Ann Glob Anal Geom

DOI 10.1007/s10455-015-9465-1

CrossMark

New integral formulae for two complementary orthogonal distributions on Riemannian manifolds

Magdalena Luzyiiczyk1 • Pawet Walczak1

Received: 28 January 2015 / Accepted: 19 April 2015

© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We derive and apply a new integral formula for a closed Riemannian manifold equipped with a pair of complementary orthogonal distributions (plane fields). The integrand depends on the second fundamental forms and integrability tensors of the distributions, their covariant derivatives, and of the Ricci curvature of the ambient manifold. Also, we discuss some applications of this formula and of another formula of this sort, the one obtained earlier by the second author, and show that both formulae may hold when the distributions are defined only outside a "reasonable" closed subset of the manifold under consideration.

Keywords Riemannian manifold • Distribution • Foliation • Integral formula

Mathematics Subject Classification Primary 53C15 • Secondary 53C12

Introduction

Integral formulae for foliated Riemannian manifold are almost that old as the foliation theory itself: already in 1950, Georges Reeb [12] has shown that the integral of the mean curvature of the leaves of a codimension-one foliation of a closed oriented Riemannian manifold is always equal to zero. Later on, Asimov [4] and Brito et al. [6] have shown that integrals of mean curvatures (of arbitrary higher order k) of codimension-one foliations F of closed manifolds M of constant curvature c depend only on k, c, volume and dimension of M, not on F. Next, one of this formulae has been extended to foliations of arbitrary Riemannian manifolds:

B Pawel Walczak

pawelwal@math.uni.lodz.pl

Magdalena LuZynczyk luzynczyk@math.uni.lodz.pl

1 Wydzial Matematyki i Informatyki, Uniwersytet Lódzki, Lódz, Poland

Published online: 19 May 2015

1 Springer

/ (2o2 - Ric(N))dvol = 0, (1)

where 02 is the second mean curvature of the leaves of F and N is a unit vector field orthogonal to M. In [19] (see also [11]) formula (1) has been generalized to foliations (and arbitrary distributions) of arbitrary codimension: the result of this generalization can be found here [Sect. 2, Eq. (4)] and has been applied by several authors in different contexts (see [5,7,14,17,18] etc.). Recently, first Rovenski and the second author [15] on symmetric spaces and then Andrzejewski [2] (see also [3]) on arbitrary Riemannian manifolds have found series of integral formulae for codimension-one foliations, formulae which extend (1) and all the equalities proved in [4] and [6]. For more about integral formulae for foliations, we refer to [16].

In this article, we consider a Riemannian manifold M equipped with two complementary orthogonal distributions D1 and D2. We propose a method of obtaining a series of formulae generalizing (4) and derive one of them (Sect. 2, Eq. (25)). We show that under some conditions both equations, (4) and (25), hold when our distributions D1 and D2 admit singularities, that is are defined on a closed manifold M apart from a finite union £ of closed submanifolds of sufficiently large codimension. This should be of some interest: existence of distributions on a closed manifold M depends on some topological conditions and implies existence of such distribution over open subsets of M, while the converse is not true. Finally, we collect some applications of (4) and (25): it occurs that our formulae provide obstructions for the existence of pairs of distributions satisfying particular geometrical conditions.

The main idea is (as in the case of formulae obtained earlier) to find a geometrically interesting vector field, calculate its divergence and use the classical Stokes Theorem. For distributions with singularities, we need a particular technical result (Lemma 1) which generalizes that from [8]. One of the other tools used in derivation of (25) is an analog of the classical Codazzi equation for non-integrable distributions (Proposition 1).

1 Preliminaries

Let D be a distribution on a Riemannian manifold (M, (, )). The second fundamental form B of D is defined as follows (see [13]). If X and Y are two vector fields tangent to D, then B(X, Y) is the orthogonal to D component of

2 (V xY + VyX ),

where V is the Levi-Civita connection on M . The trace H of B is called the mean curvature vector of D. Similarly, the integrability tensor T of D assigns to X and Y, as before two vector fields tangent to D, the orthogonal to D component of

2(VxY -VyX).

Hereafter, we deal with two orthogonal distributions Di and D2 on a Riemannian manifold (M, (, )). We put p = dim Di, q = dim D2 and m = dim M, and assume that p + q = m. For any v e T M, we write

where vT e D, and v1 e D2. Throughout the article, we shall use a local orthonormal frame ei,..., em adapted to Di and D2, i.e., we assume that is tangent to Di for i = 1,..., p and ea is tangent to D2 for a = p + 1,..., m.

