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 otherwhise.

## 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

[math]\displaystyle{ \Gamma_{kij}=\frac12 \left( \frac{\partial}{\partial x^j} g_{ki} +\frac{\partial}{\partial x^i} g_{kj} -\frac{\partial}{\partial x^k} g_{ij} \right) =\frac12 \left( g_{ki,j} + g_{kj,i} - g_{ij,k} \right) \,, }[/math]

and the Christoffel symbols of the second kind by

[math]\displaystyle{ \begin{align} \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{align} }[/math]

Here [math]\displaystyle{ g^{ij} }[/math] is the inverse matrix to the metric tensor [math]\displaystyle{ g_{ij} }[/math]. In other words,

[math]\displaystyle{ \delta^i{}_j = g^{ik}g_{kj} }[/math]

and thus

[math]\displaystyle{ n = \delta^i{}_i = g^i{}_i = g^{ij}g_{ij} }[/math]

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

[math]\displaystyle{ \Gamma_{kij} = \Gamma_{kji} }[/math] or, respectively, [math]\displaystyle{ \Gamma^i{}_{jk}=\Gamma^i{}_{kj} }[/math],

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

[math]\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} }[/math]

and

[math]\displaystyle{ g^{k\ell}\Gamma^i{}_{k\ell}=\frac{-1}{\sqrt{|g|}} \;\frac{\partial\left(\sqrt{|g|}\,g^{ik}\right)} {\partial x^k} }[/math]

where |g| is the absolute value of the determinant of the metric tensor [math]\displaystyle{ g_{ik} }[/math]. These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative of a vector field with components [math]\displaystyle{ v^i }[/math] is given by:

[math]\displaystyle{ v^i {}_{;j}=(\nabla_j v)^i=\frac{\partial v^i}{\partial x^j}+\Gamma^i{}_{jk}v^k }[/math]

and similarly the covariant derivative of a [math]\displaystyle{ (0,1) }[/math]-tensor field with components [math]\displaystyle{ v_i }[/math] is given by:

[math]\displaystyle{ v_{i;j}=(\nabla_j v)_i=\frac{\partial v_i}{\partial x^j}-\Gamma^k{}_{ij} v_k }[/math]

For a [math]\displaystyle{ (2,0) }[/math]-tensor field with components [math]\displaystyle{ v^{ij} }[/math] this becomes

[math]\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} }[/math]

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) [math]\displaystyle{ \phi }[/math] is just its usual differential:

[math]\displaystyle{ \nabla_i \phi=\phi_{;i}=\phi_{,i}=\frac{\partial \phi}{\partial x^i} }[/math]

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

[math]\displaystyle{ (\nabla_k g)_{ij} = 0, \quad (\nabla_k g)^{ij} = 0 }[/math]

as well as the covariant derivatives of the metric's determinant (and volume element)

[math]\displaystyle{ \nabla_k \sqrt{|g|}=0 }[/math]

The geodesic [math]\displaystyle{ X(t) }[/math] starting at the origin with initial speed [math]\displaystyle{ v^i }[/math] has Taylor expansion in the chart:

[math]\displaystyle{ X(t)^i=tv^i-\frac{t^2}{2}\Gamma^i{}_{jk}v^jv^k+O(t^3) }[/math]

## Curvature tensors

### Definitions

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

* [math]\displaystyle{ {R_{ijk}}^l=\frac{\partial\Gamma_{ik}^l}{\partial x^j}-\frac{\partial\Gamma_{jk}^l}{\partial x^i}+ \big(\Gamma_{ik}^p\Gamma_{jp}^l-\Gamma_{jk}^p\Gamma_{ip}^l\big) }[/math]
* [math]\displaystyle{ R(u,v)w=\nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_{[u,v]}w }[/math]

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

* [math]\displaystyle{ {R^i_{jkl}}=\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) }[/math]

#### Ricci curvature

* [math]\displaystyle{ R_{ik}={R_{ijk}}^j }[/math]
* [math]\displaystyle{ \operatorname{Ric}(v,w)=\operatorname{tr}(u\mapsto R(u,v)w) }[/math]

