ µσν = R λ µσν + g λ ν S µσ − g λ σ S µν + g µσ S λ ν − g µν S λ σ (6)
(4)
Commutators of Covariant Derivatives
[∇ µ , ∇ ν ] A λ = R λσµν A σ (7)
[∇ µ , ∇ ν ] A λ = R λ σµνA σ (8)
Bianchi Identity
∇ κ R λµσν − ∇ λ R κµσν + ∇ µ R κλσν = 0 (9)
∇ ν R λµσν = ∇ µ R λσ − ∇ λ R µσ (10)
∇ ν R µν = 1 2 ∇ µR (11)
Bianchi Identity for Weyl
∇ ν C λµσν = (d − 2)
∇ λ ∇ σ C λµσν = d − 2
d − 1
(∇ µ S λσ − ∇ λ S µσ
)
(12)
[
∇ 2 R µν − 1
2d g µν∇ 2 R − d − 1 ( ) d + 1
2d
∇ µ∇ ν R − Rµ λ R νλ (13)
d − 1
+ C λµσν R λσ (d + 1)
+
d(d − 1) R R µν + 1
d − 1 g µν
(R λσ R λσ − 1 )]
d R2
1

2. Euler Densities
Let M be a manifold with dimension d + 1 = 2n an even number. Normalized so that χ(S 2n ) = 2.
Euler Number
Euler Density
Curvature Two-Form
Examples
E 2n =
∫
χ(M) = d 2n x √ g E 2n (14)
∫M
= e 2n
(15)
M
1
(8π) n Γ(n + 1) ɛ µ 1 ...µ 2n
ɛ ν1 ...ν 2n
R µ 1 µ 2 ν 1 ν 2 . . . R
µ 2n−1 µ 2n ν 2n−1 ν 2n (16)
e 2n
=
1
(4π) n Γ(n + 1) ɛ a 1 ...a 2n
R a 1 a 2 ∧ . . . ∧ R
a 2n−1 a 2n (17)
R a b = 1 2 Ra b c d ec ∧ e d (18)
E 2 = 1
8π ɛ µνɛ λρ R µνλρ (19)
= 1
4π R
E 4 = 1
128π 2 ɛ µνλρ ɛ αβγδ R µναβ R λργδ (20)
= 1
32π 2 (
R µνλρ R µνλρ − 4 R µν R µν + R 2)
= 1
32π 2 Cµνλρ C µνλρ − 1 ( d − 2
8π 2 d − 1
) (
R µν R µν − d + 1
4 d R2 )
3. Hypersurfaces
Let Σ ⊂ M be a d dimensional hypersurface whose embedding is described locally by an outward-pointing, unit
normal vector n µ . Rather than keeping track of the signs associated with n µ being either spacelike or timelike,
we will just assume that n µ is spacelike. Indices are lowered and raised using g µν and g µν , and symmetrization
of indices is implied when appropriate.
First Fundamental Form / Induced Metric on Σ
h µν = g µν − n µ n ν (21)
Projection onto Σ
⊥ T µ ... ν ... = h µ λ . . . hσ ν . . . T λ ... σ ... (22)
Second Fundamental Form / Extrinsic Curvature of Σ
K µν = ⊥(∇ µ n ν ) = h λ
µ h σ
ν ∇ λ n σ = 1 2 £ nh µν (23)
Trace of Extrinsic Curvature
K = ∇ µ n µ (24)
2

