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An Intuitive Introduction to Ricci Curvature
Emanuel Milman
Technion - I.I.T.
Joram Memorial SeminarHebrew University
May 26, 2016
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Ricci Curvature and its Applications
Ricci Curvature(after Gregorio Ricci-Curbastro 1853–1925):
General Relativity (Einstein, 1915):
Ricα,β + (Λ− 12
S)gα,β =8πGc4 Tα,β .
Comparison Theorems in Riemannian Geometry.
Poincaré & Geometrization conjectures via Ricci flow (Hamilton’81, Perelman ’02).
Notion extended to more general settings (Bakry–Émery ’85,Lott–Sturm–Villani ’04).
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Differentiable Manifold MAbstract definition (charts, transition maps, Hausdorff, σ-compact).
Whitney (1936): Mn can always be embedded as smoothn-dimensional submanifold of RN (N = 2n).
∀p ∈ M, ∃Mp ⊂ M, ∃Up ⊂ Rn, ∃ diffeo Fp : Up → Mp s.t. Fp(0) = p.
Local coordinates:Up 3 ~x = (x1, . . . , xn) 7→ (F 1
p , . . . ,F Np ) = ~Fp ∈ Mp.
Identify: Up ↔ Mp via Fp,e.g. x1 on Up identified with x1 ◦ F−1
p on Mp.Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Differentiable Manifold M
−→
Whitney (1936): Mn can always be embedded as smoothn-dimensional submanifold of RN (N = 2n).
∀p ∈ M, ∃Mp ⊂ M, ∃Up ⊂ Rn, ∃ diffeo Fp : Up → Mp s.t. Fp(0) = p.
Local coordinates:Up 3 ~x = (x1, . . . , xn) 7→ (F 1
p , . . . ,F Np ) = ~Fp ∈ Mp.
Identify: Up ↔ Mp via Fp,e.g. x1 on Up identified with x1 ◦ F−1
p on Mp.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Tangent Space TpMTpM = Tangent space to M at p (abstract def or p ∈ M ⊂ RN ).
TpM ↪→ RN n-dimensional linear subspace spanned by:
∂i =∂
∂x i :=∂~Fp
∂x i , i = 1, . . . ,n.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Vector FieldVector field V ∈ C∞(TM) = smooth selection M 3 p 7→ V (p) ∈ TpM.
Smooth = do it locally in Up ⊂ Rn, identify via dxFp : TxUp → TFp(x)M.
Local coordinates: ∂∂x i are linearly independent in Mp (as Fp diffeo.):
V =n∑
i=1
V i ∂
∂x i = V i ∂
∂x i ; Identify V ↔ Vα.
Remark: any V 6= 0 is locally ∂∂x1 , but not any (linearly independent)
V1,V2 are locally ∂∂x1 ,
∂∂x2 (iff [V1,V2] = 0 since ∂2Fp
∂x1∂x2 =∂2Fp
∂x2∂x1 ).Hence, we will only consider tuples of coordinate vector fields.
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Vector FieldVector field V ∈ C∞(TM) = smooth selection M 3 p 7→ V (p) ∈ TpM.
Smooth = do it locally in Up ⊂ Rn, identify via dxFp : TxUp → TFp(x)M.
Local coordinates: ∂∂x i are linearly independent in Mp (as Fp diffeo.):
V =n∑
i=1
V i ∂
∂x i = V i ∂
∂x i ; Identify V ↔ Vα.
Remark: any V 6= 0 is locally ∂∂x1 , but not any (linearly independent)
V1,V2 are locally ∂∂x1 ,
∂∂x2 (iff [V1,V2] = 0 since ∂2Fp
∂x1∂x2 =∂2Fp
∂x2∂x1 ).Hence, we will only consider tuples of coordinate vector fields.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Cotangent Space T ∗p M
T ∗p M = Cotangent space to M at p = linear dual to TpM.Elements called co(tangent)-vectors.
Co-vector field (a.k.a. differential 1-form) w ∈ C∞(T ∗M):
∀V ∈ C∞(TM) w(V ) ∈ C∞(M).
