Geometric Meaning of
Ricci Curvature Tensor
and Ricci Scalar
(Video 24: Ricci Tensor)
(Videos 22,23: Riemann Curvature Tensor)
(Link in the description)
𝑅 𝑠, 𝑣 𝑣 ⋅ 𝑠
𝐾 𝑠, 𝑣 =
2
𝑠⋅𝑠 𝑣⋅𝑣 − 𝑠⋅𝑣
𝑣
𝑠
Take an orthonormal basis {𝑒1 , 𝑒2 , … , 𝑒𝑛 }
and direction vector 𝑣 = 𝑒𝑛
𝐷
𝑅𝑖𝑐 𝑣, 𝑣 =
𝐾 𝑒𝑖 , 𝑣 =
𝑖=1
𝑅 𝑒𝑖 ,𝑣 𝑣 ⋅𝑒𝑖
𝑒𝑖 ⋅𝑒𝑖 𝑣⋅𝑣 − 𝑒𝑖 ⋅𝑣 2
Ricci Tensor: Track “volume change” along geodesics
1. Sectional Curvature
Orthonormal
Basis Only!
2. Volume element derivative
Any Basis!
“Volume” created by orthonormal basis
𝑒2
𝑒2
𝑉=1
𝑒1
𝑉=1
𝑒3
𝑒1
“Volume” created by vectors on O.N. basis
𝑤
𝑒2
𝑢
𝑉=1
𝑒1
1
2
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2
1
2
𝑤 = 𝑤 𝑒1 + 𝑤 𝑒2
𝑉𝑜𝑙𝑢𝑚𝑒 =
1
1
𝑢
𝑤
det 2
2
𝑢 𝑤
“Volume” created by vectors in O.N. basis
𝑤
𝑢
𝑢 = 𝑢1 𝑒1 + 𝑢2 𝑒2
𝑤 = 𝑤 1 𝑒1 + 𝑤 2 𝑒2
1
𝑢
𝑉𝑜𝑙𝑢𝑚𝑒 = det 2
𝑢
𝑤 1 = 𝑢1 𝑤 2 − 𝑢 2 𝑤 1
2
𝑤
Levi-Civita Symbol
+1 𝜖12 (𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔 𝑜𝑟𝑑𝑒𝑟)
𝜖𝑖𝑗 = −1 𝜖21 𝑤𝑟𝑜𝑛𝑔 𝑜𝑟𝑑𝑒𝑟
0
𝑎𝑛𝑦 𝑖𝑛𝑑𝑒𝑥 𝑟𝑒𝑝𝑒𝑎𝑡𝑒𝑑
𝑖
𝜖𝑖𝑗 𝑢 𝑤
𝑗
1
1
1
2
= 𝜖11 𝑢 𝑤 + 𝜖12 𝑢 𝑤
2
1
2
2
+ 𝜖21 𝑢 𝑤 + 𝜖22 𝑢 𝑤
“Volume” created by vectors in O.N. basis
𝑤
1
𝑢
2
3
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2 +𝑢 𝑒3