#### Scalar curvature

* [math]\displaystyle{ R= g^{ik}R_{ik} }[/math]
* [math]\displaystyle{ R=\operatorname{tr}_g\operatorname{Ric} }[/math]

#### Traceless Ricci tensor

* [math]\displaystyle{ Q_{ik}=R_{ik}-\frac{1}{n}Rg_{ik} }[/math]
* [math]\displaystyle{ Q(u,v)=\operatorname{Ric}(u,v)-\frac{1}{n}Rg(u,v) }[/math]

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

* [math]\displaystyle{ R_{ijkl}= {R_{ijk}}^pg_{pl} }[/math]
* [math]\displaystyle{ \operatorname{Rm}(u,v,w,x)=g\big(R(u,v)w,x\big) }[/math]

#### (4,0) Weyl tensor

* [math]\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) }[/math]
* [math]\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) }[/math]

#### Einstein tensor

* [math]\displaystyle{ G_{ik}=R_{ik}-\frac{1}{2}Rg_{ik} }[/math]
* [math]\displaystyle{ G(u,v)=\operatorname{Ric}(u,v)-\frac{1}{2}Rg(u,v) }[/math]

### Identities

#### Basic symmetries

* [math]\displaystyle{ {R_{ijk}}^l=-{R_{jik}}^l }[/math]
* [math]\displaystyle{ R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij} }[/math]

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

* [math]\displaystyle{ W_{ijkl}=-W_{jikl}=-W_{ijlk}=W_{klij} }[/math]
* [math]\displaystyle{ g^{il}W_{ijkl}=0 }[/math]

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

* [math]\displaystyle{ R_{jk}=R_{kj} }[/math]
* [math]\displaystyle{ G_{jk}=G_{kj} }[/math]
* [math]\displaystyle{ Q_{jk}=Q_{kj} }[/math]

#### First Bianchi identity

* [math]\displaystyle{ R_{ijkl}+R_{jkil}+R_{kijl}=0 }[/math]
* [math]\displaystyle{ W_{ijkl}+W_{jkil}+W_{kijl}=0 }[/math]

#### Second Bianchi identity

* [math]\displaystyle{ \nabla_pR_{ijkl}+\nabla_iR_{jpkl}+\nabla_jR_{pikl}=0 }[/math]
* [math]\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 }[/math]

#### Contracted second Bianchi identity

* [math]\displaystyle{ \nabla_jR_{pk}-\nabla_pR_{jk}=-\nabla^lR_{jpkl} }[/math]
* [math]\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) }[/math]

#### Twice-contracted second Bianchi identity

* [math]\displaystyle{ g^{pq}\nabla_pR_{qk}=\frac{1}{2}\nabla_k R }[/math]
* [math]\displaystyle{ \operatorname{div}_g\operatorname{Ric}=\frac{1}{2}dR }[/math]

Equivalently:

* [math]\displaystyle{ g^{pq}\nabla_pG_{qk}=0 }[/math]
* [math]\displaystyle{ \operatorname{div}_gG=0 }[/math]

#### Ricci identity

If [math]\displaystyle{ X }[/math] is a vector field then

[math]\displaystyle{ \nabla_i\nabla_jX^k-\nabla_j\nabla_iX^k=-{R_{ijp}}^kX^p, }[/math]

which is just the definition of the Riemann tensor. If [math]\displaystyle{ \omega }[/math] is a one-form then

[math]\displaystyle{ \nabla_i\nabla_j\omega_k-\nabla_j\nabla_i\omega_k={R_{ijk}}^p\omega_p. }[/math]

More generally, if [math]\displaystyle{ T }[/math] is a (0,k)-tensor field then

[math]\displaystyle{ \nabla_i\nabla_j T_{l_1\cdots l_k}-\nabla_j\nabla_iT_{l_1\cdots l_k}={R_{ijl_1}}^pT_{pl_2\cdots l_k}+\cdots+{R_{ijl_k}}^pT_{l_1\cdots l_{k-1}p}. }[/math]

#### Remarks

A classical result says that [math]\displaystyle{ W=0 }[/math] if and only if [math]\displaystyle{ (M,g) }[/math] is locally conformally flat, i.e. if and only if [math]\displaystyle{ M }[/math] can be covered by smooth coordinate charts relative to which the metric tensor is of the form [math]\displaystyle{ g_{ij}=e^\varphi \delta_{ij} }[/math] for some function [math]\displaystyle{ \varphi }[/math] on the chart.