Local coordinates: ∂∂x j basis to TMp, dx i dual basis to T ∗Mp:
dx i (∂
∂x j ) = δij .
w = widx i , wi := w(∂
∂x i ) ; Identify w ↔ wβ .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Cotangent Space T ∗p M
T ∗p M = Cotangent space to M at p = linear dual to TpM.Elements called co(tangent)-vectors.
Co-vector field (a.k.a. differential 1-form) w ∈ C∞(T ∗M):
∀V ∈ C∞(TM) w(V ) ∈ C∞(M).
Local coordinates: ∂∂x j basis to TMp, dx i dual basis to T ∗Mp:
dx i (∂
∂x j ) = δij .
w = widx i , wi := w(∂
∂x i ) ; Identify w ↔ wβ .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Tensors
(κ, `)-tensor = multilinear map Q : (T ∗p M)⊗κ ⊗ (TpM)⊗` → F .Examples: vectors are (1,0) tensors, co-vectors are (0,1) tensors.
Tensor field = smoothly varying selectionM 3 p 7→ Q(p) ∈ Lin((T ∗p M)⊗κ ⊗ (TpM)⊗` → F).
Local coordinates:
Q = Q i1,...,iκj1,...,j`
∂
∂x i1⊗ . . .⊗ ∂
∂x iκ⊗ dx j1 ⊗ . . .⊗ dx j`
Q i1,...,iκj1,...,j` := Q(dx i1 , . . . ,dx iκ ,
∂
∂x j1, . . . ,
∂
∂x j`) , identify Q ↔ Qα1,...,ακ
β1,...,β`.
Remark: κ-contravariant / `-covariant under change of coordinates:
(Qy )ba = (Qx )j
idx i
dyadyb
dx j .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Riemannian Manifold (Mn,g)
(Mn,g) = Differentiable manifold Mn + Riemannian metric g.
gα,β = symmetric positive-def (0,2)-tensor field.Locally: g = gi,jdx i ⊗ dx j and g(u, v) = gi,juiv j ∀u, v ∈ TpM.
Meaning: smoothly varying scalar product on TpM:
|v | =√
g(v , v) length , 〈u, v〉 = g(u, v) angle.
Nash (’54-’56): Abstract (Mn,g) can always be (globally) isometricallyembedded in (RN , 〈·, ·〉), N � n:
gi,j = g(∂
∂x i ,∂
∂x j ) =
⟨∂Fp
∂x i ,∂Fp
∂x j
⟩RN
(1st-fundamental form).
What does g give rise to?
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Riemannian Manifold (Mn,g)
(Mn,g) = Differentiable manifold Mn + Riemannian metric g.
gα,β = symmetric positive-def (0,2)-tensor field.Locally: g = gi,jdx i ⊗ dx j and g(u, v) = gi,juiv j ∀u, v ∈ TpM.
Meaning: smoothly varying scalar product on TpM:
|v | =√
g(v , v) length , 〈u, v〉 = g(u, v) angle.
Nash (’54-’56): Abstract (Mn,g) can always be (globally) isometricallyembedded in (RN , 〈·, ·〉), N � n:
gi,j = g(∂
∂x i ,∂
∂x j ) =
⟨∂Fp
∂x i ,∂Fp
∂x j
⟩RN
(1st-fundamental form).
What does g give rise to?
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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1. Geodesic distance d on M
Geodesic distance (metric):d(x , y) := inf
{∫ 10 |γ
′(t)|dt ; γ : [0,1]→ M, γ(0) = x , γ(1) = y}
.Minimizers of distance always exist locally.Geodesics = paths which locally minimize distance.
Equivalently d(x , y)2 = inf{∫ 1
0 |γ′(t)|2dt
}(uniqueness under reparametrization t = t(s)).
Euler-Lagrange eqns yield 2nd order systemof ODEs. Given initial γ(0) = p, γ′(0) = v ∈ TpM⇒ ∃ local solution γ : [0, t0)→ M;if ∃γ(1), we obtain the exponential map:
TpM ⊃ Dom(expp) 3 v 7→ expp(v) := γ(1) ∈ M.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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1. Geodesic distance d on M
Geodesic distance (metric):d(x , y) := inf
{∫ 10 |γ
′(t)|dt ; γ : [0,1]→ M, γ(0) = x , γ(1) = y}
.Minimizers of distance always exist locally.Geodesics = paths which locally minimize distance.