‘Acceleration’ Vector
a µ = n ν ∇ ν n µ (25)
Surface-Forming Normal Vectors
n µ =
1
√
g νλ ∂ ν α∂ λ α ∂ µα ⇒ ⊥∇ [ µ n ν ] = 0 (26)
Covariant Derivative on Σ compatible with h µν
D µ T α ... β ... = ⊥ ∇ µT α ... β ... ∀ T = ⊥ T (27)
Intrinsic Curvature of (Σ, h)
[D µ , D ν ] A λ = R λ σµνA σ ∀ A λ = ⊥ A λ (28)
Gauss-Codazzi
Projections of the Ricci tensor
Decomposition of the Ricci scalar
⊥R λµσν = R λµσν − K λσ K µν + K µσ K νλ (29)
(
⊥ R λµσν n λ) = D ν K µσ − D σ K µν (30)
(
⊥ R λµσν n λ n σ) = − L n K µν + Kµ λ K λν + D µ a ν − a µ a ν (31)
⊥ (R µν ) = R µν + D µ a ν − a µ a ν − L n K µν − K K µν + 2K λ
µ K ν λ (32)
⊥ (R µν n µ ) = D µ K µν − D ν K (33)
R µν n µ n ν = − L n K − K µν K µν + D µ a µ − a µ a µ (34)
R = R − K 2 − K µν K µν − 2 L n K + 2 D µ a µ − 2 a µ a µ (35)
Lie Derivatives along n µ £ n K µν = n λ ∇ λ K µν + K λν ∇ µ n λ + K µλ ∇ ν n λ (36)
4. Sign Conventions for the Action
⊥ (£ n F µ ... ν ...) = £ n F µ ... ν ... ∀ ⊥F = F (37)
These conventions follow Weinberg, keeping in mind that he defines the Riemann tensor with a minus sign
relative to our definition. They are appropriate when using signature (−, +, . . . , +). The d + 1-dimensional
Newton’s constant is 2κ 2 = 16πG d+1 . The sign on the boundary term follows from our definition of the extrinsic
curvature.
Gravitational Action
I G = 1
2 κ
∫M
2 d d+1 x √ ( )
g R − 2 Λ + 1 ∫
κ 2
= 1
2 κ
∫M
2 d d+1 x √ g
∂M
(
R + K 2 − K µν K µν − 2 Λ
d d x √ γ K (38)
)
(39)
3

Gauge Field Coupled to Particles
I M = − 1 4
M
− ∑ n
+ ∑ n
∫
M
d d+1 x √ g F µν F µν (40)
m n
∫
e n
∫
(
dp
dp dxµ n(p)
dp
−g µν (x n (p)) dxµ n(p)
dp
dx ν ) 1/2
n(p)
(41)
dp
A µ (x n (p)) (42)
Gravity Minimally Coupled to a Gauge Field
∫
I = d d+1 x √ [ 1
g
2 κ 2 (R − 2 Λ) − 1 ]
4 F µν F µν + 1 ∫
κ 2 d d x √ γ K (43)
5. Hamiltonian Formulation
The canonical variables are the metric h µν on Σ and its conjugate momenta π µν . The momenta are defined
with respect to evolution in the spacelike direction n µ , so this is not the usual notion of the Hamiltonian as the
generator of time translations.
∂M
Bulk Lagrangian Density
L M = 1 (
)
2 κ 2 K 2 − K µν K µν + R − 2 Λ
(44)
Momentum Conjugate to h µν
π µν = ∂L M
∂ (£ n h µν ) = 1
2 κ 2 (
h µν K − K µν) (45)
Momentum Constraint
H µ = 1 κ 2 ⊥ (nν G µν ) = 2 D ν π µν = 0 (46)
Hamiltonian Constraint
6. Conformal Transformations
H = − 1 κ 2 nµ n ν G µν = 2 κ 2 (
π µν π µν − 1
d − 1 π2 )
+ 1
2 κ 2 (
R − 2 Λ
)
= 0 (47)
The dimension of spacetime is d + 1. Indices are raised and lowered using the metric g µν and its inverse g µν .
Metric
Christoffel
ĝ µν = e 2 σ g µν (48)
̂Γ λ µν = Γ λ µν + Θ λ µν (49)
Θ λ µν = δ λ µ ∇ ν σ + δ λ ν ∇ µ σ − g µν ∇ λ σ (50)
4

