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

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

and the Christoffel symbols of the second kind by

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

Here ${\displaystyle g^{ij}}$ 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

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

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

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

and

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

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:

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

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

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

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

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

and likewise for tensors with more indices.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

Equivalently:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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,

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

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

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

Clearly, the product satisfies

${\displaystyle A\wedge ◯B=B\wedge ◯A}$

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

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

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

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

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

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 ${\displaystyle g}$.)

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

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

Using the Kulkarni–Nomizu product:

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

In the same setting as above (and suppose ${\displaystyle \mathrm{\Sigma }}$ has dimension ${\displaystyle n}$), recall that the mean curvature vector is for each ${\displaystyle p\in \mathrm{\Sigma }}$ an element ${\displaystyle {\text{<mi fromhbox="1">H</mi>}}_p\in T_{F(p)}M}$ defined as the ${\displaystyle g}$-trace of the second fundamental form. Then

• ${\displaystyle \widetilde{\text{<mi fromhbox="1">H</mi>}}=e^{-2\phi }(\text{<mi fromhbox="1">H</mi>}-n(\mathrm{\nabla }\phi {)}^{\perp }).}$

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

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

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

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

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

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

• The principal symbol of the map ${\displaystyle g\mapsto {\mathrm{Ric}}^g}$ 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

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

• The principal symbol of the map ${\displaystyle g\mapsto R^g}$ 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

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

• 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

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/List of formulas in Riemannian geometry. Read more