Equivalently d(x , y)2 = inf{∫ 1
0 |γ′(t)|2dt
}(uniqueness under reparametrization t = t(s)).
Euler-Lagrange eqns yield 2nd order systemof ODEs. Given initial γ(0) = p, γ′(0) = v ∈ TpM⇒ ∃ local solution γ : [0, t0)→ M;if ∃γ(1), we obtain the exponential map:
TpM ⊃ Dom(expp) 3 v 7→ expp(v) := γ(1) ∈ M.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Radius of Injectivity
exponential map: TpM ⊃ Dom(expp) 3 v 7→ expp(v) ∈ M.
Injectivity radius at p ∈ M:
injp := sup{
r > 0; expp is injective on BTpM(0, r) ⊂ Dom(expp)}.
Facts: injp > 0 and expp : BTpM(0, injp)→ BM(p, injp) is 1-1 diffeo.
Injectivity radius of M:
inj(M) := infp∈M
injp.
Relevance: if inj(Mt )→ 0 then possibly singularities being formed.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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2. Canonical Identification TpM ↔ T ∗p M
Identify TpM ↔ T ∗p M by u[(v) = g(u, v) ∀u, v ∈ TpM.Locally: if u[ = uidx i , uiv i = gi,jujv i , or in other words, ui = gi,juj .
Similarly, w(v) = g(w ], v) ∀w ∈ T ∗p M, v ∈ TpM.Locally: if w ] = w i ∂
∂x i , wiv i = gi,jw jv i ⇒ wi = gi,jw j ⇒ wk = gk,iwi ,where gk,i denotes g−1 in local coordinates, i.e. gk,igi,j = δk
j .
Raising and lower indices by contracting with g allows changingtensor type (κ, `)↔ (µ, ν) if κ+ ` = µ+ ν. For example:
Q b,c,da = gb,iQ c,d
a,i .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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3. Riemannian Volume Measure VolM
Riemannian volume measure is VolM =√
det(gi,j )dx1 . . . dxn
(replace dx1 . . . dxn with dx1 ∧ . . . ∧ dxn to get volume form).
Interpretation: VolM is the induced Lebesgue measure on M ↪→ RN .Since Fp : Rn ⊃ Up → Mp ⊂ RN , we have:
Jac(Fp) =√
det(dF tpdFp) , (dF t
pdFp)i,j =
⟨∂Fp
∂x i ,∂Fp
∂x j
⟩RN
= gi,j .
Integration: integrate f : M → R in local coordinates in Mα, and thendeclare
∫M f dVolM =
∑α
∫Mα
fpαdVolM , where pα = partition of unity.
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3. Riemannian Volume Measure VolM
Riemannian volume measure is VolM =√
det(gi,j )dx1 . . . dxn
(replace dx1 . . . dxn with dx1 ∧ . . . ∧ dxn to get volume form).
Interpretation: VolM is the induced Lebesgue measure on M ↪→ RN .Since Fp : Rn ⊃ Up → Mp ⊂ RN , we have:
Jac(Fp) =√
det(dF tpdFp) , (dF t
pdFp)i,j =
⟨∂Fp
∂x i ,∂Fp
∂x j
⟩RN
= gi,j .
Integration: integrate f : M → R in local coordinates in Mα, and thendeclare
∫M f dVolM =
∑α
∫Mα
fpαdVolM , where pα = partition of unity.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
f (γ(t))− f (γ(0))
t= df (γ′(0)) = ∇v f .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
“X (γ(t))− X (γ(0))”
t= ∇X (γ′(0)) = ∇v X .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
“X (γ(t))− X (γ(0))”
t= ∇X (γ′(0)) = ∇v X .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
“X (γ(t))− X (γ(0))”
t= ∇X (γ′(0)) = ∇v X .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
“Affine Connection": specify 1st order infinitesimal transport ∇v X ofX in direction v ∈ TpM; connection along arbitrary path obtained bysolving parallel transport equation:
0 = “ddt
”X (γ(t)) = ∇γ′(t)X , X (γ(0)) = X0.