1
2
3
𝑤 = 𝑤 𝑒1 + 𝑤 𝑒2 +𝑤 𝑒3
𝑡 = 𝑡 1 𝑒1 + 𝑡 2 𝑒2 +𝑡 3 𝑒3
1
𝑢
𝑡
𝑉𝑜𝑙𝑢𝑚𝑒 = det 𝑢2
3
𝑢
Levi-Civita Symbol
+1 𝑖𝑗𝑘 𝑒𝑣𝑒𝑛 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛
𝜖𝑖𝑗𝑘 = −1 𝑖𝑗𝑘 𝑜𝑑𝑑 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛
0
𝑎𝑛𝑦 𝑖𝑛𝑑𝑒𝑥 𝑟𝑒𝑝𝑒𝑎𝑡𝑒𝑑
𝑤1
2
𝑤
𝑤3
𝑡1
𝑖 𝑗 𝑘
2 =𝜖
𝑢
𝑡
𝑖𝑗𝑘 𝑤 𝑡
𝑡3
𝜖123 = 𝜖231 = 𝜖312 = +1
𝜖132 = 𝜖213 = 𝜖321 = −1
𝜖113 = 𝜖222 = … = 0
“Volume” created by vectors in O.N. basis
𝑤
1
𝑢
𝑡
2
3
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2 +𝑢 𝑒3
1
2
3
𝑤 = 𝑤 𝑒1 + 𝑤 𝑒2 +𝑤 𝑒3
𝑡 = 𝑡 1 𝑒1 + 𝑡 2 𝑒2 +𝑡 3 𝑒3
𝑢1
𝑉𝑜𝑙𝑢𝑚𝑒 = det 𝑢2
𝑢3
𝑤1
2
𝑤
𝑤3
𝑡1
𝑖 𝑗 𝑘
2 =𝜖
𝑢
𝑡
𝑖𝑗𝑘 𝑤 𝑡
𝑡3
= 𝑢1 𝑤 2 𝑡 3 − 𝑢1 𝑤 3 𝑡 2 + 𝑢 2 𝑤 3 𝑡 1 − 𝑢 2 𝑤 1 𝑡 3 + 𝑢 3 𝑤 1 𝑡 2 − 𝑢 3 𝑤 2 𝑡 1
“Volume” created by vectors in O.N. basis
𝑤
1
𝑢
𝑡
2
3
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2 +𝑢 𝑒3
1
2
3
𝑤 = 𝑤 𝑒1 + 𝑤 𝑒2 +𝑤 𝑒3
𝑡 = 𝑡 1 𝑒1 + 𝑡 2 𝑒2 +𝑡 3 𝑒3
𝑢1
𝑉𝑜𝑙𝑢𝑚𝑒 = det 𝑢2
𝑢3
𝑤1
2
𝑤
𝑤3
𝑡1
𝑖 𝑗 𝑘
2 =𝜖
𝑢
𝑡
𝑖𝑗𝑘 𝑤 𝑡
𝑡3
Levi-Civita Symbol 𝜖𝑖𝑗𝑘 is a tool for getting the volume
of a “parallelogram”-shape spanned by vectors
(only works with components in orthonormal basis)
“Volume” created by vectors in O.N. basis
