List of formulas in Riemannian geometry
# 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

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

    $\Gamma _{kij}={\frac {1}{2}}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,$

and the Christoffel symbols of the second kind by

    ${\begin{aligned}\Gamma ^{m}_{ij}&=g^{mk}\Gamma _{kij}\\&={\frac {1}{2}}\,g^{mk}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\,g^{mk}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,.\end{aligned}}$

Here $g^{ij}$ is the inverse matrix to the metric tensor $g_{ij}$. In other words,

    $\delta ^{i}_{j}=g^{ik}g_{kj}$

and thus

    $n=\delta ^{i}_{i}=g^{i}_{i}=g^{ij}g_{ij}$

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

    $\Gamma _{kij}=\Gamma _{kji}$ or, respectively, $\Gamma ^{i}_{jk}=\Gamma ^{i}_{kj}$,

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

    $\Gamma ^{i}_{ki}={\frac {1}{2}}g^{im}{\frac {\partial g_{im}}{\partial x^{k}}}={\frac {1}{2g}}{\frac {\partial g}{\partial x^{k}}}={\frac {\partial \log {\sqrt {|g|}}}{\partial x^{k}}}$

and

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

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

The covariant derivative of a vector field with components $v^{i}$ is given by:

    $v^{i}_{;j}=(\nabla _{j}v)^{i}={\frac {\partial v^{i}}{\partial x^{j}}}+\Gamma ^{i}_{jk}v^{k}$

and similarly the covariant derivative of a $(0,1)$-tensor field with components $v_{i}$ is given by:

    $v_{i;j}=(\nabla _{j}v)_{i}={\frac {\partial v_{i}}{\partial x^{j}}}-\Gamma ^{k}_{ij}v_{k}$

For a $(2,0)$-tensor field with components $v^{ij}$ this becomes

    $v^{ij}_{;k}=\nabla _{k}v^{ij}={\frac {\partial v^{ij}}{\partial x^{k}}}+\Gamma ^{i}_{k\ell }v^{\ell j}+\Gamma ^{j}_{k\ell }v^{i\ell }$

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) $\phi$ is just its usual differential:

    $\nabla _{i}\phi =\phi _{;i}=\phi _{,i}={\frac {\partial \phi }{\partial x^{i}}}$

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

    $(\nabla _{k}g)_{ij}=0,\quad (\nabla _{k}g)^{ij}=0$

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

    $\nabla _{k}{\sqrt {|g|}}=0$

The geodesic $X(t)$ starting at the origin with initial speed $v^{i}$ has Taylor expansion in the chart:

    $X(t)^{i}=tv^{i}-{\frac {t^{2}}{2}}\Gamma ^{i}_{jk}v^{j}v^{k}+O(t^{3})$

## Curvature tensors

### Definitions

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

* ${R_{ijk}}^{l}={\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}-{\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}+{\big\Gamma _{ik}^{p}\Gamma _{jp}^{l}-\Gamma _{jk}^{p}\Gamma _{ip}^{l}{\big )}$
* $R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w$

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

* ${R_{jkl}^{i}}={\frac {\partial \Gamma _{lj}^{i}}{\partial x^{k}}}-{\frac {\partial \Gamma _{kj}^{i}}{\partial x^{l}}}+{\big\Gamma _{kp}^{i}\Gamma _{lj}^{p}-\Gamma _{lp}^{i}\Gamma _{kj}^{p}{\big )}$

**Ricci curvature**

* $R_{ik}={R_{ijk}}^{j}$
* $\operatorname {Ric} (v,w)=\operatorname {tr} (u\mapsto R(u,v)w)$

**Scalar curvature**

* $R=g^{ik}R_{ik}$
* $R=\operatorname {tr} _{g}\operatorname {Ric}$

**Traceless Ricci tensor**

* $Q_{ik}=R_{ik}-{\frac {1}{n}}Rg_{ik}$
* $Q(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{n}}Rg(u,v)$

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

* $R_{ijkl}={R_{ijk}}^{p}g_{pl}$
* $\operatorname {Rm} (u,v,w,x)=g{\bigR(u,v)w,x{\big )}$