So need to specify ∇v X , ∀v ∈ TpM,X ∈ TM. By linearity in v andadditivity + Leibnitz rule in X , enough to specify ∇∂i∂j = Γk
i,j∂k .Yields system of 1st order linear ODEs⇒ ∃ solution (on entire γ).
∇ called “Covariant Derivative": X is (1,0) tensor, and ∇X is (1,1).Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
“X (γ(t))− X (γ(0))”
t= ∇X (γ′(0)) = ∇v X .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
“Affine Connection": specify 1st order infinitesimal transport ∇v X ofX in direction v ∈ TpM; connection along arbitrary path obtained bysolving parallel transport equation:
0 = “ddt
”X (γ(t)) = ∇γ′(t)X , X (γ(0)) = X0.
So need to specify ∇v X , ∀v ∈ TpM,X ∈ TM. By linearity in v andadditivity + Leibnitz rule in X , enough to specify ∇∂i∂j = Γk
i,j∂k .Yields system of 1st order linear ODEs⇒ ∃ solution (on entire γ).
∇ called “Covariant Derivative": X is (1,0) tensor, and ∇X is (1,1).Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
“X (γ(t))− X (γ(0))”
t= ∇X (γ′(0)) = ∇v X .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
“Affine Connection": specify 1st order infinitesimal transport ∇v X ofX in direction v ∈ TpM; connection along arbitrary path obtained bysolving parallel transport equation:
0 = “ddt
”X (γ(t)) = ∇γ′(t)X , X (γ(0)) = X0.
So need to specify ∇v X , ∀v ∈ TpM,X ∈ TM. By linearity in v andadditivity + Leibnitz rule in X , enough to specify ∇∂i∂j = Γk
i,j∂k .Yields system of 1st order linear ODEs⇒ ∃ solution (on entire γ).
∇ called “Covariant Derivative": X is (1,0) tensor, and ∇X is (1,1).Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Affine Connection, Covariant DerivativeDerivative of function in direction of v ∈ TpM:
γ(t) := expp(tv) , limt→0
“X (γ(t))− X (γ(0))”
t= ∇X (γ′(0)) = ∇v X .
What about derivative of vector field X in direction v ∈ TpM?Need to connect between tangent spaces along a path γ : [0,1]→ M.“Connection". Transporting vector along γ called “Parallel Transport".
“Affine Connection": specify 1st order infinitesimal transport ∇v X ofX in direction v ∈ TpM; connection along arbitrary path obtained bysolving parallel transport equation:
0 = “ddt
”X (γ(t)) = ∇γ′(t)X , X (γ(0)) = X0.
So need to specify ∇v X , ∀v ∈ TpM,X ∈ TM. By linearity in v andadditivity + Leibnitz rule in X , enough to specify ∇∂i∂j = Γk
i,j∂k .Yields system of 1st order linear ODEs⇒ ∃ solution (on entire γ).
∇ called “Covariant Derivative": X is (1,0) tensor, and ∇X is (1,1).Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 27: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/27.jpg)
4. Levi-Civita Connection
Choice of affine connection (Γki,j ) may be arbitrary.
We would like a connection which is:
Compatible with metric g = parallel transport is local isometry(preserves lengths and angles):
∇v g(X ,Y ) = g(∇v X ,Y ) + g(X ,∇v Y ) (i.e. ∇g = 0).
Torsion free (= doesn’t spin vectors unnecessarily):
∇∂i∂j = ∇∂j∂i ∀i , j .
Levi-Civita (’29): ∃ unique connection / covariant derivative as above.[Γk
i,j =12
gk,a(∂
∂x i ga,j +∂
∂x j gi,a −∂
∂xk gi,j
)].
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 28: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/28.jpg)
5. Sectional Curvature - DefinitionAssume (M,g) is 2-D surface, locally isometrically embedded in R3.Let k1, k2 be the principle curvatures at p ∈ M = eigenvalues of Hess fwhere f : TpM → R s.t. M = graph(f ) (locally near p).
Gauss Theorema Egregium (1827): Gauss Curvature K = k1 · k2 isintrinsic, does not depend on isometric embedding in RN , only on g.
