Extrinsic curvatures of distributions of arbitrary codimension

Journal of Geometry and Physics 60 (2010) 708–713

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Extrinsic curvatures of distributions of arbitrary codimension Krzysztof Andrzejewski a,b,∗ , Paweł G. Walczak a,c a

Institute of Mathematics, Polish Academy of Sciences ul. Śniadeckich 8, 00-956 Warszawa, Poland

b

Department of Theoretical Physics II, University of Łódź ul. Pomorska 149/153, 90 - 236 Łódź, Poland

c

Faculty of Mathematics and Informatics, University of Łódź ul. Banacha 22, 90-238 Łódź, Poland

article

info

Article history: Received 13 August 2009 Accepted 13 January 2010 Available online 25 January 2010 MSC: 53C12 53C15

abstract In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira [F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 18 (2000) 371–383]. We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures. © 2010 Elsevier B.V. All rights reserved.

Keywords: Foliations rth mean curvature Newton transformation Distributions

1. Introduction Using some special forms Γr Brito and Naveira [1] defined higher order extrinsic curvatures of distributions and they computed the total rth mean curvature SrT of certain distributions on closed spaces of constant curvature. They generalize the ones for foliations [2–6]. On the other hand, many authors (see, among the others, [7–11]) have recently investigated higher order mean curvatures and higher order mean curvature vector fields of hypersurfaces using the Newton transformations of the second fundamental form. Especially, the papers [9,10] are devoted to submanifolds of codimension greater than one. In this paper we show that these methods can be also applied successfully for distributions of arbitrary codimension. Namely, using the generalized Newton transformation Tr we define rth mean curvature Sr and (r + 1)th mean curvature vector field Sr of a distribution D. We show that they agree with the ones from [1] (Theorem 1). Since most of the interesting and useful integral formulae in Riemannian geometry are obtained by computing the divergence of certain vector fields and applying the divergence theorem, we compute the divergence of (r + 1)th mean curvature vector field of a distribution which is orthogonal to a totally geodesic foliation in a manifold of constant sectional curvature (Theorem 3). Using this quantity we obtain a recurrence formula for the total mean curvatures (Corollary 1) and consequently we get another proof of the main theorem from [1] (Theorem 2). The paper is organized as follows. Section 2 provides some preliminaries. The main results of the paper are contained in Section 3. Throughout the paper everything (manifolds, distribution, metrics, etc.) is assumed to be C ∞ -differentiable and oriented and we usually work with Sr instead of its normalized counterpart Hr .

∗

Corresponding author at: Department of Theoretical Physics II, University of Łódź ul. Pomorska 149/153, 90 - 236 Łódź, Poland. Tel.: +48 42 665 50 74. E-mail addresses: [email protected] (K. Andrzejewski), [email protected] (P.G. Walczak).

0393-0440/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2010.01.003

K. Andrzejewski, P.G. Walczak / Journal of Geometry and Physics 60 (2010) 708–713

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2. Preliminaries Let M be an m-dimensional oriented, connected Riemannian manifold. On M we consider a distribution D, n = dim D and a distribution F which is the orthogonal complement of D, l = dim F = m − n. We assume that they are orientable and transversally orientable. Let h·, ·i represent a metric on M and ∇ denote the Levi-Civita connection of the metric. Let Γ (D) denote the set of all vector fields tangent to D. If v is a vector tangent to M, then we write

v = v> + v⊥ , where v > is tangent to D and v ⊥ to F . Define the second fundamental form B of the distribution D, by B(X , Y ) = (∇Y X )⊥ , where X , Y are vector fields tangent to D and similarly the second fundamental form of F . Throughout this paper, we will use the following index convention 1 ≤ i, j, . . . ≤ n, n + 1 ≤ α, β, . . . ≤ m, and 1 ≤ A, B, . . . ≤ m. Repeated indices denote summation over their range. Let us take a local orthonormal frame {eA } adapted to D, F , i.e., {ei } are tangent to D and {eα } are tangent to F . Moreover, {eA },{ei } and {eα } are compatible with the orientation of M , D and F , respectively. Let {θ i } and {θ α } be their dual frame and ωAB (eC ) = −h∇eC eA , eB i Define the second fundamental form (or the shape operator) Aα of D with respect to eα , by Aα (X ) = −(∇X eα )> , for X tangent to D. Then, using the notation j Aα ei = Aα i ej

and Bij = B(ei , ej ),

we have i Bij = Aα j eα .

