# Tautness for riemannian foliations on non-compact manifolds

**ABSTRACT** For a riemannian foliation F{\mathcal{F}} on a closed manifold M, it is known that F{\mathcal{F}} is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form

km\kappa_\mu (relatively to a suitable riemannian metricμ) is zero (cf. Álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally

orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group Hn(M/F)H^{n}\,(M\,/\,{\mathcal{F}}) , where n = codimFn = {\rm codim}\,{\mathcal{F}} (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471,

1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group H0km(M/F)H^{0}_{\kappa_\mu}(M\,/\,{\mathcal{F}}) , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results

for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation

on a compact manifold (CERF).

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**ABSTRACT:**In the first part, we compute the Haefliger cohomology of uniform complete Riemannian foliations by showing the duality between the Haefliger cohomology and the basic cohomology. In the second part, we will give a characterization of strongly tenseness condition of foliated manifolds in terms of the Haefliger cohomology. By combining the results of both parts, we prove that any uniform complete Riemannian foliation on any possibly noncompact manifold is strongly tense.09/2012; - SourceAvailable from: Martintxo Saralegi-Aranguren[Show abstract] [Hide abstract]

**ABSTRACT:**We prove that the basic intersection cohomology IH*[`(p)](M / F){{\mathbb H}^{*}_{\overline{p}}({M / \mathcal{F}})} , where F{\mathcal F} is the singular foliation determined by an isometric action of a Lie group G on the compact manifold M, is finite dimensional. KeywordsBasic intersection cohomology–Lie group actionsMathematische Zeitschrift 01/2014; · 0.88 Impact Factor - SourceAvailable from: Martintxo Saralegi-Aranguren[Show abstract] [Hide abstract]

**ABSTRACT:**We prove that the basic intersection cohomology $IH^*_\overline{p}(M/\mathcal{F}$, where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, verifies the Poincaré Duality Property.Preprint. 01/2014;

Page 1

arXiv:math/0505675v2 [math.DG] 19 Nov 2005

Tautness for riemannian foliations

on non-compact manifolds

Jos´ e Ignacio Royo Prieto∗

Universidad del Pa´ ıs Vasco.

Martintxo Saralegi-Aranguren†

Universit´ e d’Artois

Robert Wolak‡

Uniwersytet Jagiellonski

1 F´ evrier 2008

Abstract

For a riemannian foliation F on a closed manifold M, it is known that F is taut (i.e.

the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean

curvature form κµ (relatively to a suitable riemannian metric µ) is zero (cf. [1]). In the

transversally orientable case, tautness is equivalent to the non-vanishing of the top basic

cohomology group H

[9]) this last condition is equivalent to the non-vanishing of the basic twisted cohomology

group H

When M is not compact, the tautness class is not even defined in general. In this work,

we recover the previous study and results for a particular case of riemannian foliations on

non compact manifolds: the regular part of a singular riemannian foliation on a compact

manifold (CERF).

n(M/F), where n = codim F (cf. [10]). By the Poincar´ e Duality (cf.

0

κµ(M/F), when M is oriented.

The study of taut foliations (foliations where all the leaves are minimal submanifolds for

some riemannian metric) has been an important part of the research in regular foliations on

riemannian manifolds: F. Kamber and Ph. Tondeur (cf. [8]), H. Rummler (cf. [15]), as well as

D. Sullivan (cf. [20]) were the first ones to present algebraical or variational characterizations of

such foliations. A. Haefliger’s paper [7] proved to be an important step in the development of the

theory. He showed that “being taut” is a transverse property, i.e. it depends only on the holonomy

pseudogroup of the foliation F. This led Y. Carri` ere (cf. [3]) to propose a characterization of taut

riemannian foliations on a compact manifold M as those foliations for which the top dimensional

basic cohomology group H

concise presentation of the history of the basic cohomology and tautness we refer to V. Sergiescu’s

appendix [18] in [14], which shows a close relation between the finiteness of basic cohomology,

Poincar´ e Duality Property in basic cohomology and tautness.

n(M/F) is non-trivial, i.e. isomorphic to R, being n = codim F. For a

∗Departamento de Matem´ atica Aplicada. Escuela Superior de Ingenier´ ıa. Universidad del Pa´ ıs Vasco. Alameda

de Urquijo s/n. 48013 Bilbao. Espa˜ na. joseignacio.royo@ehu.es. Partially supported by the UPV - EHU grant

127.310-E-14790/2002 by a PostGrant from the Gobierno Vasco - Eusko Jaurlaritza and by the MCyT of the

Spanish Government.

†UPRES-EA 2462 LML. Facult´ e Jean Perrin. Universit´ e d’Artois. Rue Jean Souvraz SP 18. 62 307 Lens Cedex

- France. saralegi@euler.univ-artois.fr. Partially supported by the UPV - EHU grant 127.310-E-14790/2002

‡Instytut Matematyki. Uniwersytet Jagiellonski. Wl. Reymonta4, 30 059 Krakow - Poland. wolak@im.uj.edu.pl

Partially supported by the KBN grant 2 PO3A 021 25.

1

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

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The problem was positively solved by X. Masa in [10], in the following way: if M is compact

and oriented and F is riemannian and transversally oriented, then

(1)

F is taut if and only if H

n(M/F) = R.

The work of A.´Alvarez (cf. [1]) removes the orientability condition on M and gives another

characterization of taut riemannian foliations on a compact manifold. He constructs a cohomo-

logical class κ ∈ H

F. This class is defined from the mean curvature form κµof a bundle-like metric µ, which we can

suppose to be basic due to D. Dom´ ınguez (cf. [5]).

We have a third cohomological characterization of the tautness of F. The Poincar´ e Duality of

[9] implies that this property is equivalent to the non-vanishing of the basic twisted cohomology

group H

1(M/F), the tautness class, whose vanishing is equivalent to the tautness of

0

κ(M/F) when M is orientable.