Riemann Tensor
̂R λ µρν = R λ µρν + δ λ ν ∇ µ ∇ ρ σ − δ λ ρ ∇ µ ∇ ν σ + g µρ ∇ ν ∇ λ σ − g µν ∇ ρ ∇ λ σ (51)
+ δ λ ρ ∇ µ σ ∇ ν σ − δ λ ν ∇ µ σ ∇ ρ σ + g µν ∇ ρ σ ∇ λ σ − g µρ ∇ ν σ ∇ λ σ (52)
(
)
+ g µρ δ λ ν − g µν δ λ ρ ∇ α σ ∇ α σ (53)
Ricci Tensor
̂R µν = R µν − g µν ∇ 2 σ − (d − 1) ∇ µ ∇ ν σ + (d − 1) ∇ µ σ ∇ ν σ (54)
− (d − 1) g µν ∇ λ σ ∇ λ σ (55)
Ricci Scalar
̂R ( )
= e −2 σ R − 2 d ∇ 2 σ − d (d − 1) ∇ µ σ ∇ µ σ
(56)
Schouten Tensor
Ŝ µν = S µν − ∇ µ ∇ ν σ + ∇ µ σ ∇ ν σ − 1 2 g µν∇ λ σ ∇ λ σ (57)
Weyl Tensor
Ĉ λ µρν = C λ µρν (58)
Normal Vector
̂n µ = e −σ n µ ̂n µ = e σ n µ (59)
Extrinsic Curvature
̂K ( )
µν = e σ K µν + h µν n λ ∇ λ σ
̂K ( )
= e −σ K + d n λ ∇ λ σ
(60)
(61)
7. Small Variations of the Metric
Consider a small perturbation to the metric of the form g µν → g µν + δg µν . All indices are raised and lowered
using the unperturbed metric g µν and its inverse. All quantities are expressed in terms of the perturbation to
the metric with lower indices, and never in terms of the perturbation to the inverse metric. As in the previous
sections, ∇ µ is the covariant derivative on M compatible with g µν and D µ is the covariant derivative on a
hypersurface Σ compatible with h µν .
Inverse Metric
g µν → g µν − g µα g νβ δg αβ + g µα g νβ g λρ δg αλ δg βρ + . . . (62)
Square Root of Determinant
(
√ √ g → g 1 + 1 )
2 gµν δg µν + . . .
(63)
5

Variational Operator
δ(g µν ) = δg µν δ 2 (g µν ) = δ(δg µν ) = 0 (64)
)
δ(g µν ) = −g µα g νβ δg αβ δ 2 (g µν ) = δ
(−g µλ g νρ δg λρ = 2 g µα g νβ g λρ δg αλ δg βρ (65)
F(g + δg) = F(g) + δF(g) + 1 2 δ 2 F(g) + . . . + 1 n! δ n F(g) + . . . (66)
Christoffel (All Orders)
δ Γ λ µν = 1 2 gλρ ( ∇ µ δg ρν + ∇ ν δg µρ − ∇ ρ δg µν
)
δ 2 Γ λ µν = −g λα g ρβ δg αβ
(
∇ µ δg ρν + ∇ ν δg µρ − ∇ ρ δg µν
)
δ n Γ λ µν = n 2 δ n−1 ( g λρ) ( ∇ µ δg ρν + ∇ ν δg µρ − ∇ ρ δg µν
)
(67)
(68)
(69)
Riemann Tensor
δ R λ µσν = ∇ σ δ Γ λ µν − ∇ ν δ Γ λ µσ (70)
Ricci Tensor
δ R µν = ∇ λ δ Γ λ µν − ∇ ν δ Γ λ µλ (71)
= 1 )
(∇ λ ∇ µ δg λν + ∇ λ ∇ ν δg µλ − g λρ ∇ µ ∇ ν δg λρ − ∇ 2 δg µν (72)
2
Ricci Scalar
δ R = −R µν δg µν + ∇ µ ( ∇ ν δg µν − g λρ ∇ µ δg λρ
)
(73)
Surface Forming Normal Vector
δ n µ = 1 2 n µ n ν n λ δg νλ = 1 2 δg µν n ν + c µ (74)
c µ = 1 2 n µ n ν n λ δg νλ − 1 2 δg µν n ν = − 1 2 h λ
µ δg λν n ν (75)
Extrinsic Curvatures
δ K µν = 1 2 nα n β δg αβ K µν − n µ c λ h λν − n ν c λ h µλ + 1 ( )
2 δg λρ n ρ n µ K λ ν + n ν Kµ
λ
− 1 2 h µ λ hν ρ n α ( )
∇ λ δg αρ + ∇ ρ δg λα − ∇ α δg λρ
(76)
δ K = − 1 2 Kµν δg µν − 1 2 nµ ( ∇ ν δg µν − g νλ ∇ µ δg νλ
)
+ D µ c µ (77)
6