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.

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

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

and the Christoffel symbols of the second kind by

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

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

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

and thus

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

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

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

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

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

and

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

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

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

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

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

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

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

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

and likewise for tensors with more indices.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• ${\displaystyle {R_{ijk}}^{l}=-{R_{jik}}^{l}}$
• ${\displaystyle R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij}}$

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

• ${\displaystyle W_{ijkl}=-W_{jikl}=-W_{ijlk}=W_{klij}}$
• ${\displaystyle g^{il}W_{ijkl}=0}$

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

• ${\displaystyle R_{jk}=R_{kj}}$
• ${\displaystyle G_{jk}=G_{kj}}$
• ${\displaystyle Q_{jk}=Q_{kj}}$

• ${\displaystyle R_{ijkl}+R_{jkil}+R_{kijl}=0}$
• ${\displaystyle W_{ijkl}+W_{jkil}+W_{kijl}=0}$

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

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

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

Equivalently:

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

If ${\displaystyle X}$ is a vector field then

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Clearly, the product satisfies

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

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

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

${\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}}$

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

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

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

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

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

Using the Kulkarni–Nomizu product:

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle v\mapsto {\frac {\xi _{j}\xi _{k}v_{il}+\xi _{i}\xi _{l}v_{jk}-\xi _{i}\xi _{k}v_{jl}-\xi _{j}\xi _{l}v_{ik}}{2}}=-{\frac {1}{2}}(\xi \otimes \xi ){~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}v.}$

• The principal symbol of the map ${\displaystyle g\mapsto \operatorname {Ric} ^{g}}$ assigns to each ${\displaystyle \xi \in T_{p}^{\ast }M}$ an endomorphism of the space of symmetric 2-tensors on ${\displaystyle T_{p}M}$ given by

${\displaystyle v\mapsto v(\xi ^{\sharp },\cdot )\otimes \xi +\xi \otimes v(\xi ^{\sharp },\cdot )-(\operatorname {tr} _{g_{p}}v)\xi \otimes \xi -|\xi |_{g}^{2}v.}$

• The principal symbol of the map ${\displaystyle g\mapsto R^{g}}$ assigns to each ${\displaystyle \xi \in T_{p}^{\ast }M}$ an element of the dual space to the vector space of symmetric 2-tensors on ${\displaystyle T_{p}M}$ by

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

• Liouville equations

• 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