Examples: K (S2(r)) ≡ 1/r2, K (Cylinder/R2) ≡ 0, K (H2) = −1.
Def (Riemann, 1854): Let (Mn,g), E ↪→ TpM a 2-D plane.
Sectp(E) = Kp(E) := Gauss-Curvp{
expp(v) ; v ∈ E ∩ Dom(expp)}.
Examples: Sect(Sn) ≡ 1, Sect(Rn) ≡ 0, Sect(Hn) ≡ −1 (locally, iff).Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 29: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/29.jpg)
5. Sectional Curvature - DefinitionAssume (M,g) is 2-D surface, locally isometrically embedded in R3.Let k1, k2 be the principle curvatures at p ∈ M = eigenvalues of Hess fwhere f : TpM → R s.t. M = graph(f ) (locally near p).
Gauss Theorema Egregium (1827): Gauss Curvature K = k1 · k2 isintrinsic, does not depend on isometric embedding in RN , only on g.
Examples: K (S2(r)) ≡ 1/r2, K (Cylinder/R2) ≡ 0, K (H2) = −1.
Def (Riemann, 1854): Let (Mn,g), E ↪→ TpM a 2-D plane.
Sectp(E) = Kp(E) := Gauss-Curvp{
expp(v) ; v ∈ E ∩ Dom(expp)}.
Examples: Sect(Sn) ≡ 1, Sect(Rn) ≡ 0, Sect(Hn) ≡ −1 (locally, iff).Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 30: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/30.jpg)
5. Sectional Curvature - DefinitionAssume (M,g) is 2-D surface, locally isometrically embedded in R3.Let k1, k2 be the principle curvatures at p ∈ M = eigenvalues of Hess fwhere f : TpM → R s.t. M = graph(f ) (locally near p).
Gauss Theorema Egregium (1827): Gauss Curvature K = k1 · k2 isintrinsic, does not depend on isometric embedding in RN , only on g.
Examples: K (S2(r)) ≡ 1/r2, K (Cylinder/R2) ≡ 0, K (H2) = −1.
Def (Riemann, 1854): Let (Mn,g), E ↪→ TpM a 2-D plane.
Sectp(E) = Kp(E) := Gauss-Curvp{
expp(v) ; v ∈ E ∩ Dom(expp)}.
Examples: Sect(Sn) ≡ 1, Sect(Rn) ≡ 0, Sect(Hn) ≡ −1 (locally, iff).Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 31: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/31.jpg)
5. Sectional Curvature - Geometric Meaning
Let v ,w ∈ TpM, |v | = |w | = 1.
Scenario 1 Scenario 2
1. d2(expp(sv), expp(tw)) = s2+t2−2st cos(θ)−Kp(v ∧ w)
3s2t2 sin2(θ)+O((s2+t2)5/2).
2. Let ps = expp(sv), let τsw = parallel transport of w along geodesic p → ps . Then:
d(expp(tw), expps(t τsw)) = s
(1−
Kp(v ∧ w)
2t2 + O(t3 + t2s)
).
⇒ Sectional curvature controls distance distortion of geodesics.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 32: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/32.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 33: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/33.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 34: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/34.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 35: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/35.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 36: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/36.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 37: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/37.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 38: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/38.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 39: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/39.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 40: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/40.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 41: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/41.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 42: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/42.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
![Page 43: Homepage of Emanuel Milman - An Intuitive Introduction to ......An Intuitive Introduction to Ricci Curvature Emanuel Milman Technion - I.I.T. Joram Memorial Seminar Hebrew University](https://reader035.fdocument.pub/reader035/viewer/2022070223/613ef438c500cf75ab3636e9/html5/thumbnails/43.jpg)
5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
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5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
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5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
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5. Sectional Curvature - Riemann Curvature Tensor
Riemann Curvature Tensor = (0,4) tensor R constructed from{Kp(E) ; 2-D E ↪→ TpM} by polarization such that:
∀ orthonormal u, v ∈ TpM Rp(u, v ,u, v) = Kp(u∧v) + symmetries.
e.g. Rp(u, v ,u, v) = Kp(u ∧ v)|u ∧ v |2, |w1 ∧ w2|2 = det(〈wi ,wj〉).Rp(u, v ,w , z) = . . . (18 terms) . . .
