Integral formulae for foliated manifolds with singularities Dedicated

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Integral formulae
for foliated manifolds with singularities
Dedicated to Professor A. A. Borisenko on his 70th birthday
Pawel G. Walczak
Katedra Geometrii, Uniwersytet Lódzki
[email protected]
Kharkiv, MAGT, September 12 – 16, 2016
Pawel G. Walczak Katedra Geometrii, Uniwersytet Lódzki
[email protected]
Integral formulae for foliated manifolds with singularities
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1. General setting
M = a closed Riemannian manifold of dimension n + 1
Σ = a union of finitely many closed, pairwise disjoint submanifolds
of codim ≥ k
F = a codimension-1 foliation F
everything smooth and oriented
N = the positive oriented unit normal
A = −∇N = the shape (Weingarten) operator (on T F)
k1 , . . . kn = the principal curvatures = the eigenvalues of A
σj = the j-th mean curvature = the j-th elementary symmetric
function of ki ’s, j = 1, . . . n
Paweł G. Walczak Katedra Geometrii, Uniwersytet Łódzki
[email protected]
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2. General idea
X = a “geometrically interesting” vector field on M r Σ
f = div X calculated in terms of geometry (intrinsic/extrinsic
curvatures etc.)
Then Hölder Inequality + some calculations imply the following
integral formula:
Lemma
If (k − 1)(p − 1) ≥ 1 and
R
M
kX kp d vol < ∞, then
Z
f d vol = 0.
(1)
M
Why interesting? Because Equation (1) implies, for example,
obstructions for the existence of foliations with given geometric
properties (geodesic, minimal, umbilical etc.)
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3. First formula
Theorem
(G. Reeb, ∼ 1950 for Σ = ∅)
Z
σ1 = − div X
and
σ1 d vol = 0.
(2)
M
if only k ≥ 2.
Proof.
Apply our Lemma to X = N.
Corollary
(P. W. 1984, G. Oshikiri 1997) If Σ = ∅, then:
f = σ1 for some Riemannian metric g ⇐⇒ either f = 0 everywhere
on M or f (x)f (y ) < 0 for suitable x, y ∈ M
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4. On space forms
Question
How to get the following from our general setting?
Theorem
(Asimov, Brito - Langevin - Rosenberg, ∼ 1980)
If M is of constant curvature, then
Z
σk d vol = ck ,
(3)
M
ck being a constant independent of F.
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5. A bit of algebra
B = (B1 , B2 , . . . , Bm ) – a sequence of quadratic n × n matrices (or,
linear transformations of a given vector space)
Definition
Invariants σλ (B) are given by
det(I + t1 B1 + . . . tm Bm ) =
X
σλ (B) · t λ ,
(4)
λ
λm .
where λ = (λ1 , . . . , λm ) is a multiindex and t λ = t1λ1 · . . . · tm
Definition
For λ as above,
kλk = λ1 + 2λ2 + . . . + λm .
(5)
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6. On symmetric spaces
Define:
RN = R(·, N)N : T F → T F
B2k =
(−1)k k
(2k)! RN ,
B2k+1 =
(−1)k
k
(2k+1)! RN A
Theorem
(V. Rovenski and P. W., ∼ 2008, Math. Ann. 2012)
If M is locally symmetric and Σ = ∅, then for any m > 0 one has
Z X
σλ (B) d vol = 0.
(6)
M kλk=m
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7. On arbitrary manifolds, no singularities
Tr – Newton transformations (of A : T F → T F):
T0 = I ,
Tr = σr I − ATr −1
for 1 ≤ r ≤ n = dim M.
(7)
Calculating div Xr , where Xr = Tr (∇N N) + σr +1 N we obtain:
Theorem
(K. Andrzejewski and P. W., Ann. Glob. Anal. Geom. 2010) If
Σ = ∅, then – for all r –
Z X
r
M
trF (R((−A)j−1 ∇N N)Tr −j ) − (r + 2)σr +2
j=1
+ trF (R(N)Tr ) d vol,
(8)
where R(Z ) : T F 3 X 7→ R(X , Z )N ∈ T F.
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8. On arbitrary manifolds, with singularities
Theorem
Assume that
(i) a codimension one foliation F is defined on M r Σ, Σ being a
union of finitely many, pairwise disjoint, closed submanifolds Si of
codimensions ki ≥ k ≥ 2,
(ii) the norm kAk of the Weingarten operator A of F is
L(r +1)p -integrable on M for some r ∈ N and p > 1 such that
(k − 1)(p − 1) ≥ 1, and
(iii) the curvature κ = k∇N Nk of the 1-dimensional foliation by
integral curves of the unit normal N of F is L(r +1)p -integrable.
Then, equality (8) holds.
Remark
Both, Eqs. (6) and (8), imply (3).
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9. Particular case: r = 2
Corollary
(T. Nora, Thése, 1983, for Σ = ∅)
If (k − 1)(p − 1) ≥ 1 and kAk, κ ∈ L3p then
Z
(2σ2 − Ric(N, N))d vol = 0.
(9)
M
Corollary
(R. Lanevgin, P.W., Geom. Ded., 2008, for Σ = 0)
Under the above integrability assumptions, codimension-one
foliations on manifolds of negative Ricci curvature are far from
being umbilical:
Z X
(∃C > 0) (∀F)
|ki − kj |n+1 ≥ C > 0.
(10)
M i<j
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10. Bibliography
K. Andrzejewski and P. Walczak, The Newton transformation
and new integral formulae for foliated manifolds, Ann. Glob.
Anal. Geom. 37 (2010), 103 – 111.
D. Asimov, Average gaussian curvature of leaves of foliations,
Bull. Amer. Math. Soc. 84 (1978), 131 – 133.
F. Brito, R. Langevin and H. Rosenberg, Intégrales de courbure
sur les variétées feuilletées, J. Diff. Geom. 16, 19 – 50.
T. Nora, Seconde forme fondamentale d’une application et
d’un feuilletage, Thése, l’Univ. de Limoges, 1983.
G. Reeb, Sur la courbure moyenne des variétés intégrales d’une
équation de Pfaff, C. R. Acad. Sci. Paris 231, (1950), 101 –
102.
V. Rovenski and P. Walczak, Integral formulae for foliated
symmetric spaces, Math. Ann. 352 (2012), 223 – 237.
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Thank You!
Dziękuję!
Cnacu6o!
Cnacu6i!
νχαριστ ω!
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Integral formulae for foliated manifolds with singularities Dedicated

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