## Gradient, divergence, Laplace–Beltrami operator

The gradient of a function [math]\displaystyle{ \phi }[/math] is obtained by raising the index of the differential [math]\displaystyle{ \partial_i\phi dx^i }[/math], whose components are given by:

[math]\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} }[/math]

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

[math]\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}. }[/math]

The Laplace–Beltrami operator acting on a function [math]\displaystyle{ f }[/math] is given by the divergence of the gradient:

[math]\displaystyle{ \begin{align} \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} + \frac12 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{align} }[/math]

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

[math]\displaystyle{ \nabla_k A^{ik}= \frac{1}{\sqrt{|g|}} \frac{\partial (A^{ik}\sqrt{|g|})}{\partial x^k}. }[/math]

The Hessian of a map [math]\displaystyle{ \phi: M \rightarrow N }[/math] is given by

[math]\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}. }[/math]

## Kulkarni–Nomizu product

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

[math]\displaystyle{ A_{ij} = A_{ji} \qquad \qquad B_{ij} = B_{ji} }[/math]

Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted [math]\displaystyle{ A {~\wedge\\!\\!\\!\\!\\!\\!\\!\\!\;\bigcirc~} B }[/math]. The defining formula is

[math]\displaystyle{ \left(A {~\wedge\\!\\!\\!\\!\\!\\!\\!\\!\;\bigcirc~} B\right)_{ijkl} = A_{ik}B_{jl} + A_{jl}B_{ik} - A_{il}B_{jk} - A_{jk}B_{il} }[/math]

Clearly, the product satisfies

[math]\displaystyle{ A {~\wedge\\!\\!\\!\\!\\!\\!\\!\\!\;\bigcirc~} B = B {~\wedge\\!\\!\\!\\!\\!\\!\\!\\!\;\bigcirc~} A }[/math]

## In an inertial frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations [math]\displaystyle{ g_{ij}=\delta_{ij} }[/math] and [math]\displaystyle{ \Gamma^i{}_{jk}=0 }[/math] (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.

[math]\displaystyle{ R_{ik\ell m}=\frac{1}{2}\left( \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell} \+ \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m} \- \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m} \- \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right) }[/math]

[math]\displaystyle{ R^\ell{}_{ijk}= \frac{\partial}{\partial x^j} \Gamma^\ell{}_{ik}-\frac{\partial}{\partial x^k}\Gamma^\ell{}_{ij} }[/math]

## Conformal change

Let [math]\displaystyle{ g }[/math] be a Riemannian or pseudo-Riemanniann metric on a smooth manifold [math]\displaystyle{ M }[/math], and [math]\displaystyle{ \varphi }[/math] a smooth real-valued function on [math]\displaystyle{ M }[/math]. Then

[math]\displaystyle{ \tilde g = e^{2\varphi}g }[/math]

is also a Riemannian metric on [math]\displaystyle{ M }[/math]. We say that [math]\displaystyle{ \tilde g }[/math] is (pointwise) conformal to [math]\displaystyle{ g }[/math]. 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 [math]\displaystyle{ \tilde g }[/math], while those unmarked with such will be associated with [math]\displaystyle{ g }[/math].)