R = Ri,j,k,`dx i ⊗ dx j ⊗ dxk ⊗ dx` , Ri,j,k,` = R(∂i , ∂j , ∂k , ∂`).
Differential Definition of (1,3) version (Gauss Theorema Egregium):
[∇∂j ,∇∂i ]∂k = ∇∂j∇∂i∂k −∇∂i∇∂j∂k = R `i,j,k ∂`.
(this is indeed a tensor - linear in the third vector-field!).Curvature = lack of commutation of covariant derivative.
∇∂j∇∂i Z −∇∂i∇∂j Z =∂
∂s∂
∂t
∣∣∣∣s=t=0
τ−1s∂jτ−1
t∂iτs∂j τt∂i Z .
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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6. Ricci Curvature - Definition
Geometrically: given v ∈ TpM, |v | = 1, complete to orthonormalbasis v ,w1, . . . ,wn−1, and define:
Ricci curvature: Ricp(v) :=n−1∑k=1
Sectp(v ∧ wk ).
Ric(u, v) = symmetric (0,2) Ricci curvature tensor obtained bypolarization s.t. Ric(v , v) = Ric(v) for all |v | = 1.
Algebraically: Rici,j = trgRi,∗,j,∗ = gk,`Ri,k,j,` = R ki,k,j (non-trivial tr):
Ric(u, v) =n∑
k=1
R(u,ek , v ,ek ) ∀ orthonormal basis ek (R(v , v , ∗, ∗) = 0).
(symmetries of R yield equivalence + symmetry of Ric).
Example: RicSn (v , v) = (n − 1) |v |2 i.e. RicSn = (n − 1)g.
Scalar Curvature: S = trgRic = g i,jRici,j = Ric ii (scalar function).
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6. Ricci Curvature - Definition
Geometrically: given v ∈ TpM, |v | = 1, complete to orthonormalbasis v ,w1, . . . ,wn−1, and define:
Ricci curvature: Ricp(v) :=n−1∑k=1
Sectp(v ∧ wk ).
Ric(u, v) = symmetric (0,2) Ricci curvature tensor obtained bypolarization s.t. Ric(v , v) = Ric(v) for all |v | = 1.
Algebraically: Rici,j = trgRi,∗,j,∗ = gk,`Ri,k,j,` = R ki,k,j (non-trivial tr):
Ric(u, v) =n∑
k=1
R(u,ek , v ,ek ) ∀ orthonormal basis ek (R(v , v , ∗, ∗) = 0).
(symmetries of R yield equivalence + symmetry of Ric).
Example: RicSn (v , v) = (n − 1) |v |2 i.e. RicSn = (n − 1)g.
Scalar Curvature: S = trgRic = g i,jRici,j = Ric ii (scalar function).
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6. Ricci Curvature - Definition
Geometrically: given v ∈ TpM, |v | = 1, complete to orthonormalbasis v ,w1, . . . ,wn−1, and define:
Ricci curvature: Ricp(v) :=n−1∑k=1
Sectp(v ∧ wk ).
Ric(u, v) = symmetric (0,2) Ricci curvature tensor obtained bypolarization s.t. Ric(v , v) = Ric(v) for all |v | = 1.
Algebraically: Rici,j = trgRi,∗,j,∗ = gk,`Ri,k,j,` = R ki,k,j (non-trivial tr):
Ric(u, v) =n∑
k=1
R(u,ek , v ,ek ) ∀ orthonormal basis ek (R(v , v , ∗, ∗) = 0).
(symmetries of R yield equivalence + symmetry of Ric).
Example: RicSn (v , v) = (n − 1) |v |2 i.e. RicSn = (n − 1)g.
Scalar Curvature: S = trgRic = g i,jRici,j = Ric ii (scalar function).
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6. Ricci Curvature - Definition
Geometrically: given v ∈ TpM, |v | = 1, complete to orthonormalbasis v ,w1, . . . ,wn−1, and define:
Ricci curvature: Ricp(v) :=n−1∑k=1
Sectp(v ∧ wk ).