With this notation, the second fundamental forms Bi of Di (i = 1, 2) are defined as follows:

B,(X 1, Yi) = 2 (Vx, Yi + Vy,Xi)1, B2(X2, Y2) = 1 (VX2 Y2 + Vy2X2)T

for vector fields Xi and Yi tangent to Di.

Similarly, the integrability tensors Ti of Di (i = 1, 2) are defined by

Ti(X 1, Yi) = 2[X1, Yi]1, T2(X2, Y2) = 1 [X2, Y2]t

for vector fields Xi , Yi e Di . Therefore, the distribution Di is integrable (and defines a foliation) if and only if Ti = 0.

Then, the mean curvature vectors Hi of Di are given by

Hi = Trace B, = ^ B,(ei, ei) = ^(Veiei)L ii

H2 = Trace B2 = ^ B2(ea, ea) = ^(Veaea)T. ai

Let us define also the Weingarten operators A1 : D, x D2 ^ D, and A2 : D2 x D, ^ D2 of our distributions D, and D2, respectively, by

<A,(X, N), Y) = <B,(X, Y), N) for X, Y e D,, N e D2,

(A2(X', N'), Y') = (B2(X', Y'), N') for X', Y' e D2, N' e D,.

Finally, let us define the transformations

C, = Ai(-, Hi) : Di ^ Di, and C2 = A2O, H2) : D2 ^ D2.

Since Ci, i = 1, 2, are endomorphisms, we can compose

Ck = Ci O ... O Ci

k-times and consider the vector fields

Zk = Ck (H2) + Ck (Hi). (2)

Certainly, Zk, k = 0, 1, 2,..., are global vector fields on M defined by means of geometry of M, D, and D2.

2 The formulae

For k = 0 in (2), we get immediately

Zo = H, + H2.

The divergence div(H1 + H2) was calculated in [19]:

div(H1 + H2) = K(D1, D2) + IIB1II2 +IIB2II2

- I IH11 I 2 -I I H2 I I 2 -I I T1 I I 2 -I I T2 I I 2, (3)

where K (D1, D2) is a generalization of the Ricci curvature given by

K(D1, D2) = X<R(ei, e")e"> ei)

and called the mixed scalar curvature. If M is closed and oriented, integrating both sides of (3) and using the Stokes Theorem, we get the integral formula

f {K(Di, D2) +| | Bi| | 2 + | | B2| | 2 Jm

- 11 Hi | | 2 - | | H2 | | 2 - | | Ti | | 2 - | | T2 | | 2) dvol = 0. (4)

Now, we shall take k = 1 in (2). As we shall see, the calculation and the resulting integral formula (25) below, are rather complicated, but still applicable. Certainly, one can work also with the fields Zk for k > 1 to get integral formulae of the form

/ div Zk dvol = 0

with div Zk expressed in terms of Bi, Ti, Hi, R and their covariant derivatives, but it seems to us that the formulae obtained in this way will be even more complicated and less interesting than (25), so we decided not to proceed this way, but to concentrate on collecting applications of (4) and (25).

First, consecutive calculations yield the following:

div Z1 = div (A1(H2, H1) + A2(H1, H2))

= ZK (A1H2, H1)), ei) + £ (Ve, (A2(H1, H2)), ei)

^ (Vea (A1(H2, H1)) , ea) + £ (Vea (A2(H1, H2)) , ea)

= ^ (Vei (A1(H2, H1)) , ei) + £ (Vea (A2(H1, H2)) , ea)

^ [ei <A2 (H1, H2), ei) - (A2 (H1, H2), Vee)]

^ [ea <A1(H2, H1), ea) - (A1(H2, H1), Veaea)],

div Z1 = X (Vei (A1(H2, H1)), ei) ^ (Vea (A2(H1, H2)), ea)

■ X [(A2(H1, H2), (Veiei+ (A2(H1, H2), (Veiei)T(] i

■X[(A1(H2, H1), {Veaea)T + (A1(H2, H1), (Veaea

= T.(Vei (Ai(H2, Hi)), ei) + X(Vea (Ai(Hi, H2)), ea)

