List of formulas in Riemannian geometry

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

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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 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})}

## Curvature tensors

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### Definitions

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#### (3,1) Riemann curvature tensor

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* 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

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* 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

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* 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

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* 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

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* 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

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* 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

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* 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

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* 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

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#### Basic symmetries

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* 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

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* 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

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* ∇ 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

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* ∇ 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

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* 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

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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}.}

#### Remarks

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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

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The gradient of a function  ϕ {\displaystyle \phi } is obtained by raising the index of the differential  ∂ i ϕ d x i {\displaystyle \partial _{i}\phi dx^{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 {\displaystyle \nabla ^{i}\phi =\phi ^{;i}=g^{ik}\phi _{;k}=g^{ik}\phi _{,k}=g^{ik}\partial _{k}\phi =g^{ik}{\frac {\partial \phi }{\partial x^{k}}}}

The divergence of a vector field with components  V m {\displaystyle V^{m}} is

∇ m V m = ∂ V m ∂ x m + V k ∂ log ⁡ | g | ∂ x k = 1 | g | ∂ ( V m | g | ) ∂ x m . {\displaystyle \nabla _{m}V^{m}={\frac {\partial V^{m}}{\partial x^{m}}}+V^{k}{\frac {\partial \log {\sqrt {|g|}}}{\partial x^{k}}}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (V^{m}{\sqrt {|g|}})}{\partial x^{m}}}.}

The Laplace–Beltrami operator acting on a function  f {\displaystyle 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 {\displaystyle {\begin{aligned}\Delta f&=\nabla _{i}\nabla ^{i}f={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial x^{j}}}\left(g^{jk}{\sqrt {|g|}}{\frac {\partial f}{\partial x^{k}}}\right)\\\&=g^{jk}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}+{\frac {\partial g^{jk}}{\partial x^{j}}}{\frac {\partial f}{\partial x^{k}}}+{\frac {1}{2}}g^{jk}g^{il}{\frac {\partial g_{il}}{\partial x^{j}}}{\frac {\partial f}{\partial x^{k}}}=g^{jk}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}-g^{jk}\Gamma ^{l}{}_{jk}{\frac {\partial f}{\partial x^{l}}}\end{aligned}}}

The divergence of an antisymmetric tensor field of type  ( 2 , 0 ) {\displaystyle (2,0)} simplifies to

∇ k A i k = 1 | g | ∂ ( A i k | g | ) ∂ x k . {\displaystyle \nabla _{k}A^{ik}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (A^{ik}{\sqrt {|g|}})}{\partial x^{k}}}.}

The Hessian of a map  ϕ : M -> N {\displaystyle \phi :M\rightarrow 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 . {\displaystyle \left(\nabla \left(d\phi \right)\right)_{ij}^{\gamma }={\frac {\partial ^{2}\phi ^{\gamma }}{\partial x^{i}\partial x^{j}}}-^{M}\Gamma ^{k}{}_{ij}{\frac {\partial \phi ^{\gamma }}{\partial x^{k}}}+^{N}\Gamma ^{\gamma }{}_{\alpha \beta }{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}{\frac {\partial \phi ^{\beta }}{\partial x^{j}}}.}

## Kulkarni–Nomizu product

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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

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An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations  g i j = δ i j {\displaystyle g_{ij}=\delta _{ij}} and  Γ i j k = 0 {\displaystyle \Gamma ^{i}{}_{jk}=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 ) {\displaystyle R_{ik\ell m}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{\ell }}}+{\frac {\partial ^{2}g_{k\ell }}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{i\ell }}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{\ell }}}\right)}
R ℓ i j k = ∂ ∂ x j Γ ℓ i k − ∂ ∂ x k Γ ℓ i j {\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}}

## Conformal change

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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

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* 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

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* 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

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* 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

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* 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

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* 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

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* 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

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* 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

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* ∗ ~ 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

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* 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

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* 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

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* 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

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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

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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

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Let  M {\displaystyle M} be a smooth manifold and let  g t {\displaystyle g_{t}} be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives  v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are themselves as differentiable as necessary for the following expressions to make sense.  v = ∂ g ∂ t {\displaystyle v={\frac {\partial g}{\partial t}}} is a one-parameter family of symmetric 2-tensor fields.