Ric(u, v) = symmetric (0,2) Ricci curvature tensor obtained bypolarization s.t. Ric(v , v) = Ric(v) for all |v | = 1.
Algebraically: Rici,j = trgRi,∗,j,∗ = gk,`Ri,k,j,` = R ki,k,j (non-trivial tr):
Ric(u, v) =n∑
k=1
R(u,ek , v ,ek ) ∀ orthonormal basis ek (R(v , v , ∗, ∗) = 0).
(symmetries of R yield equivalence + symmetry of Ric).
Example: RicSn (v , v) = (n − 1) |v |2 i.e. RicSn = (n − 1)g.
Scalar Curvature: S = trgRic = g i,jRici,j = Ric ii (scalar function).
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6. Ricci Curvature - Geometric Meaning
v ∈ dω ⊂ Sn−1(TpM) ⇒ voln−1(expp(εdω)) = voln−1(dω)εn−1(
1−Ricp(v)
6ε2 + O(ε3)
).
⇒ Ricp(v) controls distortion of voln−1 in direction v ∈ TpM. Intuition:
det(I + εA) = 1 + εtr(A) + o(ε) = averaging n − 1 distance distortions.
In normal (geodesic) coordinates (i.e. x = x i ei ∈ TpM and Fp(x) = expp(x)):gi,j = δi,j − 1
3 Rp,i,q,j xpxq +O(|x |3) ⇒ det(gi,j ) = 1− 13 Ricp,qxpxq +O(|x |3).
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Applications of Ricci lower bounds
Given complete connected (Mn,g) with Ric ≥ K (n − 1)g, manycomparison thms to model space (Mn
K ,gK ) with Sect ≡ K :
Surface Area vs. Volume of geodesic balls (Bishop–Gromov).
Isoperimetric inequalities (Lévy–Gromov).
First eigenvalue of Laplacian (Lichnerowicz / Cheng).
Diameter ≤ π/√
K if K > 0 (Bonnet–Meyers).
Parabolic estimates comparison (Li–Yau, Bakry–Émery).
Heat-kernel comparison (Cheeger–Yau).
many, many more ...
Typically: comparison inqs reversed if (Mn,g) satisfies Sect ≤ K !
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Example: Bishop–Gromov Theorem
Comparison between complete (Mn,g) and (MnK ,gK ), K ∈ R.
Set V∗(R) = Vol(B(p,R)) and S∗(R) = Voln−1(∂B(p,R)) in (Mn∗ ,g∗).
Gromov (’80):
Ric ≥ K (n−1)g ⇒ S(R)
V (R)≤ SK (R)
VK (R)⇔ V (R)
V (r)≤ VK (R)
VK (r)∀0 < r < R.
Letting r → 0, obtain Bishop’s Thm (’63):
Ric ≥ K (n − 1)g ⇒ V (R) ≤ VK (R) ∀R > 0.
However, to obtain reverse comparison (Günther ’60), require:
Sect ≤ K ⇒ V (R)
V (r)≥VK (R)
VK (r)∀0 < r < R ≤ injp.
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Example: Bishop–Gromov Theorem
Comparison between complete (Mn,g) and (MnK ,gK ), K ∈ R.
Set V∗(R) = Vol(B(p,R)) and S∗(R) = Voln−1(∂B(p,R)) in (Mn∗ ,g∗).
Gromov (’80):
Ric ≥ K (n−1)g ⇒ S(R)
V (R)≤ SK (R)
VK (R)⇔ V (R)
V (r)≤ VK (R)
VK (r)∀0 < r < R.
Letting r → 0, obtain Bishop’s Thm (’63):
Ric ≥ K (n − 1)g ⇒ V (R) ≤ VK (R) ∀R > 0.
However, to obtain reverse comparison (Günther ’60), require:
Sect ≤ K ⇒ V (R)
V (r)≥VK (R)
VK (r)∀0 < r < R ≤ injp.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Example: Bishop–Gromov Theorem
Comparison between complete (Mn,g) and (MnK ,gK ), K ∈ R.
Set V∗(R) = Vol(B(p,R)) and S∗(R) = Voln−1(∂B(p,R)) in (Mn∗ ,g∗).