1
1
2
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2
𝑤 = 𝑤 1 𝑒1 + 𝑤 2 𝑒2
𝑉𝑜𝑙𝑢𝑚𝑒 =
1
1
𝑢
𝑤
det 2
2
𝑢 𝑤
= 𝜖𝑖𝑗
𝑖
𝑗
𝑢𝑤
𝜕
𝜕𝑥 𝜕
𝜕𝑥 2 𝜕
=
+
1
2
1
1
1
𝑒1 = 𝐹1 𝑒1 + 𝐹1 𝑒2 𝜕𝑥
𝜕𝑥 𝜕𝑥
𝜕𝑥 1 𝜕𝑥 2
1 𝜕
2 𝜕
𝜕
𝜕𝑥
𝜕𝑥
1
2
𝑒2 = 𝐹2 𝑒1 + 𝐹2 𝑒2 2 = 2 1 + 2 2
𝜕𝑥
𝜕𝑥 𝜕𝑥
𝜕𝑥 𝜕𝑥
𝑉𝑜𝑙𝑢𝑚𝑒 =
1
1
𝐹1 𝐹2
det 2
2
𝐹1 𝐹2
= det 𝐹
𝑖 𝑗
= 𝜖𝑖𝑗 𝐹1 𝐹2
𝑉𝑜𝑙𝑢𝑚𝑒 =
det
𝜕𝑥 1
𝜕𝑥 1
𝜕𝑥 2
𝜕𝑥 1
𝜕𝑥 1
𝜕𝑥 2
𝜕𝑥 2
𝜕𝑥 2
= det 𝐽
“Volume” created by arbitrary basis
𝑔𝑖𝑗 = 𝑒𝑖 ⋅ 𝑒𝑗
𝑎
𝑏
𝑔𝑖𝑗 = 𝐹𝑖 𝑒𝑎 ⋅ 𝐹𝑗 𝑒𝑏
𝑎 𝑏
𝑔𝑖𝑗 = 𝐹𝑖 𝐹𝑗 𝑒𝑎 ⋅ 𝑒𝑏
𝑎 𝑏
𝑔𝑖𝑗 = 𝐹𝑖 𝐹𝑗 𝑔𝑎𝑏
Basis Volume
= det 𝐹
= det 𝑔
det 𝑔 = det 𝐹 det 𝐹 det 𝑔 = det 𝐹
2
“Volume” created by arbitrary basis
𝜕
𝜕
𝑔𝑖𝑗 = 𝑖 ⋅ 𝑗
𝜕𝑥 𝜕𝑥
𝑎
𝑏
𝜕𝑥 𝜕
𝜕𝑥 𝜕
𝑔𝑖𝑗 =
⋅
𝑖
𝑎
𝜕𝑥 𝜕𝑥
𝜕𝑥 𝑗 𝜕𝑥 𝑏
𝜕𝑥 𝑎 𝜕𝑥 𝑏
𝜕
𝜕
𝑔𝑖𝑗 = 𝑖 𝑗
⋅
𝜕𝑥 𝜕𝑥 𝜕𝑥 𝑎 𝜕𝑥 𝑏
𝑎 𝑏
𝑔𝑖𝑗 = 𝐽𝑖 𝐽𝑗 𝑔𝑎𝑏
Basis Volume
= det 𝐽
= det 𝑔
det 𝑔 = det 𝐽
2
det 𝑔
“Volume” created by arbitrary basis
1
2
𝑒2
𝑒1 = 𝐹1 𝑒1 + 𝐹1 𝑒2
1
2
𝑒
=
𝐹
𝑒
+
𝐹
𝑒
2
1
2
2
2
𝑒
1
𝑉𝑜𝑙𝑢𝑚𝑒 = det
1
𝐹1
2
𝐹1
1
𝐹2
2
𝐹2
= det 𝐹 = det 𝐽 =
deg 𝑔
“Volume” created by arbitrary basis
𝑤
1
𝑒2
𝑢
2
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2
1
2
𝑤 = 𝑤 𝑒1 + 𝑤 𝑒2
𝑒1
𝑉𝑜𝑙𝑢𝑚𝑒 =
1
𝑢
deg 𝑔 ⋅ det 2
𝑢
1
𝑤
2
𝑤
“Volume” created by arbitrary basis
𝑤
1
𝑒2