- IA2(Hi, H2), X {Vee- |Ai(H2, Hi), Ç (Veaea

divZ1 = X{(VeiAi)(H2, Hi), ei) + ^((VeaA2)(Hi, H2), ea)

- {A2(Hi, H2), Hi) - {Ai(H2, Hi), H2). (5)

Next, for the second fundamental form B of an arbitrary distribution D (in particular, D = Di or D = D2) and arbitrary vectors X, Y, Z e D, applying the connections (all of them being denoted by V) induced by the Levi-Civita connection on M in different bundles (D, D1 etc.) we get

(VxB) (Y, Z) - (VyB) (X, Z)

= 2 ((VxVyZ)1 - (VyVxZ)1 - (V[xj]T Z)1 - (Vz[X, Y]T)X

+ (Vx(VzY- (VY(VzX

- {V(VxZ)T Y)1 + {V(Vyz)TX)^ . (6) Applying the identity [X, Y] = VxY - VyX, we obtain the relations

{V(VYZ)TX)1 = (VX(VyZ)tY + [(VyZ)t, X]1, (7)

{V(VXZ)TY)1 = (VY(VxZ)tY + [(VxZ)T, Y]1, (8)

(VX (VzY = [Vx [(VyZ)1 + [Z, Y J1)]1, (9)

(VY(VzX= [VY {(VxZ)1 + [Z, X]1)]1, (i0)

2[Vz ([X, Yf)]1 = iVx,y]TZ)1 + T (z, (VxY)T) - T (Z, (VyX)t) . (11)

Comparing equalities (6)-(7) with the definition of the curvature tensor R we obtain the following

Proposition 1 For any vectors X, Y and Z belonging to a distribution D on a Riemannian manifold M on has

(VxB) (Y, Z) - (VyB) (X, Z) = (R(X, Y), Z)1 + 2 (Vt(x,y)Z)1 + (VxT) (Z, Y) - (VyT) (Z, X). (12)

Equation (12) can be considered as the Codazzi equation for D, an analog of the Codazzi equation for surfaces and arbitrary submanifolds of Riemannian spaces.

Coming back to our pair of distributions Di and D2, we observe that for the operator Ai : Di x D2 ^ Di, vector fields X, Y, Z in Di and N in D2 we have

(VxAi(Y, N}, Z) = X <Ai(Y, N}, Z) - ^Ai(Y, N}, (VxZ}T)

= [(VxBi(Y, Z}}1, N + (Bi(Y, Z}, (VxN- ^Bi(Y, (VxZ}T}, n) (i3)

and similarly

(VyAi(x, N}, Z)

= ((VyBi(x, Z}}1, N + (Bi(x, Z}, (VyN- (Bi (x, (VyN}t}, N) . (i4)

Applying the Codazzi equation to the operator Bi and comparing equalities (i2), (i3), (i4), we obtain another equation for the operator Ai and vector fields x, Y, Z in Di and N in D2:

<(VxAi} (Y, N}, Z) - <(VyAi} (x, N}, Z) = ((R(x, Y} Z}1 + 2 (Vri(x,Y}Z)1 + (VxTi} (Z, Y} - (VyTi} (Z, x}, n) = ((R(x, Y} Z}1, N + 2((Vt1(x,y}Z)1, N) + ((VxTi} (Z, Y}1, N - ((VyTi} (Z, x}1, N. (15)

For the second fundamental form B2 : D2 x D2 ^ Di, operator A2 : D2 x Di ^ D2, xYZ' in D2 and N' in Di we get similarly

(Vx' B2}(Y', Z'} - (Vy' B2} (xZ'} = (R(x', Y'} Z')T + 2 (Vtkx'.yoZ')T + (Vx>T2} (Z', Y'} - (VyT2} (Z', x'} (16)

((Vx'A2} (Y', N'}, Z') - ((Vy'A2} (x', N'}, Z') = ((R(x', Y'} Z')J , N')

+ 2((Vtkx'.yOZ')T , N') + ((Vx-T2} (Z', Y'}T, N') - ((Vr T2} (Z', x'}T, N'). (17)

Moreover, applying Eq. (15) to x = Z = ei, Y = H2 and N = Hi we obtain consecutively

X(V*Ai(H2, Hi}, ei)

= Z (((Ve, Ai}(H2, Hi}, ei) + A ((VeiH2}T, Hi) , e^ + (Ai (H2, (VeiHi, ei))