**(4,0) Weyl tensor**

* $W_{ijkl}=R_{ijkl}-{\frac {1}{n(n-1)}}R{\bigg_{ik}g_{jl}-g_{il}g_{jk}{\big )}-{\frac {1}{n-2}}{\bigQ_{ik}g_{jl}-Q_{jk}g_{il}-Q_{il}g_{jk}+Q_{jl}g_{ik}{\big )}$
* $W(u,v,w,x)=\operatorname {Rm} (u,v,w,x)-{\frac {1}{n(n-1)}}R{\bigg(u,w)g(v,x)-g(u,x)g(v,w){\big )}-{\frac {1}{n-2}}{\bigQ(u,w)g(v,x)-Q(v,w)g(u,x)-Q(u,x)g(v,w)+Q(v,x)g(u,w){\big )}$

**Einstein tensor**

* $G_{ik}=R_{ik}-{\frac {1}{2}}Rg_{ik}$
* $G(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{2}}Rg(u,v)$

### Identities

**Basic symmetries**

* ${R_{ijk}}^{l}=-{R_{jik}}^{l}$
* $R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij}$

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

* $W_{ijkl}=-W_{jikl}=-W_{ijlk}=W_{klij}$
* $g^{il}W_{ijkl}=0$

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

* $R_{jk}=R_{kj}$
* $G_{jk}=G_{kj}$
* $Q_{jk}=Q_{kj}$

**First Bianchi identity**

* $R_{ijkl}+R_{jkil}+R_{kijl}=0$
* $W_{ijkl}+W_{jkil}+W_{kijl}=0$

**Second Bianchi identity**

* $\nabla _{p}R_{ijkl}+\nabla _{i}R_{jpkl}+\nabla _{j}R_{pikl}=0$
* $(\nabla _{u}\operatorname {Rm} )(v,w,x,y)+(\nabla _{v}\operatorname {Rm} )(w,u,x,y)+(\nabla _{w}\operatorname {Rm} )(u,v,x,y)=0$

**Contracted second Bianchi identity**

* $\nabla _{j}R_{pk}-\nabla _{p}R_{jk}=-\nabla ^{l}R_{jpkl}$
* $(\nabla _{u}\operatorname {Ric} )(v,w)-(\nabla _{v}\operatorname {Ric} )(u,w)=-\operatorname {tr} _{g}{\big(x,y)\mapsto (\nabla _{x}\operatorname {Rm} )(u,v,w,y){\big )}$

**Twice-contracted second Bianchi identity**

* $g^{pq}\nabla _{p}R_{qk}={\frac {1}{2}}\nabla _{k}R$
* $\operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR$

Equivalently:

* $g^{pq}\nabla _{p}G_{qk}=0$
* $\operatorname {div} _{g}G=0$

**Ricci identity**

If $X$ is a vector field then

    $\nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},$

which is just the definition of the Riemann tensor. If $\omega$ is a one-form then

    $\nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.$

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

    $\nabla _{i}\nabla _{j}T_{l_{1}\cdots l_{k}}-\nabla _{j}\nabla _{i}T_{l_{1}\cdots l_{k}}={R_{ijl_{1}}}^{p}T_{pl_{2}\cdots l_{k}}+\cdots +{R_{ijl_{k}}}^{p}T_{l_{1}\cdots l_{k-1}p}.$

**Remarks**

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 $g_{ij}=e^{\varphi }\delta _{ij}$ for some function $\varphi$ on the chart.

## Gradient, divergence, Laplace–Beltrami operator

The gradient of a function $\phi$ is obtained by raising the index of the differential $\partial _{i}\phi dx^{i}$, whose components are given by:

    $\nabla ^{i}\phi =\phi ^{;i}=g^{ik}\phi _{;k}=g^{ik}\phi _{,k}=g^{ik}\partial _{k}\phi =g^{ik}{\frac {\partial \phi }{\partial x^{k}}}$

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

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

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

    ${\begin{aligned}\Delta f&=\nabla _{i}\nabla ^{i}f={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial x^{j}}}\left(g^{jk}{\sqrt {|g|}}{\frac {\partial f}{\partial x^{k}}}\right)\\&=g^{jk}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}+{\frac {\partial g^{jk}}{\partial x^{j}}}{\frac {\partial f}{\partial x^{k}}}+{\frac {1}{2}}g^{jk}g^{il}{\frac {\partial g_{il}}{\partial x^{j}}}{\frac {\partial f}{\partial x^{k}}}=g^{jk}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}-g^{jk}\Gamma ^{l}_{jk}{\frac {\partial f}{\partial x^{l}}}\end{aligned}}$