The situation is more complicated when the manifold M is not compact. For example, the

mean curvature form κµmay be a basic form without being closed (cf. [4]).

We consider in this work a particular case of a riemannian foliation F on a non-compact

manifold M: the Compactly Embeddable Riemannian Foliations or CERFs. In this context, we

have a compact manifold N endowed with a singular riemannian foliation H (in the sense of

[14]) in such a way that M is the regular stratum of N with F = H|M. We consider a class of

bundle-like metrics on M for which we construct a tautness class κ = [κµ] ∈ H

is independent of the choice of µ. We prove that the tautness of F is equivalent to any of the

following three properties:

1(M/F) which

- κ = 0,

- H

0

κµ(M/F) ?= 0,

- H

n

c(M/F) = R, where n = codim F, when F is transversally oriented.

Notice that in the second characterization we have eliminated the orientation hypothesis. We also

prove that the cohomology groups H

The standard method to prove the equivalence of the second and third conditions is to use

the Poincar´ e Duality Property (PDP in short). However, although reasonable, the PDP for the

basic cohomology has been proved neither for (M,F) nor for (N,H). So, we shall proceed by

proving that the first condition is equivalent to either of the two remaining ones. The proof of the

equivalence of the first and second condition is purely algebraic. To obtain the second equivalence

we use Molino’s desingularisation (? N,? H) of (N,H). The key point is the following.

F is taut ⇐⇒? H is taut.

Note that this equivalence cannot be extended to the singular riemannian foliation H itself, since

the existence of leaves with different dimensions implies the non-existence of “minimal metrics.”

Then the comparison of the corresponding basic cohomology groups completes the proof. In this

way we have avoided in the proof any reference to the PDP of the basic cohomology of the foliated

manifolds we are studying.

0

κ(M/F) and H

n

c(M/F) are 0 or R.

In the sequel M and N are connected, second countable, Haussdorff, without boundary and

smooth (of class C∞) manifolds of dimension m. All the maps considered are smooth unless

something else is indicated.

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3

1Riemannian foliations1

The framework category of this work is that of CERFs. They are riemannian foliations embedded

in singular riemannian foliations on compact manifolds. Before introducing this notion, we need

to recall some important facts about singular riemannian foliations.

1.1

partition H by connected immersed submanifolds, called leaves, verifying the following properties:

The SRFs . A Singular Riemannian Foliation2(SRF for short) on a manifold N is a

I- The module of smooth vector fields tangent to the leaves is transitive on each leaf.

II- There exists a riemannian metric µ on N, called adapted metric, such that each geodesic

that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets.

The first condition implies that (N,H) is a singular foliation in the sense of [19] and [21]. Notice

that the restriction of H to a saturated open subset produces an SRF. Each (regular) Riemannian

Foliation (RF for short) is an SRF, but the first interesting examples are the following:

- The orbits of the action by isometries of a Lie group.

- The closures of the leaves of a regular riemannian foliation.

1.2

a stratification SHof N whose elements are called strata. The restriction of H to a stratum S

is a RF HS. The strata are ordered by: S1 ? S2 ⇔ S1 ⊂ S2. The minimal (resp. maximal)

strata are the closed strata (resp. open strata). Since N is connected, there is just one open

stratum, denoted RH. It is a dense subset. This is the regular stratum, the other strata are the

singular strata. The family of singular strata is written S

the dimension of the biggest leaves of H.

The depth of SH, written depth SH, is defined to be the largest i for which there exists a chain

of strata S0≺ S1≺ ··· ≺ Si. So, depth SH= 0 if and only if the foliation H is regular.

The depth of a stratum S ∈ SH, written depthHS, is defined to be the largest i for which there

exists a chain of strata S0≺ S1≺ ··· ≺ Si= S. So, depthHS = 0 (resp. depthHS = depth SH)

if and only if the stratum S is minimal (resp. regular). For each i ∈ Z we write

Stratification. Classifying the points of N following the dimension of the leaves one gets

sin

H. The dimension of the foliation H is

Σi= Σi(N) = ∪{S ∈ SH| depthHS ≤ i}.

We have Σ<0 = ∅, Σdepth SH−1= N\RH and Σi = N if i ≥ depth SH. The union of closed

(minimal) strata is Σ0.

1.3

Compactly Embeddable Riemannian Foliation (or CERF) if there exists a connected compact

manifold N, endowed with an SRF H, and a foliated imbedding (M,F) ⊂ (N,H) such that M is

the regular stratum of SH, that is, M = RH. We shall say that (N,H) is a zipper of (M,F). Both

manifolds, M and N, are connected or not at the same time.

When M is compact and F is regular then (M,F) is a CERF and (M,F) the zipper. But in the

general case, the zipper may not be unique. The foliated manifold (M,F) = (S3×]0,1[,Hopf×I)),

with Hopf the Hopf foliation of S3and I the foliation of ]0,1[ by points, is a CERF possessing

two natural zippers (Ni,Hi), i = 1,2:

The CERFs. Consider a riemannian foliation F on a manifold M. We say that F is a

1For the notions related to riemannian foliations we refer the reader to [14, 23].

2For the notions related to singular riemannian foliations we refer the reader to [2, 13, 14, ?].

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

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- N1= S4and H1is given by the orbits of the S

S4= {(z1,z2,t) ∈ C × C × R/|/|z1|2+ |z2|2+ t2= 1}.

1-action: z ·(z1,z2,t) = (z ·z1,z ·z2,t), where

- N2= CP2and H2is given by the orbits of the S

1-action: z · [z1,z2,z3] = [z · z1,z · z2,z3].

We consider in the sequel a manifold M endowed with a CERF F and we fix (N,H) a zipper.

We present the Molino’s desingularisation of (N,H) in several steps.

