Wikipedia list article
This is a list of formulas encountered in Riemannian geometry.
Christoffel symbols, covariant derivative
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)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea446b24cf5ab9092a23c551264570b48d4b62c5)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee13f8aeeae2235914d97657b3aba5ab2e043cdc)
Here
is the inverse matrix to the metric tensor
. In other words,
![{\displaystyle \delta ^{i}{}_{j}=g^{ik}g_{kj}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a79cce42b3921781d8b4ebe0bd1bd606ca81ea4e)
and thus
![{\displaystyle n=\delta ^{i}{}_{i}=g^{i}{}_{i}=g^{ij}g_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc4b0ab7c869c4547401420b80df3ca846e7578)
is the dimension of the manifold.
Christoffel symbols satisfy the symmetry relations
or, respectively,
,
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d3f9c527aab76a2b96342745698e3023002a9bb)
and
![{\displaystyle g^{k\ell }\Gamma ^{i}{}_{k\ell }={\frac {-1}{\sqrt {|g|}}}\;{\frac {\partial \left({\sqrt {|g|}}\,g^{ik}\right)}{\partial x^{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3475d0f96aaea1f97ff0bb0c7308d6dce0b50169)
where |g| is the absolute value of the determinant of the metric tensor
. These are useful when dealing with divergences and Laplacians (see below).
The covariant derivative of a vector field with components
is given by:
![{\displaystyle v^{i}{}_{;j}=(\nabla _{j}v)^{i}={\frac {\partial v^{i}}{\partial x^{j}}}+\Gamma ^{i}{}_{jk}v^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f987f8e9074f946814e0e4c0299129bd05e8762b)
and similarly the covariant derivative of a
-tensor field with components
is given by:
![{\displaystyle v_{i;j}=(\nabla _{j}v)_{i}={\frac {\partial v_{i}}{\partial x^{j}}}-\Gamma ^{k}{}_{ij}v_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49ae0c4df5438695f8c7dc66ccf90e35ce4ab76f)
For a
-tensor field with components
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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09fa89dab4b73b2400a7d6e808dbf189e730b26)
and likewise for tensors with more indices.
The covariant derivative of a function (scalar)
is just its usual differential:
![{\displaystyle \nabla _{i}\phi =\phi _{;i}=\phi _{,i}={\frac {\partial \phi }{\partial x^{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd2ddb286a182cbe3e519a508c739b7f8e79d97)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2604244502a216535f1891f57a1cd736eb1714b4)
as well as the covariant derivatives of the metric's determinant (and volume element)
![{\displaystyle \nabla _{k}{\sqrt {|g|}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/927e087252212a0fd8934e8d5b68ef9ee85da5c1)
The geodesic
starting at the origin with initial speed
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d99763576bc9f1f877b3a5d78a69a691274e30a9)
Curvature tensors
Definitions
![{\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}-{\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}+\sum _{p=1}^{n}{\big (}\Gamma _{ik}^{p}\Gamma _{jp}^{l}-\Gamma _{jk}^{p}\Gamma _{ip}^{l}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6267749a1571539b0753e18e2b29fc6e7ac92ec6)
![{\displaystyle R(u,v)w=\nabla _{v}\nabla _{u}w-\nabla _{u}\nabla _{v}w-\nabla _{[v,u]}w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b29cc983fd7fb9338f34379a51ac1b7172220a)
![{\displaystyle R_{ik}=\sum _{j=1}^{n}{R_{ijk}}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7fa25e6d5936ef3b61a2f64d8fc1d9eae2d2b7)
![{\displaystyle \operatorname {Ric} (u,v)=\operatorname {tr} (v\mapsto R(u,v)w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/342958b81938272051df7ca1d09b3f1de8429517)
![{\displaystyle R=\sum _{i=1}^{n}\sum _{k=1}^{n}g^{ik}R_{ik}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ff7156c58a4b8603565838833d794bc99d60ef)
![{\displaystyle R=\operatorname {tr} _{g}\operatorname {Ric} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6520945e066d36fc944c608c41c07554d0067514)
Traceless Ricci tensor
![{\displaystyle Q_{ik}=R_{ik}-{\frac {1}{n}}Rg_{ik}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3979393638d3ae0d685e676c32aa48cc1110cda1)