= Z (((VH2 Ai) (e,, Hi}, e,) + ((R(e,, H2} et}1, Hi) + 2 ((V^e,^i)1, Hi) i

+ ((VeiTi) (ei, H2}1, Hi) - ((VH2Ti) (ei, ei}1, Hi) + (Ai ((VeiH2}T, Hi) , e^ + (Ai(H2, (VeHi}1) , ei))

= Z , Hi), ei) - (Ax ((Vh2ei)T, Hi) , e^ - (Ai (et, (VH2Hi)1) , et)

+ ((R(ei, H2)ei)1 , Hi) + 2((V^,h2) ei)X , Hi)

+ ((Ve(Ti) (ei, H2)1, Hi) + (Ai ((VeH2)T, Hi) , e^ + (Ai (H2, (VeiHi)L) , e^) ,

Y,[VeiAi(.H2, Hi), ei)

= £ (H2 <Ai(ei, Hi), ei) - (Ai(ei, Hi), (VH2ei)T) - (Bi ((VH2ei)T, e^ , Hi)

- (fix (ei, ei), (VH2Hi)x) + (Ai ((VeiH2)J, Hi) , e^ + (Ai (H2, (VeiHi, et) + ((R(ei, H2) ei )1 , Hi) + 2 ((V^, H2 )ei )1 , Hi) + ((V^Ti) (ei, H2 Hi))

Y,(VeiAi(H2, Hi), ei)

= H2IIHi||2 - (Hx,(Vh2Hi)1)

+ Z (((R(ei, H2) ei)1, Hi) + 2((V^e,H2)ei)X , Hi) i

+ ((VeiTi) (ei, H2Hi) - 2(Bi ((VH2ei)T, e^ , Hi) + (Ai ((Ve,H2)T, Hi) , ei) + (Ai(H2, (VeHi)1) , ei)) . (18)

Analogously, applying Eq. (17) to X' = Z' = ea, 7' = Hi and N' = H2 we get

^(Vea A2(Hi, H2), ea) = HiII H2II2 - (H2, (Vhi H2)T)

+ Z (((R(ea, Hi) e„)T , H2) + ^(Vr2(ea,Hi)e„)T , H2)

+ ((VeaT2) (ea, Hi)T, H2) - 2(B2 ((Vhiea)1, e^ , H2) + (A2 [(Vea Hi)1, H2) , ea) + (A2( Hi, (Ve„ H2)T) , e^) . (19)

Next, let us observe that

£ (Ai ((Vei H2)T, Hi) , e^ = X (Ai (ey, Hi), ei) ^ (Ve, H2V)

= £ (Ai (ei, Hi), ej) [ej ,(VeiH2)J) = ^H1, V.TH2) (20)

and similarly

((VeaHi)1, H2) , ea) — ^(Á2(ea, H2), ep)(ep, (Ve„Hi}1)

— IaH2, V^H2). (21)

The well-known properties of the curvature tensor R imply the equality

X ((R(ei, H2) ei)1 , HA + X ((R(ea, Hi) e„)T , H2) = <Ric(H2), Hi). (22)

Finally, it is easy to show that

XK1 (VH2e,)T, eA — 0 and £ A2 (Vh, ea)1, ea) — 0. (23)

Applying Eqs. (18)-(23) to (5) we end up with the following Proposition 2 In the situation considered above, one has

div (Ai(H2, Hi) + A2(Hi, H2)) = <Ric(H2), Hi) + (Hi, (VH2Hi)^ + (H2, (Vh,H2)t) + (rr1 (V.Ti) (•, H2), Hi) + (rrT (V.72) (•, Hi), H2) + (ah1 , VT H2) + (Ah2 , V1 H2)

+ Z(A^H2, (VeHi)1) , e^j + X(A^Hi, (VeaH2)J) , ea)

+ 2X( (VTi (ei ,H2) e^1, Hi) + (VT2(ea ,Hi)ea)T , H2j

- <A2(Hi, H2), Hi) - <Ai(H2, Hi), H2). (24)

Integrating (24) and applying the Stokes Theorem, we get our integral formula:

Theorem 1 For arbitrary orthogonal complementary distributions D1 and D2 on a closed oriented Riemannian manifold M one has