Note that, matrices Aα ij and Bij are not symmetric with respect to i, j if D is not integrable. In spite of this, for even r ∈ {1, . . . , n}, we can define rth mean curvature Sr of the distribution D by Sr =

1 r!

i ...i

j

j

j

δj11...jrr hBi11 , Bi22 i · · · hBirr −−11 , Bjirr i, i ...i

where the generalized Kronecker symbol δj11...jrr is +1 or −1 according as the i’s are distinct and the j’s are even or odd permutation of the i’s, and is 0 in all other cases. By convention, we put S0 = 1 and Sn+1 = 0. Moreover, for even r ∈ {0, . . . , n − 1} we define (r + 1)th mean curvature vector field Sr +1 of D by Sr +1 =

1

i ...i

(r + 1)!

j

j

j

j

δj11...jrr++11 hBi11 , Bi22 i · · · hBirr −−11 , Bjirr iBirr ++11 ,

and Sn+1 = 0. If D is of codimension one, then Sr +1 = Sr +1 N where N is a unit vector field orthogonal to D, see [12]. The normalized rth mean curvature Hr of a distribution D is defined by Hr = Sr

 n  −1

.

r Obviously, the functions Sr (Hr respectively) are smooth on the whole M. If the distribution D is integrable, then for any point p ∈ M, Sr (p) coincides with the rth mean curvature at p of the leaf L of a foliation which passes through p [7,9]. Now, we introduce the operator Tr : Γ (D) → Γ (D) which generalizes the Newton transformation of the shape operator for hypersurfaces and foliations (see, among the others, [12,8–11]). For even r ∈ {1, . . . , n}, we set 1 i1 ...ir i j1 j2 j δ hB , B i · · · hBirr −−11 , Bjirr i, r ! j1 ...jr j i1 i2 and by convention T0 = I. Note that Tn = 0. We also set for a fixed index α Tr ij =

Trα−1 j =

1

i

(r − 1)!

i ...i

δj11...jrr−−11j hBi11 , Bi22 i · · · hBirr −−33 , Birr −−22 iAα irr −−11 . i

j

j

j

j

j

In the following lemma, we provide some relations between the rth mean curvature (vector field) and the operator Tr . Lemma 1. For any even integer r ∈ {1, . . . , n} we have Sr =

1 r

Tr(Trα−1 Aα ), 1

Tr(Tr Aα )eα , r +1 Tr(Tr ) = (n − r )Sr , S r +1 =

Tr = Sr I − Aα Trα−1 ,

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K. Andrzejewski, P.G. Walczak / Journal of Geometry and Physics 60 (2010) 708–713

and when r is odd, for each α , we have n−r

tr (Trα ) =

Tr(Tr −1 Aα ),

r

where Tr = TrD = (·)ii . j

Proof. The proof of lemma is quite similar to the one for submanifolds [9,10], we must only be more careful because Bi need not be a symmetric matrix.  On the other hand, Brito and Naveira [1] have introduced n-forms Γr for even r = 2s as follows:

ε(σ )(ωσ (1)β1 ∧ ωσ (2)β1 ) ∧ · · · ∧ (ωσ (2s−1)βs ∧ ωσ (2s)βs ) ∧ θ σ (2s+1) ∧ · · · ∧ θ σ (n) ,

X

Γr =

(1)

σ ∈Σn

where Σn is the group of permutations of the set {1, . . . , n}, ε(σ ) stands for the sign of the permutation σ . Furthermore, they define the total rth extrinsic mean curvature SrT of a distribution D on a compact manifold M as SrT =

Z

1 r !(n − r )!

Γr ∧ ν, M

where ν = θ n+1 ∧ · · · ∧ θ m . This suggests that we should have 1 r !(n − r )!

Γr ∧ ν = Sr Ω ,

where Ω is volume element of (M , h, i). We will show this equality in the next section. 3. Main results Using definitions and notations as in Preliminaries, we obtain the following theorem which states that, SrT defined by Brito and Naveira [1] is indeed the total mean curvature of the distribution in our sense. Theorem 1. If r = 2s and Sr is rth mean curvature of the distribution D and Γr is defined by (1), then we have 1 r !(n − r )!

Γr ∧ ν = Sr Ω .