### Levi-Civita connection

* [math]\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} }[/math]
* [math]\displaystyle{ \widetilde{\nabla}_XY=\nabla_XY+d\varphi(X)Y+d\varphi(Y)X-g(X,Y)\nabla \varphi }[/math]

### (4,0) Riemann curvature tensor

* [math]\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) }[/math] where [math]\displaystyle{ T_{ij}=\nabla_i\nabla_j\varphi-\nabla_i\varphi\nabla_j\varphi+\frac{1}{2}|d\varphi|^2g_{ij} }[/math]

Using the Kulkarni–Nomizu product:

* [math]\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|^2g \right) }[/math]

### Ricci tensor

* [math]\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} }[/math]
* [math]\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 }[/math]

### Scalar curvature

* [math]\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 }[/math]
* if [math]\displaystyle{ n\neq 2 }[/math] this can be written [math]\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] }[/math]

### Traceless Ricci tensor

* [math]\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} }[/math]
* [math]\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 }[/math]

### (3,1) Weyl curvature

* [math]\displaystyle{ {\widetilde{W}_{ijk}}^l={W_{ijk}}^l }[/math]
* [math]\displaystyle{ \widetilde{W}(X,Y,Z)=W(X,Y,Z) }[/math] for any vector fields [math]\displaystyle{ X,Y,Z }[/math]

### Volume form

* [math]\displaystyle{ \sqrt{\det \widetilde{g}}=e^{n\varphi}\sqrt{\det g} }[/math]
* [math]\displaystyle{ d\mu_{\widetilde{g}}=e^{n\varphi}\,d\mu_g }[/math]

### Hodge operator on p-forms

* [math]\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} }[/math]
* [math]\displaystyle{ \widetilde{\ast}=e^{(n-2p)\varphi}\ast }[/math]

### Codifferential on p-forms

* [math]\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} }[/math]
* [math]\displaystyle{ \widetilde{d^\ast}=e^{-2\varphi}d^\ast-(n-2p)e^{-2\varphi}\iota_{\nabla\varphi} }[/math]

### Laplacian on functions

* [math]\displaystyle{ \widetilde{\Delta}\Phi=e^{-2\varphi}\Big(\Delta\Phi + (n-2)g(d\varphi,d\Phi)\Big) }[/math]

### Hodge Laplacian on p-forms

* [math]\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) }[/math]

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 [math]\displaystyle{ (M,g) }[/math] is Riemannian and [math]\displaystyle{ F:\Sigma\to(M,g) }[/math] is a twice-differentiable immersion. Recall that the second fundamental form is, for each [math]\displaystyle{ p\in M, }[/math] a symmetric bilinear map [math]\displaystyle{ h_p:T_p\Sigma\times T_p\Sigma\to T_{F(p)}M, }[/math] which is valued in the [math]\displaystyle{ g_{F(p)} }[/math]-orthogonal linear subspace to [math]\displaystyle{ dF_p(T_p\Sigma)\subset T_{F(p)}M. }[/math] Then

* [math]\displaystyle{ \widetilde{h}(u,v)=h(u,v)-(\nabla\varphi)^\perp g(u,v) }[/math] for all [math]\displaystyle{ u,v\in T_pM }[/math]

Here [math]\displaystyle{ (\nabla\varphi)^\perp }[/math] denotes the [math]\displaystyle{ g_{F(p)} }[/math]-orthogonal projection of [math]\displaystyle{ \nabla\varphi\in T_{F(p)}M }[/math] onto the [math]\displaystyle{ g_{F(p)} }[/math]-orthogonal linear subspace to [math]\displaystyle{ dF_p(T_p\Sigma)\subset T_{F(p)}M. }[/math]

### Mean curvature of an immersion

In the same setting as above (and suppose [math]\displaystyle{ \Sigma }[/math] has dimension [math]\displaystyle{ n }[/math]), recall that the mean curvature vector is for each [math]\displaystyle{ p\in\Sigma }[/math] an element [math]\displaystyle{ \textbf H_p\in T_{F(p)}M }[/math] defined as the [math]\displaystyle{ g }[/math]-trace of the second fundamental form. Then

* [math]\displaystyle{ \widetilde{\textbf H}=e^{-2\varphi}(\textbf H-n(\nabla\varphi)^\perp). }[/math]

Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature [math]\displaystyle{ H }[/math] in the hypersurface case is

* [math]\displaystyle{ \widetilde{H}=e^{-\varphi}(H-n\langle\nabla\varphi,\eta\rangle) }[/math]

where [math]\displaystyle{ \eta }[/math] is a (local) normal vector field.