![{\displaystyle Q(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{n}}Rg(u,v)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b982f3e53f2a8f50c3f5c047f2262fba7d54c08)
(4,0) Riemann curvature tensor
![{\displaystyle R_{ijkl}=\sum _{p=1}^{n}{R_{ijk}}^{p}g_{pl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e03477f9d385c312d6a88898ed042ab8de9725e)
![{\displaystyle \operatorname {Rm} (u,v,w,x)=g{\big (}R(u,v)w,x{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d697a4f0fdeea16ecd365b954ce09e38914f1cd6)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9adb93754df4c6e9cc5b124611148637fe8fead2)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa21919a039b2c6165d02b093abd0457b68104f8)
![{\displaystyle G_{ik}=R_{ik}-{\frac {1}{2}}Rg_{ik}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4520157c8f1266c89dee81b06bc586b99bf4d0d)
![{\displaystyle G(u,v)=\operatorname {Ric} (u,v)-{\frac {1}{2}}Rg(u,v)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c256d9d56f42c6398a4aa8c44eb118785b14a831)
Identities
See Proofs involving Christoffel symbols for some proofs
Basic symmetries
![{\displaystyle {R_{ijk}}^{l}=-{R_{jik}}^{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a690286043d55a0b1fc1d45cb99d82a483b2f455)
![{\displaystyle R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1b9e36c7f032e863f216607ec5d4ed487628c2)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab180013355531a47e776581fcfda9cc19fd08)
![{\displaystyle \sum _{i=1}^{n}\sum _{l=1}^{n}g^{il}W_{ijkl}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/583c6da08992858c4f944dbe1ce541301eb27723)
The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
![{\displaystyle R_{jk}=R_{kj}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e8bbd4d1e15989e116a895bb9082b61da879ca)
![{\displaystyle G_{jk}=G_{kj}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7fda98b945d7a9239bde706a5e849032617a598)
![{\displaystyle Q_{jk}=Q_{kj}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ddf4da94a0bde1e397711611901384bb85a3d2)
First Bianchi identity
![{\displaystyle R_{ijkl}+R_{jkil}+R_{kijl}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97bd5d4fb8a18298cabcc8efe94e136333fc91da)
![{\displaystyle W_{ijkl}+W_{jkil}+W_{kijl}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b07df471913e1630ca3a5e6b99176371f84dcbf)
Second Bianchi identity
![{\displaystyle \nabla _{p}R_{ijkl}+\nabla _{i}R_{jpkl}+\nabla _{j}R_{pikl}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a56c30c477f1ccf04a1b23659f07f460e2ff6e3)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb51554dc31801f9928c181bc4216890dfc8bc37)
Contracted second Bianchi identity
![{\displaystyle \nabla _{j}R_{pk}-\nabla _{p}R_{jk}=-\nabla ^{l}R_{jpkl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91d301b4f50afe3c06a6e098226e289b033ece7a)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c925ce392a67ff27cf393110c2e89108e751b7ae)
Twice-contracted second Bianchi identity
![{\displaystyle \sum _{p=1}^{n}\sum _{q=1}^{n}g^{pq}\nabla _{p}R_{qk}={\frac {1}{2}}\nabla _{p}R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b83902834d873db6257a6fbe3a2ee8c4b890a9d3)
![{\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ded9120fbb941a9d9476d0854f9e28dcd5b60d35)
Equivalently:
![{\displaystyle \sum _{p=1}^{n}\sum _{q=1}^{n}g^{pq}\nabla _{p}G_{qk}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd32c5de740f0a3c1e9d2ba5dec0aa38ac31e0e)
![{\displaystyle \operatorname {div} _{g}G=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd8b1f9a49f40eae37bae6342bddb0d67139648)
Ricci identity
If
is a vector field then
![{\displaystyle \nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b9b1f0cb7e49505197e6d91894f98140dca7bf)
which is just the definition of the Riemann tensor. If
is a one-form then
![{\displaystyle \nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/730eb2523e734c3766ba6945c2ad812ed26ac0bc)
More generally, if
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b16b56ae281eb25411a709298e4919afd9587a)
A classical result says that
if and only if
is locally conformally flat, i.e. if and only if
can be covered by smooth coordinate charts relative to which the metric tensor is of the form
for some function
on the chart.
