List of formulas in Riemannian geometry

Template:Short description 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

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

and the Christoffel symbols of the second kind by

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

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. Additionally, we can take the trace of contravariant tensors with respect to $g$ as follows (and similarly for covariant ones): let $T$ be a $(2,0)$ tensor, then its trace with respect to $g$ is

${\mathrm{tr}}_{g}T=\mathrm{tr}(v\mapsto g^{-1}T(v,-))$

where $g^{-1}$ is the isomorphism between the cotangent space and the tangent space.

Christoffel symbols satisfy the symmetry relations

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

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

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

and

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

where Template:Abs is the absolute value of the determinant of the matrix of scalar coefficients 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:

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

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

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

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

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

and likewise for tensors with more indices.

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

${\mathrm{\nabla }}_i\varphi ={\varphi }_{;i}={\varphi }_{,i}=\frac{\partial \varphi }{\partial x^i}$

Because the Levi-Civita connection is metric-compatible, the covariant derivative of the metric vanishes,

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

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

${\mathrm{\nabla }}_k\sqrt{|g|}=0$

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

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

A coordinates-free formula for the Levi-Civita connection (albeit being implicit) is the following: $g(Z,{\mathrm{\nabla }}_{Y}X)=\frac{1}{2}(X(g(Y,Z))+Y(g(X,Z))-Z(g(X,Y))+g(Y,[Z,X])+g(X,[Z,Y])-g(Z,[X,Y]))$

Curvature tensors

Definitions

(3,1) Riemann curvature tensor

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

(3,1) Riemann curvature tensor

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

Ricci curvature

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

Scalar curvature

• $R=g^{ik}{R}_{ik}$
• $R={\mathrm{tr}}_g\mathrm{Ric}$

Traceless Ricci tensor

• $Q_{ik}=R_{ik}-\frac{1}{n}R{g}_{ik}$
• $Q(u,v)=\mathrm{Ric}(u,v)-\frac{1}{n}Rg(u,v)$

(4,0) Riemann curvature tensor

• $R_{ijkl}={R_{ijk}}^p{g}_{pl}$
• $\mathrm{Rm}(u,v,w,x)=g(R(u,v)w,x)$

(4,0) Weyl tensor

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

Einstein tensor

• $G_{ik}=R_{ik}-\frac{1}{2}R{g}_{ik}$
• $G(u,v)=\mathrm{Ric}(u,v)-\frac{1}{2}Rg(u,v)$

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

• ${\mathrm{\nabla }}_p{R}_{ijkl}+{\mathrm{\nabla }}_i{R}_{jpkl}+{\mathrm{\nabla }}_j{R}_{pikl}=0$
• $({\mathrm{\nabla }}_u\mathrm{Rm})(v,w,x,y)+({\mathrm{\nabla }}_v\mathrm{Rm})(w,u,x,y)+({\mathrm{\nabla }}_w\mathrm{Rm})(u,v,x,y)=0$

Contracted second Bianchi identity

• ${\mathrm{\nabla }}_j{R}_{pk}-{\mathrm{\nabla }}_p{R}_{jk}={\mathrm{\nabla }}^l{R}_{jpkl}$
• $({\mathrm{\nabla }}_u\mathrm{Ric})(v,w)-({\mathrm{\nabla }}_v\mathrm{Ric})(u,w)=-{\mathrm{tr}}_g((x,y)\mapsto ({\mathrm{\nabla }}_x\mathrm{Rm})(u,v,w,y))$

Twice-contracted second Bianchi identity

• $g^{pq}{\mathrm{\nabla }}_p{R}_{qk}=\frac{1}{2}{\mathrm{\nabla }}_{k}R$
• ${\mathrm{div}}_g\mathrm{Ric}=\frac{1}{2}dR$

Equivalently:

• $g^{pq}{\mathrm{\nabla }}_p{G}_{qk}=0$
• ${\mathrm{div}}_{g}G=0$

Ricci identity

If $X$ is a vector field then

${\mathrm{\nabla }}_i{\mathrm{\nabla }}_j{X}^k-{\mathrm{\nabla }}_j{\mathrm{\nabla }}_i{X}^k=-{R_{ijp}}^k{X}^p,$

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

${\mathrm{\nabla }}_i{\mathrm{\nabla }}_j{\omega }_k-{\mathrm{\nabla }}_j{\mathrm{\nabla }}_i{\omega }_k={R_{ijk}}^p{\omega }_p.$

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

${\mathrm{\nabla }}_i{\mathrm{\nabla }}_j{T}_{l_1\cdots l_k}-{\mathrm{\nabla }}_j{\mathrm{\nabla }}_i{T}_{l_1\cdots l_k}={R_{ij{l}_1}}^p{T}_{p{l}_2\cdots l_k}+\cdots +{R_{ij{l}_k}}^p{T}_{l_1\cdots l_{k-1}p}.$

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

Gradient, divergence, Laplace–Beltrami operator

The gradient of a function $\varphi$ is obtained by raising the index of the differential ${\partial }_i\varphi d{x}^i$, whose components are given by:

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

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

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

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

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

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

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

The Hessian of a map $\varphi :M\to N$ is given by

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

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,

$A_{ij}=A_{ji}{B}_{ij}=B_{ji}$

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

${\left(A\wedge ◯B\right)}_{ijkl}=A_{ik}{B}_{jl}+A_{jl}{B}_{ik}-A_{il}{B}_{jk}-A_{jk}{B}_{il}$

Clearly, the product satisfies

$A\wedge ◯B=B\wedge ◯A$

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 ${\mathrm{\Gamma }}^i{}_{jk}=0$ (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.

