This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
Christoffel symbols, covariant derivative 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. Additionally, we can take the trace of contravariant tensors with respect to g {\displaystyle g} as follows (and similarly for covariant ones): let T {\displaystyle T} be a ( 2 , 0 ) {\displaystyle (2,0)} tensor, then its trace with respect to g {\displaystyle g} is
tr g T = tr ( v ↦ g − 1 T ( v , − ) ) {\displaystyle \operatorname {tr} _{g}T=\operatorname {tr} (v\mapsto g^{-1}T(v,-))}
where g − 1 {\displaystyle g^{-1}} is the isomorphism between the cotangent space and the tangent space. 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 matrix of scalar coefficients 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 derivative of the metric vanishes,
( ∇ 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})}
A coordinates-free formula for the Levi-Civita connection (albeit being implicit) is the following:
g ( Z , ∇ Y X ) = 1 2 ( X ( g ( Y , Z ) ) + Y ( g ( X , Z ) ) − Z ( g ( X , Y ) ) + g ( Y , [ Z , X ] ) + g ( X , [ Z , Y ] ) − g ( Z , [ X , Y ] ) ) {\displaystyle g(Z,\nabla _{Y}X)={\frac {1}{2}}{\Big (}X(g(Y,Z))+Y(g(X,Z))-Z(g(X,Y))+g(Y,[Z,X])+g(X,[Z,Y])-g(Z,[X,Y]){\Big )}}
Curvature tensors
Definitions
(3,1) Riemann curvature tensor
R i j k l = ∂ Γ j k l ∂ x i − ∂ Γ i k l ∂ x j + ( Γ j k p Γ i p l − Γ i k p Γ j p l ) {\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}-{\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}+{\big (}\Gamma _{jk}^{p}\Gamma _{ip}^{l}-\Gamma _{ik}^{p}\Gamma _{jp}^{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}
(3,1) Riemann curvature tensor
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 )}}
Ricci curvature
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)}
Scalar curvature
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
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
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 )}}
(4,0) Weyl tensor
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 )}}
Einstein tensor
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
Basic symmetries
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
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
∇ 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
∇ j R p k − ∇ p R j k = ∇ l R j