𝑢
2
𝑢 = 𝑢 𝑒1 + 𝑢 𝑒2
1
2
𝑤 = 𝑤 𝑒1 + 𝑤 𝑒2
𝑒1
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝑖
deg 𝑔 𝜖𝑖𝑗 𝑢 𝑤
“Volume Form” components
𝑗
“Volume” created by arbitrary vectors
𝑒2
𝑒2
𝑒2
𝑒1
𝑉=1
𝑒1
𝑒1
1
2
𝑢 = 𝑎1 𝑒1 + 𝑎1 𝑒2
1
2
𝑤 = 𝑎2 𝑒1 + 𝑎2 𝑒2
𝑉 = det 𝐽 𝑉 = det 𝐽 det 𝐴
𝑉 = det 𝑔 𝑉 = det 𝑔 det 𝐴
nd
2 Derivative of Volume spanned by vectors
𝑤
𝑡
𝑉=
𝑣
𝑢
deg 𝑔 𝜖𝑖𝑗𝑘
𝑖
𝑗
𝑘
𝑢𝑤 𝑡
nd
2 Derivative of Volume spanned by vectors
𝑠2
𝑉=
𝑠1
𝑣
𝑉=
deg 𝑔
𝑖 𝑗 𝑘
𝜖𝑖𝑗𝑘 𝑠1 𝑠2 𝑠3
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝜇𝑖
𝐷
𝑖=1 𝑠𝑖
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝜇𝑖
𝐷
𝑖=1 𝑠𝑖
𝑠3
∇𝑣 ∇𝑣 𝑉
𝑑2
𝑑 𝑑
= 2𝑉 =
𝑑𝜆
𝑑𝜆 𝑑𝜆
𝑑 𝑉
𝑑 𝑑
=
2
𝑑𝜆
𝑑𝜆 𝑑𝜆
2
𝑑 𝑉
=
2
𝑑𝜆
𝜇
𝑠𝑖 𝑖
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝜇
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝑖=1
𝑑 𝑉
𝑑
𝜇𝑗
=
𝑠𝑗
2
𝑑𝜆
𝑑𝜆
2
Volume form is invariant
𝐷
2
𝐷
𝑠𝑖 𝑖
𝑖=1
𝑖≠𝑗
𝐷
𝜇𝑗
𝑠𝑗
𝐷
𝜇𝑖
𝑠𝑖
deg 𝑔 𝜖𝜇1 …𝜇𝐷 +
𝑖=1
𝑖≠𝑗
𝜇𝑗
𝜇𝑗
𝑦 𝑧
𝑥
𝑠𝑗 = −𝑅𝑥𝑦𝑧 𝑣 𝑠𝑗 𝑣
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
𝜇𝑖
𝑠𝑖
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝑖=1
𝑖≠𝑗,𝑘
∇𝑣 ∇𝑣 𝑠 = −𝑅 𝑠, 𝑣 𝑣
2
𝑑 𝑉
=
2
𝑑𝜆
𝐷
𝐷
𝜇𝑗
𝑦
−𝑅𝑥𝑦𝑧 𝑣 𝑥 𝑠𝑗 𝑣 𝑧
𝜇
𝑠𝑖 𝑖