$R_{ik\ell m}=\frac{1}{2}(\frac{{\partial }^2{g}_{im}}{\partial x^k\partial x^{\ell }}+\frac{{\partial }^2{g}_{k\ell }}{\partial x^i\partial x^m}-\frac{{\partial }^2{g}_{i\ell }}{\partial x^k\partial x^m}-\frac{{\partial }^2{g}_{km}}{\partial x^i\partial x^{\ell }})$

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

Conformal change

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

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

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

• ${\tilde{\mathrm{\Gamma }}}_{ij}^k={\mathrm{\Gamma }}_{ij}^k+\frac{\partial \phi }{\partial x^i}{\delta }_j^k+\frac{\partial \phi }{\partial x^j}{\delta }_i^k-\frac{\partial \phi }{\partial x^l}{g}^{lk}{g}_{ij}$
• ${\tilde{\mathrm{\nabla }}}_{X}Y={\mathrm{\nabla }}_{X}Y+d\phi (X)Y+d\phi (Y)X-g(X,Y)\mathrm{\nabla }\phi$

(4,0) Riemann curvature tensor

• ${\tilde{R}}_{ijkl}=e^{2\phi }{R}_{ijkl}+e^{2\phi }(g_{ik}{T}_{jl}+g_{jl}{T}_{ik}-g_{il}{T}_{jk}-g_{jk}{T}_{il})$ where $T_{ij}={\mathrm{\nabla }}_i{\mathrm{\nabla }}_j\phi -{\mathrm{\nabla }}_i\phi {\mathrm{\nabla }}_j\phi +\frac{1}{2}|d\phi {|}^2{g}_{ij}$

Using the Kulkarni–Nomizu product:

• $\widetilde{\mathrm{Rm}}=e^{2\phi }\mathrm{Rm}+e^{2\phi }g\wedge ◯(\mathrm{Hess}\phi -d\phi \otimes d\phi +\frac{1}{2}|d\phi {|}^{2}g)$

Ricci tensor

• ${\tilde{R}}_{ij}=R_{ij}-(n-2)({\mathrm{\nabla }}_i{\mathrm{\nabla }}_j\phi -{\mathrm{\nabla }}_i\phi {\mathrm{\nabla }}_j\phi )-(\mathrm{\Delta }\phi +(n-2)|d\phi {|}^2)g_{ij}$
• $\widetilde{\mathrm{Ric}}=\mathrm{Ric}-(n-2)(\mathrm{Hess}\phi -d\phi \otimes d\phi )-(\mathrm{\Delta }\phi +(n-2)|d\phi {|}^2)g$

Scalar curvature

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

Traceless Ricci tensor

• ${\tilde{R}}_{ij}-\frac{1}{n}\tilde{R}{\tilde{g}}_{ij}=R_{ij}-\frac{1}{n}R{g}_{ij}-(n-2)({\mathrm{\nabla }}_i{\mathrm{\nabla }}_j\phi -{\mathrm{\nabla }}_i\phi {\mathrm{\nabla }}_j\phi )+\frac{(n-2)}{n}(\mathrm{\Delta }\phi -|d\phi {|}^2)g_{ij}$
• $\widetilde{\mathrm{Ric}}-\frac{1}{n}\tilde{R}\tilde{g}=\mathrm{Ric}-\frac{1}{n}Rg-(n-2)(\mathrm{Hess}\phi -d\phi \otimes d\phi )+\frac{(n-2)}{n}(\mathrm{\Delta }\phi -|d\phi {|}^2)g$

(3,1) Weyl curvature

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

Volume form

• $\sqrt{det\tilde{g}}=e^{n\phi }\sqrt{detg}$
• $d{\mu }_{\tilde{g}}=e^{n\phi }d{\mu }_g$

Hodge operator on p-forms

• ${\tilde{\ast }}_{i_1\cdots i_{n-p}}^{j_1\cdots j_p}=e^{(n-2p)\phi }{\ast }_{i_1\cdots i_{n-p}}^{j_1\cdots j_p}$
• $\tilde{\ast }=e^{(n-2p)\phi }\ast$

Codifferential on p-forms

• ${\widetilde{d^{\ast }}}_{j_1\cdots j_{p-1}}^{i_1\cdots i_p}=e^{-2\phi }(d^{\ast }{)}_{j_1\cdots j_{p-1}}^{i_1\cdots i_p}-(n-2p)e^{-2\phi }{\mathrm{\nabla }}^{i_1}\phi {\delta }_{j_1}^{i_2}\cdots {\delta }_{j_{p-1}}^{i_p}$
• $\widetilde{d^{\ast }}=e^{-2\phi }{d}^{\ast }-(n-2p)e^{-2\phi }{\iota }_{\mathrm{\nabla }\phi }$