1.4

of the riemannian manifold (N,H,µ). So it possesses a tubular neighborhood (TS,τS,S). Recall

that associated with this neighborhood we have the following smooth maps:

Foliated tubular neighborhood. A singular stratum S ∈ S

sin

H is a proper submanifold

+ The radius map ρS: TS → [0,1[, which is defined fiberwise by z ?→ |z|. Each t ?= 0 is a

regular value of the ρS. The pre-image ρ−1

S(0) is S.

+ The contraction HS: TS× [0,1] → TS, which is defined fiberwise by (z,r) ?→ r · z. The

restriction (HS)t: TS→ TSis an imbedding for each t ?= 0 and (HS)0≡ τS.

These maps verify ρS(r · u) = r · ρS(u) and τS(r · u) = τS(u). This tubular neighborhood can be

chosen so that the two following important properties are verified (cf. [14]):

(a) Each (ρ−1

S(t),H) is a SRF, and

(b) Each (HS)t: (TS,F) → (TS,F) is a foliated map.

When this happens, we shall say that (TS,τS,S) is a foliated tubular neighborhood of S. The

hypersurface DS = ρ−1

depth SHTS.

There is a particular type of singular stratum we shall use in this work. A stratum S is a

boundary stratum if codimNH = codimSHS− 1. The reason for this name is well illustrated

by the following example. The usual S

foliation H with two singular leaves, two fixed points of the action. These points are the boundary

strata and we have N/H = [0,1]. The boundary ∂(N/H) is given by the boundary strata. In fact,

the link of a boundary stratum is a sphere with the one leaf foliation (see, for example, [16] for

the notion of link).

S(1/2) is the core of the tubular neighborhood. We have depth SHD<

1-action on S2by rotations defines a singular riemannian

In the sequel, we shall use the partial blow up

LS: (DS× [0,1[,H × I) → (TS,H),

which is the foliated smooth map defined by LS(z,t) = 2t · z. Here, I denotes the pointwise

foliation. The restriction

(2)

LS: (DS×]0,1[,H × I) → (TS\S,H)

is a foliated diffeomorphism.

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

5

1.5

endowed with two tubular neighborhoods TS1and TS2. We shall need TS2\TS1to be a tubular

neighborhood of S2\TS1, but this is not always achieved. To guarantee this property, we introduce

the following notion, which is inspired in the abstract stratified objects of [11, 22].

A family of foliated tubular neighborhoods {TS| S ∈ S

of (N,H) if, for each pair of singular strata S1,S2with S1? S2, we have

Foliated Thom-Mather system. In the proof of Lemma 2.3.1 we find two strata S1? S2

sin

F} is a foliated Thom-Mather system

(3)ρS1= ρS1◦τS2

onTS1∩ TS2= τ−1

S2(TS1∩ S2).

In these conditions we have the property:

(4)ρS1(r · z)

? ?? ?

HS2(z,r)

= ρS1(z),

for each r ∈ [0,1] and u ∈ TS1∩ TS2. We conclude that the restriction

?

is a foliated tubular neighborhood of S2\ρ−1(I) on N\ρ−1

The foliated diffeomorphism (2) becomes

(5)τS2:TS2\ρ−1

S1(I) ≡ τ−1

S2(S2\ρ−1

S1(I))

?

−→?S2\ρ−1

S1(I), where I ⊂ [0,1[ is a closed subset.

S1(I)?

(6)

LS2: (DS2\ρ−1

S1(I))×]0,1[,H × I) → ((TS2\S2)\ρ−1

S1(I),H)

Proposition 1.5.1 Each compact manifold endowed with an SRF possesses a foliated Thom-

Mather system.

Proof. See Appendix.

♣

1.6

kind but with smaller depth (see [13] and also [17]). The main idea is to replace each point of the

closed strata by a sphere.

We suppose that depth SH> 0. The union of closed (minimal) strata we denote by Σ0. We

choose T0a disjoint family of foliated tubular neighborhoods of the closed strata. The union of

the associated cores is denoted by D0. Let L0: (D0× [0,1[,H × I) → (T0,H) be the associated

partial blow up. The blow up of (N,H,µ) is

Blow up. The Molino’s blow up of the foliation H produces a new foliation? H of the same

L: (? N,? H, ? µ) −→ (N,H,µ)

where

- The manifold? N is

? N =

??

D0×] − 1,1[

???

(N\Σ0) × {−1,1}

???

∼,

where (z,t) ∼ (L0(z,|t|),t/|t|). Notice that D0×] − 1,1[ and (N\Σ0) × {−1,1} are open

subsets of? N with

(D0×] − 1,1[) ∩ ((N\Σ0) × {−1,1}) = D0× (] − 1,0[∪]0,1[).

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- The foliation? H is determined by

Here, I denotes the 0-dimensional foliation of ] − 1,1[.

? H|D0×]−1,1[= H|D0× I and

? H|(N\Σ0)×{−1,1}= H|N\Σ0× I.

- The riemannian metric ? µ is

where f :? N → [0,1] is the smooth map defined by

f(v) =

f · (µ|D0+ dt2) + (1 − f) · µ|N\Σ0,

?

ξ(|t|) if v = (z,t) ∈ D0×] − 1,1[

if v = (z,j) ∈ (N\ρ−1

0

0([0,3/4]) × {−1,1},

with ξ: [0,1] → [0,1] a smooth map verifying ξ ≡ 1 on [0,1/4] and ξ ≡ 0 on [3/4,1[.

- The map L is defined by

L(v) =

?

L0(z,|t|) if v = (z,t) ∈ D0×] − 1,1[

z if v = (z,j) ∈ (N\Σ0) × {−1,1},

Notice that the blow up of (N,H,µ) depends just on the choice of ξ. So, we fix from now on such

a ξ.

1.6.1Remarks.

(a) The blow up of (N × R,H × I,µ + dt2) is just L × Identity : (? N × R,? H × I, ? µ + dt2) −→

(b) The manifold? N is connected and compact, the foliation? H is an SRF and ? µ is an adapted

(c) The map L is a foliated continuous map whose restriction to (? N\L−1(Σ0),? H) ≡ ((N\Σ0)×

(d) We shall denote by M1the regular stratum R? Hand F1the restriction of? H to M1. In

inclusion L−1(M) = M × {−1,1} ⊂ M1. We choose S1: M → M1a smooth foliated imbedding

with L◦S1= Identity. There are two of them.