Proof. Using the following expression for the generalized Kronecker symbol i ...i

δj11...jrr

i1 δj 1 = ... δ ir j1

··· .. . ···

i δjr1 .. = X ε(τ )δ i1 · · · δ ir , jτ (1 ) jτ (r ) . τ ∈Σr ir δjr

we have Sr =

=

=

=

1

i ...ir

δ1

r ! j1 ...jr 1 X r!

j1 ...jr distinct

2s−1 αs 2s Aα1 i11 Aα1 i22 · · · Aαs i2s A i2s −1

j

i ...i

j1 ...jr distinct

τ ∈Σr

1 X X r!

j1 ...jr distinct

j

2s−1 αs 2s δj11...jrr Aα1 i11 Aα1 i22 · · · Aαs i2s A i2s −1

1 X X r!

j

j

τ ∈Σr

j

j

j

j

2s−1 αs 2s ε(τ )δjτ1(1) · · · δjiτr (r ) Aα1 i11 Aα1 i22 · · · Aαs i2s A i2s −1

i

j

j

j

ε(τ )Aα1 j1τ (1) Aα1 j2τ (2) · · · Aαs jτ2s(−2s1−1) Aαs j2s . τ (2s) j

j

j

j

j

(2)

On the other hand, by the definition of ωiα , we deduce

ωiα (ej ) = hei , ∇ej eα i >= −Aα ij , thus

ωiα = −Aα ij θ j + Xβiα θ β .

(3)

K. Andrzejewski, P.G. Walczak / Journal of Geometry and Physics 60 (2010) 708–713

711

From (1) and (3) we have

Γr ∧ ν =

X σ ∈Σn

=

X

ε(σ )(Aα1 σj1(1) Aα1 σj2(2) · · · Aαs σj2s(−2s1−1) Aαs σj2s(2s) θ j1 ∧ · · · ∧ θ j2s ) ∧ θ σ (2s+1) ∧ · · · ∧ θ σ (2s) ∧ ν

σ ∈Σn

=

X

ε(σ )

τ ∈Σ {σ (1)...σ (2s)}

X

X

σ ∈Σn

τ ∈Σ {σ (1)...σ (2s)}

  (1) σ (1) αs σ (2s) αs σ (2s−1) α1 σ (2) ∧ · · · ∧ θ σ (n) ∧ ν ε(τ )Aα1 στ (σ (1)) A τ (σ (2)) · · · A τ (σ (2s−1)) A τ (σ (2s)) θ

(1) α1 σ (2) ε(τ )Aα1 στ (σ (1)) A τ (σ (2))

X

X

= (n − 2s)!

σ :{1...2s}→{1...n} τ ∈Σ {σ (1)...σ (2s)}

(2s−1) αs σ (2s) · · · Aαs στ (σ (2s−1)) A τ (σ (2s))

(1) α1 σ (2) ε(τ )Aα1 στ (σ (1)) A τ (σ (2))

! Ω

(2s−1) αs σ (2s) · · · Aαs στ (σ (2s−1)) A τ (σ (2s))

! Ω

! = (n − 2s)!

X

X

j1 ...j2s distinct

τ ∈Σ2s

α 1 j1

ε(τ )A

α 1 j2

jτ (1) A

jτ (2)

αs j2s−1

···A

jτ (2s−1) A

Comparing the above with (2) we complete the proof of our theorem.

αs j2s

jτ (2s)

Ω.



Brito and Naveira have also shown that in some special cases one can compute explicitly the total mean curvature SrT of the distribution D and it does not depend on D. Indeed, we have the following theorem [1]. Theorem 2. If M is a closed manifold of constant sectional curvature c ≥ 0 and F = D⊥ is a totally geodesic distribution, then we have

    n/2 l + 2s − 1 (l + 2s − 1)/2 −1 s    c vol(M ) if n is even and l is odd,   s 2s  s   T S2s = n/2 2s 2 −1 l/2 + s − 1  c s vol(M ) if n and l are even,  2 (s!) ((2s)!) s s   0, otherwise. Remark 1. Since the distribution F determines a totally geodesic foliation F on M, the constant curvature c must be nonnegative; see [13]. The next part of this section will be devoted to the calculation of the divergence of the mean curvature vector fields. Next, we will use this to find a recurrence formula for the total mean curvatures and consequently we will get an alternative proof of Theorem 2. In order to do this we need the following lemma. Lemma 2. Let {e1 , . . . , em } be a local orthonormal frame field adapted to D, F , such that (∇X ei )> (p) = 0 and (∇X eα )⊥ (p) = 0 for any vector field X on M. Then we have at the point p eα (Aβ j ) = (Aβ Aα )ij − hR(ej , eα )ei , eβ i + h(∇eα eγ )> , ej ihei , (∇eγ eβ )> i − h∇ej (∇eα eβ )> , ei i. i

Proof. Our proof starts with the observation that at p we have the following equality 0 = h∇ej ∇eα eβ , ei i + heβ , ∇ej ∇eα ei i. Thus, we have also at p

