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IntroductionScalar curvatures on foliations
The Vanishing theorem
Positive scalar curvature on foliations
Weiping Zhang
Chern Institute of Mathematics
Workshop on Geometric Analysis
Dalian, September 1st, 2016
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theoremIntroduction
Introduction
I Mn smooth manifold, gTM Riemannian metric on TM
I For any p ∈M ,
vol(BMp (r)
)vol(BRn
0 (r)) = 1− kg
TM
6(n+ 2)r2 +O
(r3).
I kgTM
the scalar curvature of gTM .
I Basic question : when exists gTM with kgTM
> 0 ?
(Every manifold Mn with n ≥ 3 carries gTM with
kgTM
< 0)
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theoremIntroduction
Introduction
I Lichnerowicz vanishing theorem. If a compact spinmanifold M carries a Riemannian metric of positive scalarcurvature, then the Hirzebruch A-genus vanishes. That is,A(M) = 0.
A(M) =⟨A(TM), [M ]
⟩=
∫M
det1/2(
RTM/4πi
sinh(RTM/4πi)
).
(a la Chern-Weil)
I spin condition essential, as A(CP 2) = −18 .
(A complex manifold is spin if c1(M) is even)
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theoremIntroduction
Introduction
I Proof : Using the Dirac operator on spin manifolds :
I D± : S±(TM) −→ S∓(TM).
I D2 = −∆ + kTM
4 > 0 (Lichnerowicz formula)
I 0 = ind(D+) = A(M). (Atiyah-Singer index theorem)
I Aim : Present a generalization to the case of foliation
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Foliation
I (M,F ) is a foliation if F ⊆ TM is an integrable subbundle
I i.e., X, Y ∈ Γ(F ) implies [X,Y ] ∈ Γ(F )
I For any x ∈M , (unique) leaf through x, denoted by Fx (asubmanifold of M)
I For any x ∈M , F |Fx = TFx
I Locally looks like fibration near each x ∈M
I “space of leaves” (M/∪x∈MFx) might be non-Hausdorff
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Leafwise scalar curvature
I gF a Euclidean metric on F
I For any x ∈M , gF induces a Riemannian metricgTFx = gF |Fx on TFx
I kgTFx
be the scalar curvature along Fx associated to gTFx
I kgF ∈ C∞(M) with
kgF
(x) = kgTFx
(x)
well-defined, called leafwise scalar curvature of gF
I When F = TM , kgTM
is the usual scalar curvature on M .
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
An open question
I Open question : If
kgF> 0
over M , whether there exists gTM such that
kgTM
> 0 ?
I Easy case : if F = T VM of a fibration π : M → B, set
gTMε =π∗gTB
ε2⊕ gTVM ,
thenkg
TMε = kg
F+O(ε2).
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Foliation : the Bott connection
I Let p⊥ : TM → F⊥ be the orthogonal projection withrespect to a splitting : TM = F ⊕ F⊥, gTM = gF ⊕ gF⊥
.
I Let ∇B be the Bott connection on F⊥ ' TM/F :for any X ∈ Γ(F ), U ∈ Γ(F⊥),
∇BXU = p⊥[X,U ].
I For any X ∈ Γ(F ), define ω(X) ∈ Γ(End(F⊥)) by that forany U, V ∈ Γ(F⊥),
〈ω(X)U, V 〉 = X〈U, V 〉 − 〈∇BXU, V 〉 − 〈U,∇BXV 〉.
I Riemannian foliation : For any X ∈ Γ(F ), ω(X) ≡ 0.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Adiabatic limit of kgTMε
I In general, set gTMε = gF ⊕ gF⊥
ε2.
I Let f1, · · · , fq be an orthonormal basis of (F, gF ) ;
h1, · · · , hq1 an orthonormal basis of (F⊥, gF⊥
).
I Set
|ω (fi)|2 =
q1∑s=1
|ω (fi)hs|2 .
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Adiabatic limit of kgTMε
II (Based on earlier computations with Kefeng Liu andYong Wang)
kgTMε = kg
F+
3
4
q∑i=1
|ω(fi)|2 −1
4
q∑i=1
(q1∑s=1
〈ω(fi)hs, hs〉
)2
−q∑i=1
q1∑s=1
fi (〈ω(fi)hs, hs〉) + 2
q∑i=1
q1∑s=1
⟨ω(fi)hs, p
⊥[fi, hs]⟩
+
q∑i=1
q1∑s=1
⟨ω(p∇TMfi fi
)hs, hs
⟩+O
(ε2).
