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 matrix of scalar coefficients 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 derivative of the metric vanishes,

(∇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)

A coordinates-free formula for the Levi-Civita connection (albeit being implicit) is the following:[1] g(Z,∇YX)=12(X(g(Y,Z))+Y(g(X,Z))−Z(g(X,Y))+g(Y,[Z,X])+g(X,[Z,Y])−g(Z,[X,Y]))

## Curvature tensors

### Definitions

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

* Rijkl=∂Γjkl∂xi−∂Γikl∂xj+(ΓjkpΓipl−ΓikpΓjpl)
* 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 Riemannian 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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1. ↑ M. P. Do Carmo, Riemannian Geometry, p. 55

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