## Variation formulas

Let [math]\displaystyle{ M }[/math] be a smooth manifold and let [math]\displaystyle{ g_t }[/math] 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 [math]\displaystyle{ v_{ij}=\frac{\partial}{\partial t}\big((g_t)_{ij}\big) }[/math] exist and are themselves as differentiable as necessary for the following expressions to make sense. [math]\displaystyle{ v=\frac{\partial g}{\partial t} }[/math] is a one-parameter family of symmetric 2-tensor fields.

* [math]\displaystyle{ \frac{\partial}{\partial t}\Gamma_{ij}^k=\frac{1}{2}g^{kp}\Big(\nabla_i v_{jp}+\nabla_jv_{ip}-\nabla_pv_{ij}\Big). }[/math]
* [math]\displaystyle{ \frac{\partial}{\partial t}R_{ijkl}=\frac{1}{2}\Big(\nabla_j\nabla_k v_{il}+\nabla_i\nabla_lv_{jk}-\nabla_i\nabla_kv_{jl}-\nabla_j\nabla_lv_{ik}\Big)+\frac{1}{2}{R_{ijk}}^pv_{pl}-\frac{1}{2}{R_{ijl}}^pv_{pk} }[/math]
* [math]\displaystyle{ \frac{\partial}{\partial t}R_{ik}=\frac{1}{2}\Big(\nabla^p\nabla_kv_{ip}+\nabla_i(\operatorname{div}v)_k-\nabla_i\nabla_k(\operatorname{tr}_gv)-\Delta v_{ik}\Big)+\frac{1}{2}R_i^pv_{pk}-\frac{1}{2} R_i{}^p{}_k{}^qv_{pq} }[/math]
* [math]\displaystyle{ \frac{\partial}{\partial t}R=\operatorname{div}_g\operatorname{div}_gv-\Delta(\operatorname{tr}_gv)-\langle v,\operatorname{Ric}\rangle_g }[/math]
* [math]\displaystyle{ \frac{\partial}{\partial t}d\mu_g=\frac{1}{2} g^{pq}v_{pq}\,d\mu_g }[/math]
* [math]\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_jv_{ip}-\nabla_pv_{ij}\Big)\frac{\partial\Phi}{\partial x^k} }[/math]
* [math]\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}_gv),d\Phi\Big) }[/math]

## 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 [math]\displaystyle{ g\mapsto\operatorname{Rm}^g }[/math] assigns to each [math]\displaystyle{ \xi\in T_p^\ast M }[/math] a map from the space of symmetric (0,2)-tensors on [math]\displaystyle{ T_pM }[/math] to the space of (0,4)-tensors on [math]\displaystyle{ T_pM, }[/math] given by

[math]\displaystyle{ v\mapsto \frac{\xi_j\xi_kv_{il}+\xi_i\xi_lv_{jk}-\xi_i\xi_kv_{jl}-\xi_j\xi_lv_{ik}}{2} = -\frac12 (\xi \otimes \xi) {~\wedge\\!\\!\\!\\!\\!\\!\\!\\!\;\bigcirc~} v. }[/math]

* The principal symbol of the map [math]\displaystyle{ g\mapsto\operatorname{Ric}^g }[/math] assigns to each [math]\displaystyle{ \xi\in T_p^\ast M }[/math] an endomorphism of the space of symmetric 2-tensors on [math]\displaystyle{ T_pM }[/math] given by

[math]\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. }[/math]

* The principal symbol of the map [math]\displaystyle{ g\mapsto R^g }[/math] assigns to each [math]\displaystyle{ \xi\in T_p^\ast M }[/math] an element of the dual space to the vector space of symmetric 2-tensors on [math]\displaystyle{ T_pM }[/math] by

[math]\displaystyle{ v\mapsto |\xi|_g^2\operatorname{tr}_gv+v(\xi^\sharp,\xi^\sharp). }[/math]

## 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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