List of formulas in Riemannian geometry

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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.

## Contents

* 1 Christoffel symbols, covariant derivative
* 2 Curvature tensors
* 2.1 Definitions
* 2.1.1 (3,1) Riemann curvature tensor
* 2.1.2 (3,1) Riemann curvature tensor
* 2.1.3 Ricci curvature
* 2.1.4 Scalar curvature
* 2.1.5 Traceless Ricci tensor
* 2.1.6 (4,0) Riemann curvature tensor
* 2.1.7 (4,0) Weyl tensor
* 2.1.8 Einstein tensor
* 2.2 Identities
* 2.2.1 Basic symmetries
* 2.2.2 First Bianchi identity
* 2.2.3 Second Bianchi identity
* 2.2.4 Contracted second Bianchi identity
* 2.2.5 Twice-contracted second Bianchi identity
* 2.2.6 Ricci identity
* 2.2.7 Remarks
* 3 Gradient, divergence, Laplace–Beltrami operator
* 4 Kulkarni–Nomizu product
* 5 In an inertial frame
* 6 Conformal change
* 6.1 Levi-Civita connection
* 6.2 (4,0) Riemann curvature tensor
* 6.3 Ricci tensor
* 6.4 Scalar curvature
* 6.5 Traceless Ricci tensor
* 6.6 (3,1) Weyl curvature
* 6.7 Volume form
* 6.8 Hodge operator on p-forms
* 6.9 Codifferential on p-forms
* 6.10 Laplacian on functions
* 6.11 Hodge Laplacian on p-forms
* 6.12 Second fundamental form of an immersion
* 6.13 Mean curvature of an immersion
* 7 Variation formulas
* 8 Principal symbol
* 9 See also
* 10 Notes
* 11 References

## Christoffel symbols, covariant derivative

In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

Γkij=12(∂∂xjgki+∂∂xigkj−∂∂xkgij)=12(gki,j+gkj,i−gij,k),

and the Christoffel symbols of the second kind by

Γmij=gmkΓkij=12gmk(∂∂xjgki+∂∂xigkj−∂∂xkgij)=12gmk(gki,j+gkj,i−gij,k).

Here gij is the inverse matrix to the metric tensor gij. In other words,

δij=gikgkj

and thus

n=δii=gii=gijgij

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

Γkij=Γkji or, respectively, Γijk=Γikj,

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

Γiki=12gim∂gim∂xk=12g∂g∂xk=∂log⁡|g|∂xk

and

gkℓΓikℓ=−1|g|∂(|g|gik)∂xk

where |g| is the absolute value of the determinant of the metric tensor gik. These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative of a vector field with components vi is given by:

vi;j=(∇jv)i=∂vi∂xj+Γijkvk

and similarly the covariant derivative of a (0,1)-tensor field with components vi is given by:

vi;j=(∇jv)i=∂vi∂xj−Γkijvk

For a (2,0)-tensor field with components vij this becomes

vij;k=∇kvij=∂vij∂xk+Γikℓvℓj+Γjkℓviℓ

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) ϕ is just its usual differential:

∇iϕ=ϕ;i=ϕ,i=∂ϕ∂xi

Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,

(∇kg)ij=0,(∇kg)ij=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 vi has Taylor expansion in the chart:

X(t)i=tvi−t22Γijkvjvk+O(t3)

## Curvature tensors

### Definitions

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

* Rijkl=∂Γikl∂xj−∂Γjkl∂xi+(ΓikpΓjpl−ΓjkpΓipl)
* R(u,v)w=∇u∇vw−∇v∇uw−∇[u,v]w

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

* Rjkli=∂Γlji∂xk−∂Γkji∂xl+(ΓkpiΓljp−ΓlpiΓkjp)

#### Ricci curvature

* Rik=Rijkj
* Ric⁡(v,w)=tr⁡(u↦R(u,v)w)

#### Scalar curvature

* R=gikRik
* R=trgRic

#### Traceless Ricci tensor

* Qik=Rik−1nRgik
* Q(u,v)=Ric⁡(u,v)−1nRg(u,v)

#### (4,0) Riemann curvature tensor

* Rijkl=Rijkpgpl
* Rm⁡(u,v,w,x)=g(R(u,v)w,x)

#### (4,0) Weyl tensor

* Wijkl=Rijkl−1n(n−1)R(gikgjl−gilgjk)−1n−2(Qikgjl−Qjkgil−Qilgjk+Qjlgik)
* W(u,v,w,x)=Rm⁡(u,v,w,x)−1n(n−1)R(g(u,w)g(v,x)−g(u,x)g(v,w))−1n−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))

#### Einstein tensor

* Gik=Rik−12Rgik
* G(u,v)=Ric⁡(u,v)−12Rg(u,v)

### Identities

#### Basic symmetries

* Rijkl=−Rjikl
* Rijkl=−Rjikl=−Rijlk=Rklij

The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:

* Wijkl=−Wjikl=−Wijlk=Wklij
* gilWijkl=0

The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:

* Rjk=Rkj
* Gjk=Gkj
* Qjk=Qkj

#### First Bianchi identity

* Rijkl+Rjkil+Rkijl=0
* Wijkl+Wjkil+Wkijl=0

#### Second Bianchi identity

* ∇pRijkl+∇iRjpkl+∇jRpikl=0
* (∇uRm⁡)(v,w,x,y)+(∇vRm⁡)(w,u,x,y)+(∇wRm⁡)(u,v,x,y)=0

#### Contracted second Bianchi identity

* ∇jRpk−∇pRjk=−∇lRjpkl
* (∇uRic⁡)(v,w)−(∇vRic⁡)(u,w)=−trg((x,y)↦(∇xRm⁡)(u,v,w,y))

#### Twice-contracted second Bianchi identity

* gpq∇pRqk=12∇kR
* divgRic⁡=12dR

Equivalently:

* gpq∇pGqk=0
* divgG=0

#### Ricci identity

If X is a vector field then

∇i∇jXk−∇j∇iXk=−RijpkXp,

which is just the definition of the Riemann tensor. If ω is a one-form then

∇i∇jωk−∇j∇iωk=Rijkpωp.

More generally, if T is a (0,k)-tensor field then

∇i∇jTl1⋯lk−∇j∇iTl1⋯lk=Rijl1pTpl2⋯lk+⋯+RijlkpTl1⋯lk−1p.

#### Remarks

A classical result says that W=0 if and only if (M,g) is locally conformally flat, i.e. if and only if M can be covered by smooth coordinate charts relative to which the metric tensor is of the form gij=eφδij for some function φ on the chart.

## Gradient, divergence, Laplace–Beltrami operator

The gradient of a function ϕ is obtained by raising the index of the differential ∂iϕdxi, whose components are given by:

∇iϕ=ϕ;i=gikϕ;k=gikϕ,k=gik∂kϕ=gik∂ϕ∂xk

The divergence of a vector field with components Vm is

∇mVm=∂Vm∂xm+Vk∂log⁡|g|∂xk=1|g|∂(Vm|g|)∂xm.

The Laplace–Beltrami operator acting on a function f is given by the divergence of the gradient:

Δf=∇i∇if=1|g|∂∂xj(gjk|g|∂f∂xk)=gjk∂2f∂xj∂xk+∂gjk∂xj∂f∂xk+12gjkgil∂gil∂xj∂f∂xk=gjk∂2f∂xj∂xk−gjkΓljk∂f∂xl

The divergence of an antisymmetric tensor field of type (2,0) simplifies to

∇kAik=1|g|∂(Aik|g|)∂xk.

The Hessian of a map ϕ:M->N is given by

(∇(dϕ))ijγ=∂2ϕγ∂xi∂xj−MΓkij∂ϕγ∂xk+NΓγαβ∂ϕα∂xi∂ϕβ∂xj.

## Kulkarni–Nomizu product

The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let A and B be symmetric covariant 2-tensors. In coordinates,

Aij=AjiBij=Bji

Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted A∧◯B. The defining formula is

(A∧◯B)ijkl=AikBjl+AjlBik−AilBjk−AjkBil

Clearly, the product satisfies

A∧◯B=B∧◯A

## In an inertial frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations gij=δij and Γijk=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.

Rikℓm=12(∂2gim∂xk∂xℓ+∂2gkℓ∂xi∂xm−∂2giℓ∂xk∂xm−∂2gkm∂xi∂xℓ)

Rℓijk=∂∂xjΓℓik−∂∂xkΓℓij

## Conformal change

Let g be a Riemannian or pseudo-Riemanniann metric on a smooth manifold M, and φ a smooth real-valued function on M. Then

g~=e2φg

is also a Riemannian metric on M. We say that g~ is (pointwise) 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.)

### Levi-Civita connection

* Γ~ijk=Γijk+∂φ∂xiδjk+∂φ∂xjδik−∂φ∂xlglkgij
* ∇~XY=∇XY+dφ(X)Y+dφ(Y)X−g(X,Y)∇φ

### (4,0) Riemann curvature tensor

* R~ijkl=e2φRijkl−e2φ(gikTjl+gjlTik−gilTjk−gjkTil) where Tij=∇i∇jφ−∇iφ∇jφ+12|dφ|2gij

Using the Kulkarni–Nomizu product:

* Rm~=e2φRm⁡−e2φg∧◯(Hess⁡φ−dφ⊗dφ+12|dφ|2g)

### Ricci tensor

* R~ij=Rij−(n−2)(∇i∇jφ−∇iφ∇jφ)−(Δφ+(n−2)|dφ|2)gij
* Ric~=Ric⁡−(n−2)(Hess⁡φ−dφ⊗dφ)−(Δφ+(n−2)|dφ|2)g