Gromov (’80):
Ric ≥ K (n−1)g ⇒ S(R)
V (R)≤ SK (R)
VK (R)⇔ V (R)
V (r)≤ VK (R)
VK (r)∀0 < r < R.
Letting r → 0, obtain Bishop’s Thm (’63):
Ric ≥ K (n − 1)g ⇒ V (R) ≤ VK (R) ∀R > 0.
However, to obtain reverse comparison (Günther ’60), require:
Sect ≤ K ⇒ V (R)
V (r)≥VK (R)
VK (r)∀0 < r < R ≤ injp.
Emanuel Milman An Intuitive Introduction to Ricci Curvature
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Images courtesy of:
www.wikipedia.org / www.wikimedia.org.
ww.science4all.org.
www.zweigmedia.com.
JefferyWinkler.com.
Blagoje Oblak, arXiv:1508.00920.
Paulo H.F Reimberg et al., arXiv:1211.5114.
Paul M. Alsing et al., arXiv:1308.4148.
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Bonus Material - 1. Jacobi Equation
If {t 7→ γs(t)}s is a family of geodesics, then its variationξ(t) := ∂γ
∂s |s=0 is called a Jacobi vector field along γ0.It satisfies the Jacobi Equation:
(∇dt
)2ξ + R(γ′(t), ξ)γ′(t) = 0.
Proof.Commuting covariant derivatives twice, we obtain by definition of R:
∇dt∇dtξ =∇dt∇dt∇dsγ =(∗)
∇dt∇ds∇dtγ =
∇ds∇dt∇dtγ + R(
∂γ
∂s,∂γ
∂t)∂γ
∂t,
where (*) follows since ∇ is torsion-free:
∇ ∂γ∂t
∂γ
∂s−∇ ∂γ
∂s
∂γ
∂t= [
∂γ
∂t,∂γ
∂s] = 0.
Conclude using (∇dt )2γ = ∇γ′(t)γ′(t) = 0 since γ is geodesic.
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Bonus Material - 2. Initial ConditionsGiven family of n − 1 Jacobi fields along unit-speed geodesict 7→ γ(t), arrange them as columns in (n − 1)× (n − 1) matrix A(t).Then column-wise:
(∇dt
)2A(t) + R(t)A(t) = 0 , R(t) := R(γ′(t), ·, γ′(t), ·).
Typically A(t) = dxF (x , t), S × R 3 (x , t) 7→ F (x , t) := expp(x)(tν(x)).Examples:
1 Exponential map from point p.S = Sn−1(TpM), p(x) ≡ p, ν(x) = x , and:
A(0) = 0 ,∇dt
A(0) = IISn−1 (x) = Id .
2 Normal map from oriented hypersurface S ⊂ M.p(x) = x , ν(x) = unit-normal to S at x , and:
A(0) = Id ,∇dt
A(0) = IIS(x) (second fundamental form).
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Bonus Material - 3. Tracing Matrix Riccati EquationDenoting U(t) = ∇
dt A(t)A−1(t), easy to check that:
(∇dt
)2A(t) + R(t)A(t) = 0 ⇔ ∇dt
U(t) + U(t)2 + R(t) = 0,
as long as A(t) is invertible (before focal point). Symmetry of R(t)and U(0) ensure symmetry of U(t), and hence U(t)2 ≥ 0.Taking traces and applying Cauchy-Schwarz 1
n−1 tr(U)2 ≤ tr(U2):
ddt
tr(U(t)) +tr(U(t))2
n − 1+ Ric(γ′(t), γ′(t)) ≤ 0.
But tr(U) = tr(∇dt A(t)A−1(t)) = ddt log det A(t) = d
dt log J(t), where J isthe Jacobian of (x , t) 7→ F (x , t). Assuming Ric ≥ K (n − 1)g, obtain:
(log J)′′ +((log J)′)2
n − 1+ K (n − 1) ≤ 0 ⇔ (J
1n−1 )′′
J1
n−1+ K ≤ 0.
By maximum principle, if J(0) = JK (0), J ′(0) = J ′K (0), comparison tomodel space follows J ≤ JK . In fact, obtain (log J)′ ≤ (log JK )′.
Emanuel Milman An Intuitive Introduction to Ricci Curvature