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Easy case : Riemannian foliation
II If ω = 0, i.e., in the case of Riemannian foliation,
I as ε→ 0,kg
TMε = kg
F+O
(ε2)
I General case : Much more complicated relations ...
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Adiabatic limit of kgTMε : the codimension one case
I Assume q1 = rk(TM/F ) = 1, then
I
kgTMε = kg
F − 1
2
q∑i=1
|ω(fi)|2 −q∑i=1
fi (〈ω(fi)h1, h1〉)
+
q∑i=1
⟨ω(p∇TMfi fi
)h1, h1
⟩+O
(ε2).
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Connes vanishing theorem
I Theorem of Connes (1986) : Let (M,F ) be a foliationsuch that M is compact and oriented, while F is spin. If
there is a metric gF on F such that kgF> 0 over M , then
A(M) = 0.
I F = TM , Lichnerowicz theorem.
I In general, M need not spin, A(M) is not a priori aninteger
I Highly non-trivial even in the q1 = 1 case.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Connes vanishing theorem
I If (M,F ) is a fibration, easy consequence of Atiyah-Singerfamilies index theorem
A(M) =
∫BA(TB)
∫Mb
A(T VM) =
∫BA(TB)ch(ind(Db))
I Connes’ proof : noncommutative (families) index theory onfoliations + cyclic cohomology
I Relies essentially on the spin structure on F
I Natural question : What happens if one assumesTM spin instead of F spin ?
I If TM spin, one has Dirac operators on M .
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
Main result
I Theorem (Zhang, arXiv : 1508.04503) : Let (M,F ) bea foliation such that M is compact and spin. If there is a
metric gF on F such that kgF> 0 over M , then A(M) = 0.
I F = TM , Lichnerowicz theorem.
I Corollary. If M4k (k > 1) is also simply connected, then
M4k carries a metric with positive scalar curvature.
I Main difficulty : ω 6= 0.
I Resolution : need Connes fibration.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
The Connes fibration
I Let π : M →M be the fibration
I For any x ∈M ,
π−1(x) = {Euclidean metrics on TxM/Fx}
I Each fiber π−1(x) is a space of non-positive curvature
I non-positive curvature property plays an essential role inConnes’ proof of his vanishing theorem
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
The Connes fibration
I F ⊆ TM lifts to an integrable subbundle F of TM.
I T V M vertical tangent bundle
I Natural splitting : TM = F ⊕ T V M ⊕ F⊥
I T V M carries a natural metric of (fiberwise) non-positivecurvature
I From F⊥ ' π∗(TM/F ), by definition, F⊥ carries acanonically induced metric
I (Any p ∈ M, determines a metric onTπ(p)M/Fπ(p) ' π∗F⊥p , which determines a metric on F⊥p )
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
The Connes fibration
I gF lifts to a metric on F
I Orthogonal splitting
TM = F ⊕ T V M ⊕ F⊥, gTM = gF ⊕ gTV M ⊕ gF⊥
I Connes : Hol(M, F ) acts almost isometrically on
T V M ⊕ F⊥
I Infinitesimally, for any X ∈ Γ(F ), Y, Z ∈ Γ(T V M),U, V ∈ Γ(F⊥), one has
X〈Y,Z〉 = 〈[X,Y ], Z〉+ 〈Y, [X,Z]〉,
X〈U, V 〉 = 〈[X,U ], V 〉+ 〈U, [X,V ]〉,
〈[X,Y ], U〉 = 0.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
Scalar curvatures on foliationsConnes fibration
The Connes fibration
I Following Connes, take an embedded section s : M ↪→ M
I Equivalently : take a metric on F⊥ ' TM/F
I F ⊕ F⊥ = π∗(TM) spin
I For any β > 0, ε > 0, rescale the metric to
TM = F ⊕ T V M ⊕ F⊥, gTM = β2gF ⊕ gTV M ⊕ gF⊥
ε2.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
The vanishing theorem
I Theorem (Zhang, arXiv : 1508.04503) : Let (M,F ) bea foliation such that M is compact and spin. If there is a
metric gF on F such that kgF> 0 over M , then A(M) = 0.