### Scalar curvature

* R~=e−2φR−2(n−1)e−2φΔφ−(n−2)(n−1)e−2φ|dφ|2
* if n≠2 this can be written R~=e−2φ[R−4(n−1)(n−2)e−(n−2)φ/2Δ(e(n−2)φ/2)]

### Traceless Ricci tensor

* R~ij−1nR~g~ij=Rij−1nRgij−(n−2)(∇i∇jφ−∇iφ∇jφ)+(n−2)n(Δφ−|dφ|2)gij
* Ric~−1nR~g~=Ric⁡−1nRg−(n−2)(Hess⁡φ−dφ⊗dφ)+(n−2)n(Δφ−|dφ|2)g

### (3,1) Weyl curvature

* W~ijkl=Wijkl
* W~(X,Y,Z)=W(X,Y,Z) for any vector fields X,Y,Z

### Volume form

* det⁡g~=enφdet⁡g
* dμg~=enφdμg

### Hodge operator on p-forms

* ∗~i1⋯in−pj1⋯jp=e(n−2p)φ∗i1⋯in−pj1⋯jp
* ∗~=e(n−2p)φ∗

### Codifferential on p-forms

* d∗~j1⋯jp−1i1⋯ip=e−2φ(d∗)j1⋯jp−1i1⋯ip−(n−2p)e−2φ∇i1φδj1i2⋯δjp−1ip
* d∗~=e−2φd∗−(n−2p)e−2φι∇φ

### Laplacian on functions

* Δ~Φ=e−2φ(ΔΦ+(n−2)g(dφ,dΦ))

### Hodge Laplacian on p-forms

* Δd~ω=e−2φ(Δdω−(n−2p)d∘ι∇φω−(n−2p−2)ι∇φ∘dω+2(n−2p)dφ∧ι∇φω−2dφ∧d∗ω)

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

Suppose (M,g) is Riemannian and F:Σ->(M,g) is a twice-differentiable immersion. Recall that the second fundamental form is, for each p∈M, a symmetric bilinear map hp:TpΣ×TpΣ->TF(p)M, which is valued in the gF(p)-orthogonal linear subspace to dFp(TpΣ)⊂TF(p)M. Then

* h~(u,v)=h(u,v)−(∇φ)⊥g(u,v) for all u,v∈TpM

Here (∇φ)⊥ denotes the gF(p)-orthogonal projection of ∇φ∈TF(p)M onto the gF(p)-orthogonal linear subspace to dFp(TpΣ)⊂TF(p)M.

### Mean curvature of an immersion

In the same setting as above (and suppose Σ has dimension n), recall that the mean curvature vector is for each p∈Σ an element Hp∈TF(p)M defined as the g-trace of the second fundamental form. Then

* H~=e−2φ(H−n(∇φ)⊥).

Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature H in the hypersurface case is

* H~=e−φ(H−n⟨∇φ,η⟩)

where η is a (local) normal vector field.

## Variation formulas

Let M be a smooth manifold and let gt be a one-parameter family of Riemanannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives vij=∂∂t((gt)ij) exist and are themselves as differentiable as necessary for the following expressions to make sense. v=∂g∂t is a one-parameter family of symmetric 2-tensor fields.

* ∂∂tΓijk=12gkp(∇ivjp+∇jvip−∇pvij).
* ∂∂tRijkl=12(∇j∇kvil+∇i∇lvjk−∇i∇kvjl−∇j∇lvik)+12Rijkpvpl−12Rijlpvpk
* ∂∂tRik=12(∇p∇kvip+∇i(div⁡v)k−∇i∇k(trgv)−Δvik)+12Ripvpk−12Ripkqvpq
* ∂∂tR=divgdivgv−Δ(trgv)−⟨v,Ric⁡⟩g
* ∂∂tdμg=12gpqvpqdμg
* ∂∂t∇i∇jΦ=∇i∇j∂Φ∂t−12gkp(∇ivjp+∇jvip−∇pvij)∂Φ∂xk
* ∂∂tΔΦ=−⟨v,Hess⁡Φ⟩g−g(div⁡v−12d(trgv),dΦ)

## Principal symbol

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↦Rmg assigns to each ξ∈Tp∗M a map from the space of symmetric (0,2)-tensors on TpM to the space of (0,4)-tensors on TpM, given by

v↦ξjξkvil+ξiξlvjk−ξiξkvjl−ξjξlvik2=−12(ξ⊗ξ)∧◯v.

* The principal symbol of the map g↦Ricg assigns to each ξ∈Tp∗M an endomorphism of the space of symmetric 2-tensors on TpM given by

v↦v(ξ♯,⋅)⊗ξ+ξ⊗v(ξ♯,⋅)−(trgpv)ξ⊗ξ−|ξ|g2v.

* The principal symbol of the map g↦Rg assigns to each ξ∈Tp∗M an element of the dual space to the vector space of symmetric 2-tensors on TpM by

v↦|ξ|g2trgv+v(ξ♯,ξ♯).

## See also

* Liouville equations

## Notes

## References

* 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

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