J' (<Ric(H2), Hi) + (Hi, (VH2Hi)1 + (H2, (Vh,Hz)T) + (Tr1 (V.Ti) (•, H2), Hi) + (TrT (V.T2) (•, Hi), H2) + (AH1, VTH2) + (ah2, V1 H2) + x(Ai(H2, (VeHi)1) , ei) + x(a^Hi, (Ve„H2)T) , ea)

+ 2X((VTi(ei,H2)ei)1, Hi) + 2X((VT2(e„,Hie)T , H2)

- <A2(Hi, H2), Hi) - <Ai(H2, Hi), H2)) d vol = 0. (25)

3 Distributions with singularities

In this section, we work with a closed Riemannian manifold M equipped with a pair of orthogonal and complementary distributions (Di, D2) defined on M \ £, £ being the union of pairwise disjoint closed submanifolds of variable codimensions >2. Briefly, we say that our distributions admit singularities at points of £. We shall show that in this case, the integral formulae (4) and (25) hold under some natural assumptions.

First, observe that M has bounded geometry, i.e., bounded sectional curvature and injec-tivity radii rx, x e M, separated away from zero. Let A be a closed submanifold of M and k = codim A > 2. Given r > 0, we denote the tube of radius r about A by NA(r) and by dNA(r) the tubular hypersurface at distance r > 0 from A. Let f : M \ A ^[0, be a function defined on M outside A.

Lemma 1 If hmnf Jg N ,r) f > 0, then Jm f2 = ro.

Proof Since the geometry of M is bounded, there exists c > 0 such that vol(9Na (r)) < c • rkfor sufficiently small r. The assumption implies that there exists e > 0 such that

f f > e

for small r. Hölder's inequality implies that t f <-

(f f • vol(gNa (r))2.

\Jg na (r) /

jg Na (r) Consequently,

/ f2 >

Ig Na (r)

if r is small enough.

Again, if r is small, then by Fubini's Theorem,

f f2 > Í f2 = f(f f^dt > E- lim f tl-k = IM JNA(r) Jo \Jg NA (t) / c ? ^0+J %

The above implies the following. Lemma 2 IfZ is a vector field on M \ A such that M II Z y2 < <x>, then

/ div Z = 0. Jm

Proof Let vr be the suitably oriented unit vector field orthogonal to dNa (r). By the Stokes Theorem and our Lemma 1 applied to f = || Z||, we get

I div Z = I (Z, vr} < / || Z|| ^ 0

Jm^.Na (r) Jd Na (r) Jd Na (r)

as r ^ 0.

Applying Lemma 2 to Z = Zk (k = 0, 1) we get

c • rk-1

Theorem 2 Let M be closed and oriented and distributions Di and D2 be defined on M \ £, codim £ > 2.

(i) If Im II Hi II2 < ro and Jm || H2 ||2 < ro, then formula (4) holds.

(ii) If fM || A1(H2, H1) ||2< ro and Jm II A2(H1, H2) ||2< ro, then formula (25) holds.

Finally, let us observe that since || Hi || < c(p, q) • || Ai || for a constant c(p, q) depending on p and q only, the inequalities in (ii) above hold for example when Jm II Ai ||6 < ro for i = 1, 2. Indeed, for, say, i = 1, we have

IIAi(H2, Hi)|| < ||A,|| • IIHiII • IIH2II < c(p, q)2||A1II2 • IIA2I and, by the Holder inequality,

J^ II Ai(H2, Hi)|2 < c( p, q )4 ^Jj Ai||6^ \jM IA2 ' ■

4 Applications

In this section, we will consider pairs of distributions Di, i = 1, 2, satisfying some geometrical conditions, write formulae (4) and (25) in particular cases and prove some (non-)existence results which follow from them. Our distributions are defined either on a closed manifold M or on M \ £, £ being the union of a finite family of embedded closed submanifolds of codimension >2.

4.1 Minimal and totally geodesic distributions

First, if Di and D2 are totally geodesic, that is B1 = 0 and B2 = 0, then H, = 0, H2 = 0 and (3) reduces to the identity

K(Di, D2) - ||Ti||2 - IIT2II2 = 0 which implies immediately the analogous equality for the integral in (4) and the following.

Corollary 1 If a closed manifold M has non-positive sectional curvature (Km < 0) and admits a pair of orthogonal complementary totally geodesic distributions, then M is flat ( Km = 0).