(e) The stratification S? H. For each non-minimal stratum S ∈ SHthere exists a unique stratum

S?∈ S? H, with L−1(S) ⊂ S?, in fact, S?=

gives S? H= {S?| S ∈ SHand non-minimal}. We have the following important properties

- depthHS − 1 = depth? HS?, for each non-minimal stratum S ∈ SH,

- L−1(Σi\Σ0) = Σi−1(? N)\L−1(Σ0)) for each i ∈ Z, and

(f) We shall use the diffeomorphism σ:? N →? N defined by

σ(v) =

(z,−j) if v = (z,j) ∈ (N\Σ0) × {−1,1}.

(N × R,H × I,µ + dt2).

metric.

{−1,1},H × I) is the canonical projection on the first factor.

fact, the foliation F1is a CERF on M1and a zipper is given by (? N,? H). Notice that we have the

??

(D0∩ S)×] − 1,1[

???

S × {−1,1}

???

∼ . This

- depth S? H< depth SH.

?

(z,−t) if v = (z,t) ∈ D0×] − 1,1[

In fact, the diffeomorphism σ is a foliated isometry verifying L◦σ = L. It induces the smooth

foliated action Φ: Z2× M1→ M1defined by ζ · v = σ(v), where ζ is the generator of Z2.

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

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1.7

(cf. 1.6.1 (b)). In the end, we have a riemannian foliated manifold (? N,? H, ? µ) and a foliated

bundle (cf. 1.6.1 (d)). Notice that? N is connected and compact. This type of construction is a

We choose S: M →? N a smooth foliated imbedding verifying N◦S = Identity. It always

Molino’s desingularisation. If the depth of S? His not 0 then the blow up can be continued

continuous map N: (? N,? H) → (N,H), whose restriction N: N−1(M) → M is a smooth trivial

Molino’s desingularisation of (N,H,µ) (cf. [13]).

exists.

2 Tautness

The tautness of a RF on a compact manifold can be detected by using the basic cohomology.

This is not the case when the manifold is not compact. In this section we recover this result for

a CERF.

We consider in the sequel a manifold M endowed with a CERF F and we fix a zipper (N,H).

We also consider a Molino’s desingularisation N: (? N,? H) → (N,H). We shall write n = codim F.

2.1 Basic cohomology. Recall that the basic cohomology H

complex Ω

vector field X tangent to F.

An open covering {U,V } of M by saturated open subsets possesses a subordinated partition

of the unity made up of basic functions (see Lemma below). For such a covering we have the

Mayer-Vietoris short sequence

We fix S: M →? N a smooth foliated imbedding verifying N◦S = Identity.

∗(M/F) of basic forms. A differential form ω is basic when iXω = iXdω = 0 for every

∗(M/F) is the cohomology of the

(7)0 → Ω

∗(M/F) → Ω

∗(U/F) ⊕ Ω

∗(V/F) → Ω

∗((U ∩ V )/F) → 0,

where the maps are defined by restriction. The third map is onto since the elements of the partition

of the unity are basic functions. Thus, the sequence is exact.

The compactly supported basic cohomology H

plex Ω

The twisted basic cohomology H

ogy of the basic complex Ω

does not depend on the choice of the cycle: we have H

isomorphism: [ω] ?→ [efω].

Given V , a Z2-invariant saturated open subset of M, we shall write

?H

= {ω ∈ H

∗

c(M/F) is the cohomology of the basic subcom-

∗(M/F) | the support of ω is compact}.

∗

κ(M/F), relatively to the cycle κ ∈ Ω

∗(M/F) relatively to the differential ω ?→ dω−κ∧ω. This cohomology

∗

κ(M/F)∼= H

∗

c(M/F) = {ω ∈ Ω

1(M/F), is the cohomol-

∗

κ+d f(M/F) through the

∗(V/F)?Z2

= {ω ∈ H

∗(V/F) | σ∗ω = ω}

?H

∗(V/F)?−Z2

∗(V/F) | σ∗ω = −ω}.

For the existence of the Mayer-Vietoris sequence (7) we need the following folk result, well-

known for compact Lie group actions and regular riemannian foliations.

Lemma 2.1.1 Any covering of M by saturated open subsets possesses a subordinated partition of

the unity made up of basic functions.

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

8

Proof. The closure L of a leaf L ∈ F is a saturated submanifold of M whose leaves are dense (cf.

[14]). So, the open subsets U and V are in fact F-saturated subsets.

The closure L of a leaf L ∈ F possesses a tubular neighborhood as in 1.4 (cf. [14]). Since the

family of these tubular neighborhoods is a basis for the family of saturated open subsets then it

suffices to construct the partition of unity relatively to the tubular neighborhoods. This is done

by using the radius maps.

♣

2.2

curvature form κµ ∈ Ω

fundamental form of the leaves and W the corresponding Weingarten map. Then,

?trace W(X) if X is orthogonal to the foliation F

0

Tautness reminder (compact case). Given a bundle-like metric µ on (M,F), the mean

1(M) is defined as follows (see for example [23]). Consider the second

κµ(X) =

if X is tangent to the foliation F.

When the manifold is compact (and then (N,H) = (M,F)), the following properties of κµare

well-known:

(a) The form κµcan be supposed to be basic, i.e., there exists a a bundle-like metric µ such

that its mean curvature form is basic (see [5]).

(b) If κµis basic, then κµis a closed form (see [23]). The Example 2.4 of [4] shows that the

compactness assumption cannot be removed: there, the mean curvature form is basic, but not

closed.

(c) The class κ = [κµ] ∈ H

This is the tautness class of F.