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
deg 𝑔 𝜖𝜇1…𝜇𝐷 +
𝑖=1
𝑖≠𝑗
𝜇
𝑠𝑖 𝑖
deg 𝑔 𝜖𝜇1…𝜇𝐷
𝑖=1
𝑖≠𝑗,𝑘
Every index except 𝜇𝑗 is used
𝐷
2
𝑑 𝑉
𝜇𝑗
𝜇𝑖
𝑥 𝜇𝑗 𝑧
=
−𝑅
𝑣
𝑠
𝑣
𝑠
𝑥𝜇𝑗 𝑧
𝑖
𝑗
𝑑𝜆2
𝑖=1
𝐷
deg 𝑔 𝜖𝜇1 …𝜇𝐷 +
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
𝜇𝑗
𝑥𝑣 𝑧
=
−𝑅
𝑣
𝑥𝜇𝑗 𝑧
𝑑𝜆2
𝐷
𝐷
𝜇𝑖
𝑠𝑖
𝑖=1
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝑖=1
𝑖≠𝑗,𝑘
𝑖≠𝑗
𝑑2 𝑉
𝜇
𝑠𝑖 𝑖
deg 𝑔 𝜖𝜇1…𝜇𝐷 +
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
𝜇𝑖
𝑠𝑖
𝑖=1
𝑖≠𝑗,𝑘
deg 𝑔 𝜖𝜇1…𝜇𝐷
2
𝑑 𝑉
𝜇𝑗
𝑥𝑣 𝑧
=
−𝑅
𝑣
𝑥𝜇𝑗 𝑧
𝑑𝜆2
𝐷
𝐷
𝜇
𝑠𝑖 𝑖
deg 𝑔 𝜖𝜇1…𝜇𝐷 +
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
deg 𝑔 𝜖𝜇1 …𝜇𝐷
𝑖=1
𝑖≠𝑗,𝑘
𝑖=1
𝑑2𝑉
𝑥 𝑧
= −𝑅𝑥𝑧 𝑣 𝑣 𝑉
2
𝑑𝜆
𝜇
𝑠𝑖 𝑖
+
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
𝜇𝑖
𝐷
𝑖=1 𝑠𝑖
𝑖≠𝑗,𝑘
deg 𝑔 𝜖𝜇1…𝜇𝐷
Conclusion: Ricci Tensor tells us how volumes
change as we move around in space
𝑑2𝑉
𝑥 𝑧
=
−𝑅
𝑣
𝑣 𝑉
𝑥𝑧
2
𝑑𝜆
𝑣
𝑠
Curved Space
+
𝜇𝑗 𝜇𝑘
𝑠𝑗 𝑠𝑘
𝜇𝑖
𝐷
𝑖=1 𝑠𝑖
𝑖≠𝑗,𝑘
𝑣
deg 𝑔 𝜖𝜇1…𝜇𝐷
𝑠
Flat Space
Conclusion: Ricci Tensor tells us how volumes change
due to the curvature of the space we’ere in
1 2 3 𝑖𝑗𝑘
det 𝐴 = 𝑎𝑖 𝑎𝑗 𝑎𝑘 𝜖
1 2 3 𝜇1 𝜇2 𝜇3
det 𝐴 = 𝑎𝜇1 𝑎𝜇2 𝑎𝜇3 𝜖
𝐷
𝑖
𝑎𝜇𝑖
det 𝐴 =
𝜖
𝑖=1
𝜇1 …𝜇𝐷
Second Derivative of Volume Element