If Di and D2 are minimal, that is H, = 0 and H2 = 0, and integrable (T, = 0 and T2 = 0), then (4) reduces to the identity

K(Di, D2) +||Bi||2 + IIB2II2 = 0. As before, this implies the analogous integral equality and the following.

Corollary 2 If a closed manifold M has non-negative sectional curvature and admits a pair of orthogonal complementary minimal distributions, then M is flat.

In both cases, this of totally geodesic and that of minimal (either integrable or not) distributions (25) reduces to the identity "0 = 0", so yields no reasonable consequences.

4.2 Umbilical distributions

Let us recall that a distribution D is said to be umbilical, when its Weingarten operator A satisfies

A(X, N) = o(N) • X, (26)

for any X e D and N L D, and some 1-form o.

Assume now that our pair (D1, D2) consists of two umbilical distributions, one of them, say D1, being integrable and denote by o¡, i = 1, 2, corresponding 1-forms. Note that, this situation is of some interest: umbilicity is a conformally invariant property, the distribution orthogonal to a Hopf fibration on the round sphere S2n+1 is totally geodesic [so, umbilical with o = 0in (26)], fibers are also totally geodesics (so, umbilical), therefore, Hopf fibrations provide pairs of umbilical distributions, one of them being integrable, on odd-dimensional spheres equipped with arbitrary locally conformally flat Riemannian structures. In the situation considered here, we obtain

<H1, N) = B1(et, ei), N) = A1(et, N), ei) i i

= X01(N)<ei, e¡) = po(N),

A1 (X, N) = 1 <H1, N) • X, p

<B1(X, Y), N) = <A1(X, N), Y) = 1 <H1, N) • <X, Y)

for X, Y e D1 and N e D2. Similarly, for D2 and vectors X, Y e D2 and N e D1 we get

A2 (Y, N) = 1 <H2, N) • Y

<B2(X, Y), N) = <A2(X, N), Y) = 1 <H2, N) • <X, Y).

Applying the above equalities, we obtain

( aH1 , vT H2)

= X (A1(e,, H1), VeTH2) i

= -||H1||2 • Trace VTH2 p

= "Z (ei, VeH) <H1, H1) pi

Z(Al(H2 AVeHl)1) , i

= -i £ e, H2) (Ve,Hx, Hi) = (vZi^,H2>eiHx, Hi

= - (Vh2Hi, Hi).

Similarly,

Moreover,

(aH2, v1 Hi) = iIIH2II2 • TrV1 Hi Hi, (Vea H2)1) , ea) = i (VHi H2, H2).

< Ai (H2, Hi), H2) = - || Hi||2 • IIH2II2 and < A2 (Hi, H2), Hi) = -|l H2II2 • | |Hi II2.

Finally, since Di is integrable, Ti = 0 and formulae (24) and (25) reduce, respectively, to div (Ai(H2, Hi) + A2(Hi, H2))

= <Ric(H2), Hi) + 2^ {{VT2(ea,Hi)ea)T , H2) + (TrT (V.T2) (•, Hi), H^

+ ^ IIHi II2 • TrVTH2 + i IIH2II2 • TrV1 Hi

+ ^P-1 (Hi, VH2 Hi) + (H2, Vhi H2)

- ^ II Hi II2 • 11H2II2 - i IIH2II2 • I I Hi I I 2 pq

and (on closed manifolds)

<Ric(H2), Hi) + 2 X((VT2(ea,Hi)ea)T , H2) + (TrT (V.T2) (., Hi), H^

+ i i| Hi I I 2 • TrVTH2 + i i| H2 | | 2 • TrV1 Hi + (Hi, VH2Hi)

+ — (H2, VHiH2) - ^IIHi||2 •IIH2II2 - ^IIH2II2 • | | Hi| I A dvol = 0.

q p q J

Also, since

IIBill2 = ^ • I|HiII2 and IIB2II2 = i •IIH2II2, pq

formulae (3) and (4) reduce, respectively, to

div(Hi + H2) = K(Di, D2) + ^-1 |Hi | | 2 + | |H2 | | 2 - | | T2 | | 2

and (on closed manifolds)

J (k(Di, D2) + 1——P||Hi||2 + ||H21|2 — 1172||2)dvol = 0.