Gradient, divergence, Laplace–Beltrami operator
The gradient of a function
is obtained by raising the index of the differential
, 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/939b9b2c9418ff9c21bd4a0e0b7bca245b004926)
The divergence of a vector field with components
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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b94995d7ec37ce2ee8778b2bafed298603623b0c)
The Laplace–Beltrami operator acting on a function
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7028f2c1e27035bc3bc1d5a81f131720173c831c)
The divergence of an antisymmetric tensor field of type
simplifies to
![{\displaystyle \nabla _{k}A^{ik}={\frac {1}{\sqrt {|g|}}}{\frac {\partial (A^{ik}{\sqrt {|g|}})}{\partial x^{k}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e33c07e64f8b7c5ecfde7ee32e071711d2158a6c)
The Hessian of a map
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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bfbc6257e5149d41a58c264d6614138be5fcc7)
Kulkarni–Nomizu product
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let
and
be symmetric covariant 2-tensors. In coordinates,
![{\displaystyle A_{ij}=A_{ji}\qquad \qquad B_{ij}=B_{ji}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/010c2e493db4b4a1842716efff75fb25a2f917d8)
Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted
. The defining formula is
Clearly, the product satisfies
![{\displaystyle A{~\wedge \!\!\!\!\!\!\bigcirc ~}B=B{~\wedge \!\!\!\!\!\!\bigcirc ~}A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d3aea7119a5d8fa11f0d9f466a70e556609e1bb)
In an inertial frame
An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations
and
(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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0750590e3c55fffa72d2bc72802b040716292607)
![{\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6868fd8b448cb1f2f3f7fa97223e897eda5587)
Conformal change ![{\displaystyle {\widetilde {g}}=e^{2\varphi }g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aedc33293541c4cb010700d34ed15985c107079)
Let
be a Riemannian or pseudo-Riemanniann metric on a smooth manifold
, and
a smooth real-valued function on
. Then
![{\displaystyle {\tilde {g}}=e^{2\varphi }g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e46a004a0ba7b2f871de262816defe6ba0f3adc1)
is also a Riemannian metric on
. We say that
is (pointwise) conformal to
. 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
, while those unmarked with such will be associated with
.)
Levi-Civita connection
![{\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_{k}}}g_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3712d39fac9a51a02c26dbc8334c0398e8664294)
![{\displaystyle {\widetilde {\nabla }}_{X}Y=\nabla _{X}Y+d\varphi (X)Y+d\varphi (Y)X-g(X,Y)\nabla \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be0c297d9720c6a0540dc08124d339b09d0e27d)
(4,0) Riemann curvature tensor
where ![{\displaystyle T_{ij}=\nabla _{i}\nabla _{j}\varphi -\nabla _{i}\varphi \nabla _{j}\varphi +{\frac {1}{2}}|d\varphi |^{2}g_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3fca970c089a3c59c92aa6f0d996274a8f62785)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24959c4f090a954d4a3438a1de5426f9af30629f)
Ricci tensor
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e17f91e534c89eff1e0ea737c1dc3c40585d8c)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84dae8f7fa2d4b997faf44c728a56a979383b3d7)
Scalar curvature
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f308e86c0282eafc7f1a0abefd41112fcb75ab13)
- if
this can be written ![{\displaystyle {\tilde {R}}=e^{-2\varphi }\left[R+{\frac {4(n-1)}{(n-2)}}e^{-(n-2)\varphi /2}\triangle \left(e^{(n-2)\varphi /2}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3acb15c55f08afd53525ca15278248131a10fdf3)
Traceless Ricci tensor
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbd9609f1aa2249f19d1687221a87f8a4bf90d7)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8546032e2d5814a971e9d172dba6f27c2b0af0bb)
(3,1) Weyl curvature
![{\displaystyle {{\widetilde {W}}_{ijk}}^{l}={W_{ijk}}^{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364d363d1cb2d2013b04096765cf973b24b7ceff)
for any vector fields ![{\displaystyle X,Y,Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf4a8b48db1a32d24aabe164b07744069093225)
Volume form
![{\displaystyle {\sqrt {\det {\widetilde {g}}}}=e^{n\varphi }{\sqrt {\det g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b337060266c26fb00e18d08b3df26454a86c3c22)
![{\displaystyle d\mu _{\widetilde {g}}=e^{n\varphi }\,d\mu _{g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b414b50e288378ea6e9afd390879b7b2d4aaf92)