The mean curvature form contains some geometric information about F. Recall that the

foliation F is taut if there exists a riemannian metric µ on M such that every leaf is a minimal

submanifold of M. It is known (see [23]) that

1(M/F) does not depend on the metric, but just on F (see [1]).

F is taut ⇐⇒ the tautness class κ vanishes.

We also have the following cohomological characterizations for the tautness of F:

F is taut ⇐⇒ H

n(M/F) ?= 0,

when F is transversally oriented. We also have that

F is taut ⇐⇒ H

0

κµ(M/F) ?= 0,

when M is oriented and F is transversally oriented.

Immediate examples of taut foliations are isometric flows (i.e. 1-foliations induced by the

orbits of a nonvanishing Killing vector field), isometric actions on compact manifolds and compact

foliations with locally bounded volume of leaves (foliations where every leaf is compact, see [6],[15]).

In the example of [4] referred above, we find a non-compact manifold where the tautness class

may not exist.

These results are not directly extendable to the framework of singular riemannian foliations,

as we can see in the following example: the usual S

riemannian foliation H with two singular leaves, two fixed points. Notice that H

there cannot exist a metric on S2such that the one dimensional orbits are geodesics (in dimension

1-action on S2by rotations defines a singular

1(N/H) = 0. But

Page 9

Tautness for riemannian foliations on ... 1 F´ evrier 2008

9

1, “minimal” implies “geodesic”). To see this, it suffices to consider a totally convex neighborhood

U of one of the fixed points: it would contain as geodesics some orbits (full circles, indeed), apart

from rays, which would contradict uniqueness of geodesics connecting any two points of U. In

higher dimensions (i.e. the cone of a sphere), one may consider the volume of every leaf. It should

be constant by minimality, which would give a positive volume to the vertex (for more details see

[12]).

2.3

would play a rˆ ole similar to that of the tautness class of a regular riemannian foliation on a regular

manifold. First, we introduce the notion of D-metric. A bundle-like metric metric µ on (M,F)

is a D-metric if the mean curvature form κµ is a basic cycle. The tautness class of F is the

cohomological class κ = [κµ] ∈ H

defined and independent of the D-metric.

Construction of κ. We would like to define a cohomological class κ ∈ H

1(M/F) which

1(M/F). The next Proposition proves that this class is well

Lemma 2.3.1 There exists a saturated open subset U ⊂ M such that

(a) the inclusion υ: U ֒→ M induces the isomorphism υ∗: H

∗(M/F) → H

∗(U/F), and

(b) the closure U (in N) is included in M.

Proof. We consider {TS | S ∈ SF} a foliated Thom-Mather system of (N,H) (cf. Proposition

1.5.1). For each i ∈ Z we write:

- τi: Ti→ Σi\Σi−1the associated foliated tubular neighborhood of Σi\Σi−1.

- ρi: Ti→ [0,1[ its radius function, and

- Dithe core of Ti.

The family {M∩T0,M\ρ−1

sion ((M ∩T0)\ρ−1

since it is foliated diffeomorphic to the inclusion

0([0,7/8])} is a saturated open covering of M. Notice that the inclu-

0([0,7/8]),F) ֒→ (M ∩T0,F) induces an isomorphism for the basic cohomology

((M ∩ D0)×]7/8,1[,F × I) ֒→ ((M ∩ D0)×]0,1[,F × I)

(cf. (2)). From the Mayer-Vietoris sequence we conclude that the inclusion M\ρ−1

induces the isomorphism

H

0([0,7/8]) ֒→ M

∗(M/F) = H

∗?M\ρ−1

0([0,7/8])/F?

(cf. (7)).

Notice now that M\ρ−1

family

0([0,7/8]) is the regular part of (N\ρ−1

0([0,7/8]),H). Moreover, the

{TS\ρ−1

0([0,7/8]) | S ∈ SF,dimS > 0}

is a foliated Thom-Mather system of (N\ρ−1

(using the foliated diffeomorphism (6) instead of that of (2)) gives

0([0,7/8]),H) (cf. (5)). The same previous argument

H

∗?M\ρ−1

0([0,7/8])/F?= H

∗?M\(ρ−1

0([0,7/8]) ∪ ρ−1

1([0,7/8]))/F?.

So, one gets the isomorphisms

H

∗(M/F) = ··· = H

∗?M\(ρ−1

0([0,7/8]) ∪ ··· ∪ ρ−1

p−1([0,7/8]))/F?,

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

10

where p = depth SH. Take U = M\(ρ−1

subset of F included on M. This gives (a).

Consider K = M\(ρ−1

We compute its closure on N:

0([0,7/8])∪···∪ρ−1

p−1([0,7/8])), which is an open saturated

0([0,6/8[) ∪ ··· ∪ ρ−1

p−1([0,6/8[)), which is a subset of M containing U.

K= M\(ρ−1

= N\?ρ−1

since N\M = Σp−1= ρ−1

therefore compact. This gives (b).

0([0,6/8]) ∪ ··· ∪ ρ−1

p−1([0,6/8])) ⊂ M\

p−1([0,6/8[)?= M\(ρ−1

0({0})∪···∪ρ−1

?

(ρ−1

0([0,6/8[))

◦∪ ··· ∪ (ρ−1

p−1([0,6/8[))

◦?

0([0,6/8[) ∪ ··· ∪ ρ−1

0([0,6/8[) ∪ ··· ∪ ρ−1

p−1([0,6/8[)),

p−1({0}). This implies that K is a closed subset of N and

♣

Proposition 2.3.2 The tautness class of a CERF exists and it does not depend on the choice of

the D-metric.

Proof. We proceed in two steps.

(i) - Existence of D-metrics. Since? N is compact then there exists a D-metric ν on? N. The

(ii) - Uniqueness of κ. Consider µ a D-metric on M and let ? κ be the tautness class of? H.

partition of unity associated to the covering {M,N\U} made up of basic functions (cf. Lemma

2.1.1). Notice that f ≡ 1 on U. So, it suffices to prove

metric S∗ν is a bundle-like metric on M. Since κS∗ν= S∗κνthen S∗ν is a D-metric on M.