* ∂ ∂ t Γ i j k = 1 2 g k p ( ∇ i v j p + ∇ j v i p − ∇ p v i j ) . {\displaystyle {\frac {\partial }{\partial t}}\Gamma _{ij}^{k}={\frac {1}{2}}g^{kp}{\Big (}\nabla _{i}v_{jp}+\nabla _{j}v_{ip}-\nabla _{p}v_{ij}{\Big )}.}
* ∂ ∂ t R i j k l = 1 2 ( ∇ j ∇ k v i l + ∇ i ∇ l v j k − ∇ i ∇ k v j l − ∇ j ∇ l v i k ) + 1 2 R i j k p v p l − 1 2 R i j l p v p k {\displaystyle {\frac {\partial }{\partial t}}R_{ijkl}={\frac {1}{2}}{\Big (}\nabla _{j}\nabla _{k}v_{il}+\nabla _{i}\nabla _{l}v_{jk}-\nabla _{i}\nabla _{k}v_{jl}-\nabla _{j}\nabla _{l}v_{ik}{\Big )}+{\frac {1}{2}}{R_{ijk}}^{p}v_{pl}-{\frac {1}{2}}{R_{ijl}}^{p}v_{pk}}
* ∂ ∂ t R i k = 1 2 ( ∇ p ∇ k v i p + ∇ i ( div ⁡ v ) k − ∇ i ∇ k ( tr g ⁡ v ) − Δ v i k ) + 1 2 R i p v p k − 1 2 R i p k q v p q {\displaystyle {\frac {\partial }{\partial t}}R_{ik}={\frac {1}{2}}{\Big (}\nabla ^{p}\nabla _{k}v_{ip}+\nabla _{i}(\operatorname {div} v)_{k}-\nabla _{i}\nabla _{k}(\operatorname {tr} _{g}v)-\Delta v_{ik}{\Big )}+{\frac {1}{2}}R_{i}^{p}v_{pk}-{\frac {1}{2}}R_{i}{}^{p}{}_{k}{}^{q}v_{pq}}
* ∂ ∂ t R = div g ⁡ div g ⁡ v − Δ ( tr g ⁡ v ) − ⟨ v , Ric ⟩ g {\displaystyle {\frac {\partial }{\partial t}}R=\operatorname {div} _{g}\operatorname {div} _{g}v-\Delta (\operatorname {tr} _{g}v)-\langle v,\operatorname {Ric} \rangle _{g}}
* ∂ ∂ t d μ g = 1 2 g p q v p q d μ g {\displaystyle {\frac {\partial }{\partial t}}d\mu _{g}={\frac {1}{2}}g^{pq}v_{pq}\,d\mu _{g}}
* ∂ ∂ t ∇ i ∇ j Φ = ∇ i ∇ j ∂ Φ ∂ t − 1 2 g k p ( ∇ i v j p + ∇ j v i p − ∇ p v i j ) ∂ Φ ∂ x k {\displaystyle {\frac {\partial }{\partial t}}\nabla _{i}\nabla _{j}\Phi =\nabla _{i}\nabla _{j}{\frac {\partial \Phi }{\partial t}}-{\frac {1}{2}}g^{kp}{\Big (}\nabla _{i}v_{jp}+\nabla _{j}v_{ip}-\nabla _{p}v_{ij}{\Big )}{\frac {\partial \Phi }{\partial x^{k}}}}
* ∂ ∂ t Δ Φ = − ⟨ v , Hess ⁡ Φ ⟩ g − g ( div ⁡ v − 1 2 d ( tr g ⁡ v ) , d Φ ) {\displaystyle {\frac {\partial }{\partial t}}\Delta \Phi =-\langle v,\operatorname {Hess} \Phi \rangle _{g}-g{\Big (}\operatorname {div} v-{\frac {1}{2}}d(\operatorname {tr} _{g}v),d\Phi {\Big )}}

## Principal symbol

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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

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* Liouville equations

## Notes

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## References

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* 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