Hodge operator on p-forms
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbdb4b3521856d0240cddca11b1a30629b82ee5)
![{\displaystyle {\widetilde {\ast }}=e^{(n-2p)\varphi }\ast }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7544324fda69bda62315dc5a62307e56f7e8bd)
Codifferential on p-forms
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/493ede180d41b46877deebf9fbea1f446660fb67)
![{\displaystyle {\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48a3db8a16c2dbf58d8fcf6ef3a168efd9527bcf)
Laplacian on functions
![{\displaystyle {\widetilde {\Delta ^{d}}}\Phi =e^{-2\varphi }{\Big (}\Delta ^{d}\Phi -(n-2)g(d\varphi ,d\Phi ){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1287bbde860d9cbda64f8afac8360c08bd506455)
Hodge Laplacian on p-forms
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2717d8f405f329f614dd869f4f64f9b133f29c7)
Second fundamental form of an immersion
Suppose
is Riemannian and
is a twice-differentiable immersion. Recall that the second fundamental form is, for each
a symmetric bilinear map
which is valued in the
-orthogonal linear subspace to
Then
for all ![{\displaystyle u,v\in T_{p}M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2a891365f8c31138253b7ed5cf0b8fe7b73838)
Here
denotes the
-orthogonal projection of
onto the
-orthogonal linear subspace to
Mean curvature of an immersion
In the same setting as above, recall that the mean curvature is for each
an element
defined as the
-trace of the second fundamental form. Then
![{\displaystyle {\widetilde {H}}=H-n(\nabla \varphi )^{\perp }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a785afa6bfc1396b24e86a7360f9c8533d16943)
Variation formulas
Let
be a smooth manifold and let
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
exist and are themselves as differentiable as necessary for the following expressions to make sense. Denote
as a one-parameter family of symmetric 2-tensor fields.
![{\displaystyle {\frac {\partial }{\partial t}}\Gamma _{ij}^{k}={\frac {1}{2}}\sum _{p=1}^{n}g^{kp}{\Big (}\nabla _{i}v_{jp}+\nabla _{j}v_{ip}-\nabla _{p}v_{ij}{\Big )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc95d7e363a3177ee410278058173fcefe591a32)
![{\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 )}+\sum _{p=1}^{n}{R_{ijk}}^{p}v_{pl}-\sum _{p=1}^{n}{R_{ijl}}^{p}v_{pk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef542eb7443461166f2166bde1312124b81d0ae3)
![{\displaystyle {\frac {\partial }{\partial t}}R_{ik}={\frac {1}{2}}{\Big (}\sum _{p=1}^{n}\nabla ^{p}\nabla _{k}v_{ip}+\nabla _{i}(\operatorname {div} v)_{k}-\nabla _{i}\nabla _{k}(\operatorname {tr} _{g}v)-\Delta v_{ik}{\Big )}-\sum _{p=1}^{n}R_{i}^{p}v_{pk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a858859226792dd1d0aff0992fcdf2ffbc5f0e3f)
![{\displaystyle {\frac {\partial }{\partial t}}R=\Delta (\operatorname {tr} _{g}v)+\operatorname {div} _{g}\operatorname {div} _{g}v-\langle v,\operatorname {Ric} \rangle _{g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d729631f2759a53f710110c69fcf9848edb365d)
![{\displaystyle {\frac {\partial }{\partial t}}d\mu _{g}=-{\frac {1}{2}}\sum _{p=1}^{n}\sum _{q=1}^{n}g^{pq}v_{pq}\,d\mu _{g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a3f04bc0bffbf646890001fab5eb77fa726ec2)
![{\displaystyle {\frac {\partial }{\partial t}}\nabla _{i}\nabla _{j}\Phi =\nabla _{i}\nabla _{j}{\frac {\partial \Phi }{\partial t}}-{\frac {1}{2}}\sum _{p=1}^{n}g^{kp}{\Big (}\nabla _{i}v_{jp}+\nabla _{j}v_{ip}-\nabla _{p}v_{ij}{\Big )}{\frac {\partial \Phi }{\partial x^{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cacad09bc3b5313039720954c5230e1f74190b)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfca13e2881d71ed319c4803405df455dfb0fc7)
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
assigns to each
a map from the space of symmetric (0,2)-tensors on
to the space of (0,4)-tensors on
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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1692256b704a32e1e7d20081f725b9329c1c14af)
- The principal symbol of the map
assigns to each
an endomorphism of the space of symmetric 2-tensors on
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/418e6549959029683a3355b85f3bbbf5c343f932)
- The principal symbol of the map
assigns to each
an element of the dual space to the vector space of symmetric 2-tensors on
by
![{\displaystyle v\mapsto |\xi |_{g}^{2}\operatorname {tr} _{g}v+v(\xi ^{\sharp },\xi ^{\sharp }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbac7f6d930f2d9e7744c00c872a672b2960805c)
See also
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