It suffices to prove [κµ] = S∗? κ. Take U as in the previous Lemma and {f,g} a subordinated

(8)υ∗[κµ] = υ∗S∗? κ

(cf Lemma 2.3.1 (a)). Consider the following diagram:

UM

N,

N−1(U)

N−1(M)

? N

--

--

??????

ι′

*

?

6

S′′

?

6

S′

?

υ

υ′

ι

N′′

N′

S

N

where υ′, ι, ι′are the natural inclusions, N′, N′′are the restrictions of N and S′, S′′are the

restrictions of S. Notice the equalities S◦υ = ι◦υ′◦S′′and υ = N′◦υ′◦S′′.

The maps υ′, N′are foliated local diffeomorphism so κυ′∗N′∗µ= υ′∗N′∗κµ. This differential form

is a cycle since µ is a D-metric. Take on? N the riemannian metric

λ = N∗f · N′∗µ + (1 − N∗f) · ν

It is a bundle-like metric since f is basic and the support of f is included on M. Notice the

equality: υ′∗ι∗λ = υ′∗N′∗µ.

We can use this metric for the computation of ? κ in the following way. Let (κλ)bbe the

Ω

? N/? H

the proof of Lemma 2.1.1). From the definition of the basic component (κλ)bwe get that the

basic part of the mean curvature form κλ, relatively to the λ-orthogonal decomposition Ω

∗?

Since N−1(U) is? H-saturated open subset of? N then it is also a? H-saturated subset of? N (see

∗?? N

?

=

?

⊕ Ω

∗?? N/? H

?⊥. It is a basic cycle and we have ? κ = [(κλ)b] (see [1]).

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

11

restriction υ′∗ι∗(κλ)bis defined by using just (N−1(U),υ′∗ι∗? H,υ′∗ι∗λ). As a consequence, we have

Finally, we get

υ′∗ι∗((κλ)b) = κυ′∗ι∗λ= κυ′∗N′∗µ= υ′∗N′∗κµ.

υ∗S∗? κ = S′′∗υ′∗ι∗? κ = S′′∗υ′∗ι∗[(κλ)b] = S′′∗[υ′∗N′∗κµ] = S′′∗υ′∗N′∗[κµ] = υ∗[κµ].

This gives (8).

♣

2.3.3Remarks.

(a) The tautness class κ of (M,F) and the tautness class ? κ of a Molino’s desingularisation

verifying N◦S = Identity.

(b) Let κ1be the tautness class of (M1,F1). This class is Z2-invariant (cf. 1.6). This comes

from the fact that the diffeomorphism σ preserves H1and therefore κσ∗µ= σ∗κµfor a D-metric µ

on M1. Thus, the metric σ∗µ is also a D-metric and we obtain: σ∗κ1= [σ∗κµ] = [κσ∗µ] = κ1.

(? N,? H) are related by the formula κ = S∗? κ, where S: M →? N is any smooth foliated embedding

2.4First characterization of tautness: vanishing of κ.

We give the first characterization of the tautness of F through the vanishing of κ. We lift the

question to the Molino’s desingularisation (? N,? H).

Lemma 2.4.1 The map S1: M → M1(cf. 1.6) induces the isomorphism:

?H

Proof. The open covering {(D0∩ M)×] − 1,1[,M × {−1,1}} of M1is a Z2-equivariant one (cf.

1.6.1 (f)). So, from Mayer-Vietoris (cf. (7)) we get the long exact sequence

?

→

H

?

where I denotes the restriction map (i.e. induced by the inclusion). Since the natural projection

(D0∩ M)×] − 1,1[→ (D0∩ M) is Z2-invariant, then we get isomorphisms

?H

H

∗(M1/F1)?Z2∼= H

∗(M/F).

··· →

H

j((D0∩ M)×] − 1,1[/F × I)

?Z2⊕

?

H

j(M × {−1,1}/F × I)

?Z2→

?

?Z2

I

−→

?

j((D0∩ M) × (] − 1,0[∪]0,1[)/F × I)

?

H

j+1(M1/F1)

?Z2

I

−→

?Z2→ ··· ,

→

H

j+1((D0∩ M)×] − 1,1[/F × I)

?Z2⊕

H

j+1(M × {−1,1}/F × I)

H

∗((D0∩ M)/F)∼=

∗((D0∩ M)×] − 1,1[/F × I)?Z2

∗((D0∩ M)/F)∼=

?H

∗((D0∩ M) × (] − 1,0[∪]0,1[)/F × I)?Z2.

We conclude that the inclusion M × {−1,1} ֒→ M1induces the isomorphism

?H

Since the natural projection P : M × {−1,1} → M is Z2-invariant, then we get isomorphism

?H

∗(M1/F1)?Z2∼=?H

∗(M × {−1,1}/F × I)?Z2.

H

∗(M/F)∼=

∗(M × {−1,1}/F × I)?Z2.

Since P◦S1= Identity then S1induces the isomorphism?H

The main result of this section is the following.

∗(M1/F)?Z2∼= H

∗(M/F).

♣

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

12

Theorem 2.4.2 Let M be a manifold endowed with a CERF F. Then, the following two state-

ments are equivalent:

(a) The foliation F is taut.

(b) The tautness class κ ∈ H

1(M/F) vanishes.

Proof. We prove the two implications.

(a) ⇒ (b). There exists a D-metric µ on M with κµ= 0. Then κ = [κµ] = 0.

(b) ⇒ (a). We proceed by induction on depth S

case of [1]. When this depth is not 0 then we can consider the Molino’s desingularisation N =

L◦N1: (? N,? H) → (N,H) of (N,H), where N1: (? N,? H) → (? N,? H) is a Molino’s desingularisation

Write S1: M1→? N a smooth foliated imbedding verifying N1◦S1= Identity. The composition

Remark 2.3.3 we get that S∗

H

By induction hypothesis (depth S? H< depth SH) we get that H1is taut. So, the restriction of H1

sin

H. When this depth is 0 we have the regular

of (? N,? H) (cf. 1.6 and 1.7).