Laplacian on functions

• $\tilde{\mathrm{\Delta }}\mathrm{\Phi }=e^{-2\phi }(\mathrm{\Delta }\mathrm{\Phi }+(n-2)g(d\phi ,d\mathrm{\Phi }))$

Hodge Laplacian on p-forms

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

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

• $\tilde{h}(u,v)=h(u,v)-(\mathrm{\nabla }\phi {)}^{\perp }g(u,v)$ for all $u,v\in T_{p}M$

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

Mean curvature of an immersion

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

• $\widetilde{\mathbf{\text{𝐇}}}=e^{-2\phi }(\mathbf{\text{𝐇}}-n(\mathrm{\nabla }\phi {)}^{\perp }).$

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

• $\tilde{H}=e^{-\phi }(H-n⟨\mathrm{\nabla }\phi ,\eta ⟩)$

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 Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives $v_{ij}=\frac{\partial }{\partial t}((g_t{)}_{ij})$ 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.

• $\frac{\partial }{\partial t}{\mathrm{\Gamma }}_{ij}^k=\frac{1}{2}{g}^{kp}({\mathrm{\nabla }}_i{v}_{jp}+{\mathrm{\nabla }}_j{v}_{ip}-{\mathrm{\nabla }}_p{v}_{ij}).$
• $\frac{\partial }{\partial t}{R}_{ijkl}=\frac{1}{2}({\mathrm{\nabla }}_j{\mathrm{\nabla }}_k{v}_{il}+{\mathrm{\nabla }}_i{\mathrm{\nabla }}_l{v}_{jk}-{\mathrm{\nabla }}_i{\mathrm{\nabla }}_k{v}_{jl}-{\mathrm{\nabla }}_j{\mathrm{\nabla }}_l{v}_{ik})+\frac{1}{2}{R_{ijk}}^p{v}_{pl}-\frac{1}{2}{R_{ijl}}^p{v}_{pk}$
• $\frac{\partial }{\partial t}{R}_{ik}=\frac{1}{2}({\mathrm{\nabla }}^p{\mathrm{\nabla }}_k{v}_{ip}+{\mathrm{\nabla }}_i(\mathrm{div}v{)}_k-{\mathrm{\nabla }}_i{\mathrm{\nabla }}_k({\mathrm{tr}}_{g}v)-\mathrm{\Delta }{v}_{ik})+\frac{1}{2}{R}_i^p{v}_{pk}-\frac{1}{2}{R}_i{}^p{}_k{}^q{v}_{pq}$
• $\frac{\partial }{\partial t}R={\mathrm{div}}_g{\mathrm{div}}_{g}v-\mathrm{\Delta }({\mathrm{tr}}_{g}v)-⟨v,\mathrm{Ric}{⟩}_g$
• $\frac{\partial }{\partial t}d{\mu }_g=\frac{1}{2}{g}^{pq}{v}_{pq}d{\mu }_g$
• $\frac{\partial }{\partial t}{\mathrm{\nabla }}_i{\mathrm{\nabla }}_j\mathrm{\Phi }={\mathrm{\nabla }}_i{\mathrm{\nabla }}_j\frac{\partial \mathrm{\Phi }}{\partial t}-\frac{1}{2}{g}^{kp}({\mathrm{\nabla }}_i{v}_{jp}+{\mathrm{\nabla }}_j{v}_{ip}-{\mathrm{\nabla }}_p{v}_{ij})\frac{\partial \mathrm{\Phi }}{\partial x^k}$
• $\frac{\partial }{\partial t}\mathrm{\Delta }\mathrm{\Phi }=-⟨v,\mathrm{Hess}\mathrm{\Phi }{⟩}_g-g(\mathrm{div}v-\frac{1}{2}d({\mathrm{tr}}_{g}v),d\mathrm{\Phi })$

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 $g\mapsto {\mathrm{Rm}}^g$ 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

  $v\mapsto \frac{{\xi }_j{\xi }_k{v}_{il}+{\xi }_i{\xi }_l{v}_{jk}-{\xi }_i{\xi }_k{v}_{jl}-{\xi }_j{\xi }_l{v}_{ik}}{2}=-\frac{1}{2}(\xi \otimes \xi )\wedge ◯v.$

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

  $v\mapsto v({\xi }^{\mathrm{♯}},\cdot )\otimes \xi +\xi \otimes v({\xi }^{\mathrm{♯}},\cdot )-({\mathrm{tr}}_{g_p}v)\xi \otimes \xi -|\xi {|}_g^{2}v.$

• 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

  $v\mapsto |\xi {|}_g^2{\mathrm{tr}}_{g}v+v({\xi }^{\mathrm{♯}},{\xi }^{\mathrm{♯}}).$

See also

• Liouville equations

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. Template:ISBN