I Proof. Instead of working near s(M) ⊂ M as usual, we
work on the whole M.
I s ◦ π : M → s(M) looks like a vector bundle over s(M).
I Apply analytic Riemann-Roch property on M.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Vanishing theorem : outline of proof
I For any p ∈ M, let dMp denote the fiberwise distant onMp = π−1(π(p))
I Denote ρ(p) = dMp(p, s ◦ π(p)).
I For any R > 0, set
MR = {p ∈ M : ρ(p) ≤ R}
I MR is a smooth manifold with boundary
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Vanishing theorem : outline of proof
I On M, consider the Dirac type operator(called sub-Dirac operator, go back to Liu-Zhang 1999)
(1) Dβ,ε : Γ(Sβ,ε(F ⊕ F⊥)⊗Λ∗(T V M))
−→ Γ(Sβ,ε(F ⊕ F⊥)⊗Λ∗(T V M))
I Set
Dβ,ε,R = Dβ,ε +c(ρ dT
V Mρ)
β R.
I No R in the usual Riemann-Roch.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Vanishing theorem : outline of proof
I Key estimate 1 :
(Dβ,ε)2 = −∆β,ε +
kgF
4β2+ o
(1
β2
).
I This step uses essentially the almost isometric property ofthe Connes fibration.
I Key estimate 2 :[Dβ,ε,
c(ρ dTV Mρ)
β R
]= O
(1
β2R
).
I This step uses essentially the nonpositive curvatureproperty of each fiber Mp
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Vanishing theorem : outline of proof
I By key estimates 1 and 2 :(Dβ,ε,R
)2=
(Dβ,ε +
c(ρ dTV Mρ)
β R
)2
= (Dβ,ε)2 +
ρ2
β2R2+O
(1
β2R
)= −∆β,ε +
kgF
4β2+
ρ2
β2R2+O
(1
β2R
)+ o
(1
β2
)I Thus, when R >> 0 and β > 0, ε > 0 are very small,(
Dβ,ε,R
)2≥ kg
F
8β2
in the interior of MR.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Vanishing theorem : outline of proof
I Proposition. There are R >> 0, β, ε > 0 sufficientlysmall, such thata). Dβ,ε,R is invertible over the interior of MR ;
b). (Dβ,ε,R)|∂MR
is invertible on the boundary ∂MR.
I Proof of b). This is because ρ = R on ∂MR. QED
I By anaytic Riemann-Roch,
0 = ind(Dβ,ε,R,+) = A(s(M)) = A(M).
I Easy modification gives a purely geometric proof ofthe Connes vanishing theorem.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Summary
I Theorem. M spin + kgF> 0 => A(M) = 0.
I if M4k (k > 1) also simply connected => kgTM
> 0.
I Open question : kgF> 0 => kg
TM> 0
I Positive anwser if dimM ≥ 5 and M simply connected.
I Theorem. On Tn, no kgF> 0.
(F = TM due to Schoen-Yau and Gromov-Lawson)
I General case still open.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Two more open questions (1)
I Theorem. M spin + kgF> 0 =>⟨
A(F )e(TM/F )p(TM/F ), [M ]⟩
= 0,
where e(TM/F ) is the Euler class of TM/F .
I Corollary. 〈A(F )e(TM/F )3, [M ]〉 = 0.
I Open question : e(TM/F )3 = 0 in general ?
I Bott : There is no integrable codimension 2 subbundle ofT (CP 2n+1) for n ≥ 2. (Reason : e(TM/F )4 = 0)
I Open question. How about n = 1, i.e. CP 3 ?I Partial answer : No if assume further that kg
F> 0.
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Two more open questions (2)
I Witten : M4 oriented, closed, b+2 > 1, spinc-structure c,
then kgTM
> 0 implies SW(M4, c) = 0,
where “SW” stands for the Seiberg-Witten invariant.
I Question : (M4, F ) foliation, · · ·
whether kgF> 0 implies SW(M4, c) = 0 ?
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
IntroductionScalar curvatures on foliations
The Vanishing theorem
A Lichnerowicz vanishing theorem on foliationsSummary
Thanks !
Weiping Zhang Chern Institute of Mathematics Workshop on Geometric Analysis Dalian, September 1st, 2016Positive scalar curvature on foliations
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