S = S1◦S1: M →? N is a smooth foliated imbedding verifying N◦S = Identity (cf. 1.6). From

1κ1= κ with κ1∈

?

1(M1/F1)

?Z2. The above Lemma gives κ1= 0.

to L−1(M) = M × {−1,1} is also taut. We conclude that F is also taut (cf. 1.6).

♣

2.4.3

related. In fact,

Remark. The proof of the above Theorem shows that the tautness of F and? H are closely

The foliation F is taut ⇐⇒ The foliation? H is taut.

The tautness class κ ∈ H

??

1(M/F) vanishes ⇐⇒

The tautness class ? κ ∈ H

1?? N/? H

?

vanishes.

2.5Second characterization of tautness: the bottom group H

We give a characterization of the tautness of F using H

case, this result comes directly from (1) and the Poincar´ e Duality of [8, 9] when M is oriented

and F is transversally oriented. In fact, we shall not need these orientability conditions.

0

κ(M/F).

0

κ(M/F). Notice that, in the compact

Theorem 2.5.1 Let M be a manifold endowed with a CERF F. Consider µ a D-metric on M.

Then, the following two statements are equivalent:

(a) The foliation F is taut.

(b) The cohomology group H

0

κµ(M/F) is R (cf. 2.1).

Otherwise, H

0

κµ(M/F) = 0.

Proof. We proceed in two steps.

(a) ⇒ (b). If F is taut then κ = [κµ] = 0. So, H

0

κµ(M/F)∼= H

0(M/F) = R.

(b) ⇒ (a). If H

The set Z(f) = f−1(0) is clearly a closed subset of M. Let us see that it is also an open subset.

0

κµ(M/F) ?= 0 then there exists a function 0 ?= f ∈ Ω

0(M/F) with df = fκµ.

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Tautness for riemannian foliations on ... 1 F´ evrier 2008

13

Take x ∈ Z(f) and consider a contractible open subset U ⊂ M containing x. So, there exists a

smooth map g: U → R with κµ= dg on U. The calculation

d(fe−g) = e−gdf − fe−gdg = e−gfκµ− e−gfκµ= 0

shows that fe−gis constant on U. Since x ∈ Z(f) then f ≡ 0 on U and therefore x ∈ U ⊂ Z(f).

We get that Z(f) is an open subset.

By connectedness we have that Z(f) = ∅ and |f| is a smooth function. From the equality

d(log|f|) =1

fdf = κµwe conclude that κ = 0 and then F is taut.

Notice that we have also proved: H

0

κµ(M/F) ?= 0 ⇒ H

0

κµ(M/F) = R. This ends the proof. ♣

2.6Third characterization of tautness: the top group H

We give a characterization of the tautness of F by using H

to a Molino’s desingularisation of F, where the result is known. But we need to formulate an

orientability condition on F.

n

c(M/F).

n

c(M/F). We lift the question

Lemma 2.6.1

The foliation F is transversally orientable ⇐⇒ The foliation? H is transversally orientable.

Proof. Since S: M → M1is a smooth foliated imbedding then we get “⇐”.

Consider O a transverse orientation on (M,F). The tubular neighborhood (T0∩M,F) inherits

the transverse orientation O. Since ((D0∩M)×]0,1[,F×I) is foliated diffeomorphic to (T0∩M,F)

then it inherits a transverse orientation, written O. This transverse orientation induces on the

product ((D0∩M)×]−1,1[,F ×I) a transverse orientation, written O. Notice that the involution

(x,t) ?→ (x,−t) reverses the orientation O.

Since

M1=(D0∩ M)×] − 1,1[

?????

M × {−1,1}

???

∼,

where (z,t) ∼ (2|t| · z,t/|t|), then it suffices to define on M1the transverse orientation O1by:

*

O on (D0∩ M)×] − 1,1[,

*

O on M × {1}, and

* −O on M × {−1}.

This gives “⇒”.

♣

Before passing to the third characterization, we need two computational Lemmas.

Lemma 2.6.2 The inclusion

Ω

∗

c(((D0∩ M)×]0,1[)/F × I) ֒→ Ω

∗

c(((D0∩ M)×] − 1,1[)/F × I)

induces an isomorphism in cohomology.

Page 14

Tautness for riemannian foliations on ... 1 F´ evrier 2008

14

Proof. For the sake of simplicity, we write E = (D0∩ M). Let f ∈ Ω

with f ≡ 0 on ] − 1,1/3] and f ≡ 1 on [2/3,1[. So, df ∈ Ω

proved if we show that the assignment [γ] ?→ [df ∧ γ] establishes the isomorphisms of degree +1

0

c(] − 1,1[) be a function

c(] − 1,1[). Lemma will be

1

c(]0,1[) ⊂ Ω

1

H

∗

c(E/F)∼= H

∗

c((E×]0,1[)/F × I) andH

∗

c(E/F)∼= H

∗

c((E×] − 1,1[)/F × I).

Let us prove the first one (the second one is proved in the same way). Consider the following

differential complexes:

?

?

?

Proceeding as in (7) we get the short exact sequence

- A

∗(]0,3/4[) =ω ∈ Ω

∗(E×]0,3/4[/F × I)

??

??

supp ω ⊂ K × [c,3/4[

for a compact K ⊂ E and 0 < c < 3/4

?

?

.

- A

∗(]1/4,1[) =ω ∈ Ω

∗(E×]1/4,1[/F × I)

supp ω ⊂ K×]1/4,c]

for a compact K ⊂ E and 1/4 < c < 1

??supp ω ⊂ K×]1/4,3/4[

.