The last formula above implies the following

Corollary 3 IfK(D1, D2) < 0 andp, q > 1,thenH1 = H2 = 0,T2 = 0 and K(D1, D2) = 0. If M is closed and Km < 0, then the distributions Di, D2 satisfying the conditions of this section do not exist.

The last statement could be compared with Theorem 4.1.2 in [9].

4.3 Constant mean curvature

Now, let us recall that a distribution D on a Riemannian manifold M has constant mean curvature whenever its mean curvature vector H satisfies

V1 H = 0,

where V1 is the connection in D1 induced by the Levi-Civita connection on M.

Coming back to a pair of distributions, let us observe that if both of them, D1 and D2 have constant mean curvature, then several terms in (24) vanish identically, therefore (24) and (25) reduce, respectively, to

div (A1(H2, H1) + A2(H1, H2)) = (Ric(H2), H1) — <A2(H1, H2), H1) — <A1(H2, H1), H2> + (Trace1 (V. T1) (., H2), H^ + (Trace1 (V. T2) (., H1), H^j

+ 2Z {(VT1(ei ,H2)ei )1 , H1) + ((VT2 (ea,H1)ea )T , H2)

and (on closed manifolds, again)

J (<Ric(H2), H1) — (A2(H1, H2), H1) — (A1 (H2, H1), H2)

+ (Trace1 (V.T1) (., H2), H^ + (TrT (V.T2) (., H1), H2) +2Z((VT1(ei,H2)et)1, H1) + 2 X((VT2(e^H^a)T , H^dvol = 0.

Certainly, there is a number of formulae which can be obtained form (24) and (25) in other geometrically interesting cases. Here, let us mention just the following one.

Proposition 3 If D1 and D2 are complementary orthogonal distributions on a Riemannian manifold M which are umbilical, integrable and have constant mean curvature, then

div (A1(H2, H1) + A2(H1, H2))

= (Ric(H2), H1) — 11|H1||2 • ||H21|2 — 11|H21|2 • 11H11|2 pq

J ^(Ric(H2), Hi)—^p + ||Hi||2 •IIH2II2) dvol = 0 (28)

when M is closed.

Finally, observe that (Ric(Hi), H2) = 0 when M is an Einstein manifold. This implies the following application of our main formula (25).

Corollary 4 If M is a closed Einstein manifold, then arbitrary pairs of distributions satisfying the conditions of Proposition 3 have to be totally geodesic.

Proof From (28), we get Hi = H2 = 0 what—together with umbilicity—yields Ai = 0 and A2 = 0.

Note that, in the situation described in Corollary 4, M is locally isometric to the Rie-mannian product of the leaves of foliations Fi and F2 determined by the distributions Di and D2 of our pair.

5 An example

An open book decomposition (OBD, for short) of a three-dimensional manifold M is a pair (B, n) where B is an oriented link in M, called the binding of the open book, and n : M \ B ^ S1 is a fibration such that, for each 9 e S1 , n—i (9) is the interior of a compact surface (with boundary) £ C M whose boundary is B. The surface £ is called the page of the open book. Since almost a century [i], it is known that every closed oriented 3-manifold has an open book decomposition.

A closed 3-manifold M with an open book decomposition (B,n) is equipped with two singular distributions Di and D2 (dim Di = 2 and dim D2 = i) defined on M \ B. Both of them are integrable, so Ti = 0 and T2 = 0, and || B2||2 = ||H2||2. Therefore, (4) reduces to (i) (for the foliation of M \ B by the pages of the open book) which holds here if only the integrals Jm || Hi ||2, i = i, 2, are finite. If the pages are taut, that is Hi = 0, then 202 = —1| A11|2 < 0 and the Ricci curvature of M cannot be positive everywhere (if only the flow orthogonal to the pages has finite total curvature). Similarly, if the pages are umbilical, then 02 > 0 and the Ricci curvature of M cannot be negative everywhere (again, if the integrals mentioned above are finite).

Also, formula (25) for distributions arising from an OBD reduces significantly and provides results analogous to those mentioned above for open book decompositions with, say, pages of constant mean curvature and constant normal curvature.

Note that, open book decompositions can be defined and studied on manifolds of higher dimension. In general, the existence of them depends on the topology of manifolds under consideration (see, for example [20]), however, Lawson [i0] proved that all odd-dimensional closed manifolds of dimension >6 admit OBDs. Certainly, our formulae can be applied to OBDs in higher dimensions as well.

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