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

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

The Hessian of a map $\phi :M\rightarrow N$ is given by

    $\left(\nabla \left(d\phi \right)\right)_{ij}^{\gamma }={\frac {\partial ^{2}\phi ^{\gamma }}{\partial x^{i}\partial x^{j}}}-^{M}\Gamma ^{k}_{ij}{\frac {\partial \phi ^{\gamma }}{\partial x^{k}}}+^{N}\Gamma ^{\gamma }_{\alpha \beta }{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}{\frac {\partial \phi ^{\beta }}{\partial x^{j}}}.$

## Kulkarni–Nomizu product

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,

    $A_{ij}=A_{ji}\qquad \qquad B_{ij}=B_{ji}$

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

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

Clearly, the product satisfies

    $A{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}B=B{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}A$

## In an inertial frame

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

    $R_{ik\ell m}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{\ell }}}+{\frac {\partial ^{2}g_{k\ell }}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{i\ell }}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{\ell }}}\right)$
    $R^{\ell }_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }_{ij}$

## Conformal change

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

    ${\tilde {g}}=e^{2\varphi }g$

is also a Riemannian metric on $M$. We say that ${\tilde {g}}$ 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 ${\tilde {g}}$, while those unmarked with such will be associated with $g$.)

### Levi-Civita connection

* ${\widetilde {\Gamma }}_{ij}^{k}=\Gamma _{ij}^{k}+{\frac {\partial \varphi }{\partial x^{i}}}\delta _{j}^{k}+{\frac {\partial \varphi }{\partial x^{j}}}\delta _{i}^{k}-{\frac {\partial \varphi }{\partial x^{l}}}g^{lk}g_{ij}$
* ${\widetilde {\nabla }}_{X}Y=\nabla _{X}Y+d\varphi (X)Y+d\varphi (Y)X-g(X,Y)\nabla \varphi$

### (4,0) Riemann curvature tensor

* ${\widetilde {R}}_{ijkl}=e^{2\varphi }R_{ijkl}-e^{2\varphi }{\bigg_{ik}T_{jl}+g_{jl}T_{ik}-g_{il}T_{jk}-g_{jk}T_{il}{\big )}$ where $T_{ij}=\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi +{\frac {1}{2}}|d\varphi |^{2}g_{ij}$

Using the Kulkarni–Nomizu product:

* ${\widetilde {\operatorname {Rm} }}=e^{2\varphi }\operatorname {Rm} -e^{2\varphi }g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}\left(\operatorname {Hess} \varphi -d\varphi \otimes d\varphi +{\frac {1}{2}}|d\varphi |^{2}g\right)$

### Ricci tensor

* ${\widetilde {R}}_{ij}=R_{ij}-(n-2){\big\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi {\big )}-{\big\Delta \varphi +(n-2)|d\varphi |^{2}{\big )}g_{ij}$
* ${\widetilde {\operatorname {Ric} }}=\operatorname {Ric} -(n-2){\big\operatorname {Hess} \varphi -d\varphi \otimes d\varphi {\big )}-{\big\Delta \varphi +(n-2)|d\varphi |^{2}{\big )}g$

### Scalar curvature

* ${\widetilde {R}}=e^{-2\varphi }R-2(n-1)e^{-2\varphi }\Delta \varphi -(n-2)(n-1)e^{-2\varphi }|d\varphi |^{2}$
* if $n\neq 2$ this can be written ${\tilde {R}}=e^{-2\varphi }\left[R-{\frac {4(n-1)}{(n-2)}}e^{-(n-2)\varphi /2}\Delta \left(e^{(n-2)\varphi /2}\right)\right]$

### Traceless Ricci tensor

* ${\widetilde {R}}_{ij}-{\frac {1}{n}}{\widetilde {R}}{\widetilde {g}}_{ij}=R_{ij}-{\frac {1}{n}}Rg_{ij}-(n-2){\big\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi {\big )}+{\frac {(n-2)}{n}}{\big\Delta \varphi -|d\varphi |^{2}{\big )}g_{ij}$
* ${\widetilde {\operatorname {Ric} }}-{\frac {1}{n}}{\widetilde {R}}{\widetilde {g}}=\operatorname {Ric} -{\frac {1}{n}}Rg-(n-2){\big\operatorname {Hess} \varphi -d\varphi \otimes d\varphi {\big )}+{\frac {(n-2)}{n}}{\big\Delta \varphi -|d\varphi |^{2}{\big )}g$

### (3,1) Weyl curvature

* ${{\widetilde {W}}_{ijk}}^{l}={W_{ijk}}^{l}$
* ${\widetilde {W}}(X,Y,Z)=W(X,Y,Z)$ for any vector fields $X,Y,Z$