- A

∗(]1/4,3/4[) =ω ∈ Ω

∗(E×]1/4,3/4[/F × I)

for a compact K ⊂ E

?

.

0 −→ Ω

∗

c(E×]0,1[/F × I) −→ A

∗(]0,3/4[) ⊕ A

∗(]1/4,1[) −→ A

∗(]1/4,3/4[) −→ 0.

The associated long exact sequence is

··· → H

i−1?A

∗(]1/4,3/4[)?

δ→ H

I?A

i

c(E×]0,1[/F × I) → H

∗(]1/4,3/4[)?→ ··· ,

i?A

∗(]0,3/4[)?⊕

⊕

H

i?A

∗(]1/4,1[)?→ H

where the connecting morphism is δ([ω]) = [df ∧ ω].

Before executing the calculation let us introduce some notation.Let β be a differential

form on Ω

i(DS×]a,b[) which does not include the dt factor. By

? c

−

β(s) ∧ ds and

? −

c

β(s) ∧

ds we denote the forms on Ω

?? c

?? −

and (? v1,...,? vi) ∈ T(x,t)(DS×]a,b[) .

i(DS×]a,b[) obtained from β by integration with respect to s,

? c

? t

that is,

−

β(s) ∧ ds

?

?

(x,t)(? v1,...,? vi) =

t

(β(x,s)(? v1,...,? vi)) ds and on the other hand

c

β(s) ∧ ds(x,t)(? v1,...,? vi)) =

c

(β(x,s)(? v1,...,? vi))ds where c ∈]a,b[, (x,t) ∈ DS×]a,b[

(i) Computing δ.

Each differential form ω ∈ A

contain dt.

Consider a cycle ω = α + β ∧ dt ∈ A

∗(Interval) can be written ω = α + β ∧ dt where α and β do not

i(]0,3/4[) with supp ω ⊂ K × [c,3/4[ for a compact

?? c/2

∗?A

K ⊂ E and 0 < c < 3/4. We have ω = α(c/2) − d

−

β(s) ∧ ds

?

= −d

?? c/2

−

β(s) ∧ ds

?

.

Since supp

? c/2

−

β(s) ∧ ds ⊂ K × [c,3/4[ then we get H

·(]1/4,1[)?= 0.

We conclude that δ is an isomorphism.

·(]0,3/4[)?= 0. In the same way, we

get H

∗?A

Page 15

Tautness for riemannian foliations on ... 1 F´ evrier 2008

15

(i) Computing H

∗

c((E×]0,1[)/F × I).

Consider a cycle ω = α + β ∧ dt ∈ A

∗(]1/4,3/4[). We have ω = α(1/2) + d

?? −

1/2

β(s) ∧ ds

?

.

Notice that supp

? −

1/2

β(s) ∧ ds ⊂ K×]1/4,3/4[ and supp α(1/2) ⊂ K. A standard procedure

shows that the operator ∆: H

an isomorphism. The inverse is ∆−1([γ]) = [γ].

So, the composition δ◦∆−1: H

exactly the operator: [γ] ?→ [df ∧ γ].

∗?A

·(]1/4,3/4[)?→ H

∗−1

c

(E/F) → H

∗−1

c

(E/F), defined by ∆([ω]) = [α(1/2)], is

∗

c((E×]0,1[)/F × I) is an isomorphism. It is

♣

The reason why we use the (−Z2)-invariant classes in the next Lemma instead of the more natu-

ral Z2-invariant classes is the following: we have?H

∗

c(]−1,0[∪]0,1[/I)?−Z2∼=

?H

∗

c(]−1,1[)/I)?−Z2

but also?H

Lemma 2.6.3 The inclusion

∗

c(] − 1,0[∪]0,1[/I)?Z2?∼=

?H

∗

c(] − 1,1[)/I)?Z2.

Ω

∗

c(((D0∩ M) × (] − 1,0[∪]0,1[))/F × I) ֒→ Ω

∗

c(((D0∩ M)×] − 1,1[)/F × I)

induces the isomorphism

?H

∗

c(((D0∩ M) × (] − 1,0[∪]0,1[))/F × I)?−Z2∼=?H

Proof. For the sake of simplicity, we write E = (D0∩ M). Consider f : ] − 1,1[→ R a smooth

function with f ≡ 0 on ] − 1,1/3] and f ≡ 1 on [2/3,1[. The function f◦σ + f − 1: ] − 1,1[→ R

is a smooth function whose support is in [−2/3,2/3]. So, [df] = −[d(f◦σ)] ∈ H

above Lemma gives

∗

c(((D0∩ M)×] − 1,1[)/F × I)?−Z2.

∗

c(] − 1,1[). The

- H

∗

c(E/F)∼= H

∗

c((E×] − 1,1[)/F × I) by [γ] ?→ [df] ∧ [γ],

- H

∗

c(E/F)⊕H

[df] ∧ γ2.

∗

c(E/F)∼= H

∗

c((E × (] − 1,0[∪]0,1[))/F × I) by ([γ1],[γ2]) ?→ [d(f◦σ)]∧[γ1]+

Notice that ζ ·df = −σ∗df = −d(f◦σ) on ]−1,1[ and ]−1,0[∪]0,1[. This gives the isomorphisms

of degree +1:

- H

∗

c(E/F)∼=?H

∗

c((E×] − 1,1[)/F × I)?−Z2by [γ] ?→ [df] ∧ [γ],

- H

∗

c(E/F)∼=?H

∗

c((E × (] − 1,0[∪]0,1[))/F × I)?−Z2by [γ] ?→ −1

We obtain the result, since on ] − 1,1[ we have −1

2[d(f◦σ)]∧[γ]+1

2[df]∧[γ].

2[d(f◦σ)] +1

2[df] =1

2[df] +1

2[df] = [df].

♣

The way to lift the question to the Molino’s desingularisation is the following.

Lemma 2.6.4

H

n

c(M1/F1) ?= 0 ⇐⇒ H

n

c(M/F) ?= 0.

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