∇𝑣 ∇𝑣 det 𝑉
𝑑2
= 2 det 𝑉
𝑑𝜆
𝐷
𝑑 𝑑
=
𝑎𝜇𝑖 𝑖
𝑑𝜆 𝑑𝜆
det 𝑔 𝜖 𝜇1…𝜇𝐷
𝑖=1
𝑉=
det 𝑔 det 𝐴
det 𝑔
𝑑
𝑗
𝐷
𝑎𝜇𝑗
𝑖
𝜇1 …𝜇𝐷 =
𝑑𝜆
𝑖=1 𝑎𝜇𝑖 𝜖
𝐷
𝑎𝜇𝑖 𝑖
𝑖=1
𝑖≠𝑗
det 𝑔 𝜖 𝜇1 …𝜇𝐷
1 2 3 𝑖𝑗𝑘
det 𝐴 = 𝑎𝑖 𝑎𝑗 𝑎𝑘 𝜖
1 2 3 𝜇1 𝜇2 𝜇3
det 𝐴 = 𝑎𝜇1 𝑎𝜇2 𝑎𝜇3 𝜖
𝐷
𝑖
𝑎𝜇𝑖
det 𝐴 =
𝜖
𝑖=1
𝜇1 …𝜇𝐷
Second Derivative of Volume Element
∇𝑣 ∇𝑣 det 𝑉
𝑑2
= 2 det 𝑉
𝑑𝜆
𝐷
𝑑 𝑑
=
𝑎𝜇𝑖 𝑖
𝑑𝜆 𝑑𝜆
det 𝑔 𝜖 𝜇1…𝜇𝐷
𝑖=1
𝑉=
det 𝑔 det 𝐴
det 𝑔
𝑑
𝑗
𝐷
𝑎𝜇𝑗
𝑖
𝜇1 …𝜇𝐷 =
𝑑𝜆
𝑖=1 𝑎𝜇𝑖 𝜖
𝐷
𝑎𝜇𝑖 𝑖
𝑖=1
𝑖≠𝑗
det 𝑔 𝜖 𝜇1 …𝜇𝐷
2
𝑑 𝑉
𝑑
𝑗
=
𝑎𝜇𝑗
2
𝑑𝜆
𝑑𝜆
2
𝑑 𝑉
=
2
𝑑𝜆
𝐷
𝑎𝜇𝑖 𝑖
det 𝑔 𝜖 𝜇1 …𝜇𝐷
𝑖=1
𝑖≠𝑗
𝐷
𝑗
𝑎𝜇𝑗
𝐷
𝑎𝜇𝑖 𝑖
det 𝑔 𝜖
𝜇1 …𝜇𝐷
+
𝑖=1
𝑖≠𝑗
∇𝑣 ∇𝑣 𝑠 = −𝑅 𝑠, 𝑣 𝑣
𝑗
𝑗
𝑥 𝑦 𝑧
𝑎𝜇𝑗 = −𝑅𝑥𝑦𝑧 𝑣 𝑎𝜇𝑗 𝑣
𝑘 𝑗
𝑎𝜇𝑘 𝑎𝜇𝑗
𝑎𝜇𝑖 𝑖
𝑖=1
𝑖≠𝑗,𝑘
det 𝑔 𝜖 𝜇1 …𝜇𝐷
𝐷
2
𝑑 𝑉
=
2
𝑑𝜆
𝑑2𝑉
=
2
𝑑𝜆
𝑑2𝑉
𝑗
𝑦
−𝑅𝑥𝑦𝑧 𝑣 𝑥 𝑎𝜇𝑗 𝑣 𝑧
𝑗
𝐷
𝑎𝜇𝑖 𝑖
𝑗
det 𝑔 𝜖 𝜇1…𝜇𝐷 + 𝑎𝜇𝑘𝑘 𝑎𝜇𝑗
𝑖=1
𝑖≠𝑗
𝑖=1
𝑖≠𝑗,𝑘
𝐷
𝐷
𝑗
−𝑅𝑥𝑗𝑧 𝑣 𝑥 𝑎𝜇𝑗 𝑣 𝑧
𝑎𝜇𝑖 𝑖
𝑗
det 𝑔 𝜖 𝜇1…𝜇𝐷 + 𝑎𝜇𝑘𝑘 𝑎𝜇𝑗
𝑖=1
𝑖≠𝑗
𝑗
𝑎𝜇𝑖 𝑖
det 𝑔 𝜖 𝜇1…𝜇𝐷
𝐷
𝑎𝜇𝑖 𝑖
𝑖=1
det 𝑔 𝜖 𝜇1…𝜇𝐷
𝑖=1
𝑖≠𝑗,𝑘
𝐷
𝑧
=
−𝑅
𝑣
𝑥𝑗𝑧
𝑑𝜆2
𝑎𝜇𝑖 𝑖
𝑗
det 𝑔 𝜖 𝜇1…𝜇𝐷 + 𝑎𝜇𝑘𝑘 𝑎𝜇𝑗
𝑎𝜇𝑖 𝑖
𝑖=1
𝑖≠𝑗,𝑘
det 𝑔 𝜖 𝜇1…𝜇𝐷
2
𝑑 𝑉
𝑗
𝑥𝑣 𝑧
=
−𝑅
𝑣
𝑥𝑗𝑧
𝑑𝜆2
𝐷
𝐷
𝑎𝜇𝑖 𝑖
𝑗
det 𝑔 𝜖 𝜇1…𝜇𝐷 + 𝑎𝜇𝑘𝑘 𝑎𝜇𝑗
det 𝑔 𝜖 𝜇1…𝜇𝐷
𝑖=1
𝑖≠𝑗,𝑘
𝑖=1
𝑑2𝑉
𝑥 𝑧
= −𝑅𝑥𝑧 𝑣 𝑣 𝑉
2
𝑑𝜆
𝑎𝜇𝑖 𝑖
+
𝑘 𝑗
𝑎𝜇𝑘 𝑎𝜇𝑗
𝐷
𝑖
𝑎
𝑖=1 𝜇𝑖
𝑖≠𝑗,𝑘
det 𝑔 𝜖 𝜇1…𝜇𝐷