### Volume form

* ${\sqrt {\det {\widetilde {g}}}}=e^{n\varphi }{\sqrt {\det g}}$
* $d\mu _{\widetilde {g}}=e^{n\varphi }\,d\mu _{g}$

### Hodge operator on p-forms

* ${\widetilde {\ast }}_{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}=e^{(n-2p)\varphi }\ast _{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}$
* ${\widetilde {\ast }}=e^{(n-2p)\varphi }\ast$

### Codifferential on p-forms

* ${\widetilde {d^{\ast }}}_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}=e^{-2\varphi }(d^{\ast })_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}-(n-2p)e^{-2\varphi }\nabla ^{i_{1}}\varphi \delta _{j_{1}}^{i_{2}}\cdots \delta _{j_{p-1}}^{i_{p}}$
* ${\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }$

### Laplacian on functions

* ${\widetilde {\Delta }}\Phi =e^{-2\varphi }{\Big\Delta \Phi +(n-2)g(d\varphi ,d\Phi ){\Big )}$

### Hodge Laplacian on p-forms

* ${\widetilde {\Delta ^{d}}}\omega =e^{-2\varphi }{\Big\Delta ^{d}\omega -(n-2p)d\circ \iota _{\nabla \varphi }\omega -(n-2p-2)\iota _{\nabla \varphi }\circ d\omega +2(n-2p)d\varphi \wedge \iota _{\nabla \varphi }\omega -2d\varphi \wedge d^{\ast }\omega {\Big )}$

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

Suppose $(M,g)$ is Riemannian and $F:\Sigma \to (M,g)$ is a twice-differentiable immersion. Recall that the second fundamental form is, for each $p\in M,$ a symmetric bilinear map $h_{p}:T_{p}\Sigma \times T_{p}\Sigma \to T_{F(p)}M,$ which is valued in the $g_{F(p)}$-orthogonal linear subspace to $dF_{p}(T_{p}\Sigma )\subset T_{F(p)}M.$ Then

* ${\widetilde {h}}(u,v)=h(u,v)-(\nabla \varphi )^{\perp }g(u,v)$ for all $u,v\in T_{p}M$

Here $(\nabla \varphi )^{\perp }$ denotes the $g_{F(p)}$-orthogonal projection of $\nabla \varphi \in T_{F(p)}M$ onto the $g_{F(p)}$-orthogonal linear subspace to $dF_{p}(T_{p}\Sigma )\subset T_{F(p)}M.$

### Mean curvature of an immersion

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

* ${\widetilde {\textbf {H}}}=e^{-2\varphi }({\textbf {H}}-n(\nabla \varphi )^{\perp }).$

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

* ${\widetilde {H}}=e^{-\varphi }(H-n\langle \nabla \varphi ,\eta \rangle )$

where $\eta$ is a (local) normal vector field.

## Variation formulas

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

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

## Principal symbol

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 $g\mapsto \operatorname {Rm} ^{g}$ assigns to each $\xi \in T_{p}^{\ast }M$ a map from the space of symmetric (0,2)-tensors on $T_{p}M$ to the space of (0,4)-tensors on $T_{p}M,$ given by

    $v\mapsto {\frac {\xi _{j}\xi _{k}v_{il}+\xi _{i}\xi _{l}v_{jk}-\xi _{i}\xi _{k}v_{jl}-\xi _{j}\xi _{l}v_{ik}}{2}}=-{\frac {1}{2}}(\xi \otimes \xi ){~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}v.$
* The principal symbol of the map $g\mapsto \operatorname {Ric} ^{g}$ assigns to each $\xi \in T_{p}^{\ast }M$ an endomorphism of the space of symmetric 2-tensors on $T_{p}M$ given by

    $v\mapsto v(\xi ^{\sharp },\cdot )\otimes \xi +\xi \otimes v(\xi ^{\sharp },\cdot )-(\operatorname {tr} _{g_{p}}v)\xi \otimes \xi -|\xi |_{g}^{2}v.$
* The principal symbol of the map $g\mapsto R^{g}$ assigns to each $\xi \in T_{p}^{\ast }M$ an element of the dual space to the vector space of symmetric 2-tensors on $T_{p}M$ by

    $v\mapsto |\xi |_{g}^{2}\operatorname {tr} _{g}v+v(\xi ^{\sharp },\xi ^{\sharp }).$