−eα (Aβ j ) + (Aβ Aα )ij − hR(ej , eα )ei , eβ i = (Aβ Aα )ij − h∇ej ∇eα ei , eβ i + h∇[ej ,eα ] ei , eβ i i

= Aβ k Aα kj − h∇ej ∇eα ei , eβ i + h∇ej eα , ek ih∇ek ei , eβ i − h∇eα ej , eγ ih∇eγ ei , eβ i i

= Aβ k Aα kj + h∇ej ∇eα eβ , ei i + h∇ej eα , ek ih∇ek ei , eβ i − h∇eα ej , eγ ih∇eγ ei , eβ i = h∇ej ∇eα eβ , ei i − h∇eα ej , eγ ih∇eγ ei , eβ i i

= h∇ej ∇eα eβ , ei i − h(∇eα eγ )> , ej ih(∇eγ eβ )> , ei i = h∇ej (∇eα eβ )> , ei i − h(∇eα eγ )> , ej ih(∇eγ eβ )> , ei i. This ends the proof.



Remark 2. Note that, using parallel transport in D and F respectively, we can always construct the frame field from Lemma 2.

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Now, we introduce for even r auxiliary notations as follows i

i

1

i

r! 1

1 r +2 = Tr jrr + +1 jr +2

i

i

1 r +2 Tr jrr + = +1 jr +2 j

i ...i

j

j

j

δj11...jrr++22 hBi11 , Bi22 i · · · hBirr −−11 , Bjirr i,

r!

i ...i

i

j

j

j

δj11...jrr++22j hBi11 , Bi22 i · · · hBirr −−11 , Bjirr i.

Lemma 3. i

1

i

i

i

i

i

1 r +2 = δjrr++22 Tr jrr ++11 − δjrr++12 Tr jrr ++12 − Tr jrr + +1 jr +2

1 r +1 −1 α r +2 A ir . Tr −2 jrr − Aα irr − 1 −1 jr +1 jr +2

i

r −1

i

ir

i

j

Proof. The proof is analogous to the one for submanifolds [9].



Now, we are ready to find the divergence of Sr +1 . Theorem 3. Let D be a distribution on a Riemannian manifold M with constant sectional curvature c and Sr (Sr +1 ) its rth mean curvature (vector field), for even r ∈ {0, 1, . . . , n}. Assume that F is a totally geodesic distribution (equivalently a totally geodesic foliation) orthogonal to D. Then div(Sr +1 ) = −(r + 2)Sr +2 +

c (n − r )(l + r ) r +1

Sr ,

where n = dim D, l = dim F . Proof. Let {e1 , . . . , em } be a frame in the neighbourhood of a point p as in Lemma 2. By Lemma 1 we have at p 1

1

h∇eβ (Tr(Tr Aα )eα ), eβ i r +1 1 Tr(Tr Aα )h∇ei eα , ei i + eα (Tr(Tr Aα )) = r +1 r +1

div(Sr +1 ) =

r +1 1

h∇ei (Tr(Tr Aα )eα ), ei i +

1

=−

Tr(Tr Aα ) Tr(Aα ) +

r +1

1 r +1

eα (Tr(Tr Aα )).

(4)

Using the definition of Tr and the symmetries of the generalized Kronecker symbol we obtain eα (Tr(Tr Aα )) = eα (Tr ij )Aα i + Tr ij eα (Aα i ) r i ...i i j j j j j j = δj11...jrr j hBi11 , Bi22 i · · · hBirr −−33 , Birr −−22 iAβ irr −−11 eα (Aβ irr )Aα ji + Tr ij eα (Aα ji ) r! j

=

1

j

1 Tr −2 jrr − Aβ ir −1 eα (Aβ ir )Aα i + Tr ij eα (Aα i ). −1 jr j

i

r −1

jr −1

ir i

jr

j

j

(5)

Now let us compute the terms on the right-hand side of (5) one by one. From Lemma 2, under our assumption (∇eα eβ )> = 0, we obtain eα (Aβ ir ) = (Aβ Aα )irr + c δαβ δirr . jr

j

j

(6)

Using Lemma 1, Lemma 3 and (6), we see that the first term on the right-hand side of (5) is of the form 1

= = =

1 Tr −2 jrr − Aβ ir −1 eα (Aβ ir )Aα i −1 jr j

i

r −1

1

jr −1

ir i

r −1 1 r −1

j

1 Tr −2 jrr − Aβ ir −1 Aα i Aβ k Aα ir + −1 jr j

i

r −1 1

jr

jr − 1

ir i

j

jr

k

1 Tr −2 jrr − Aα irr Aα i Aβ ir −1 Aβ k + −1 jr j

i

ir i

j

k

jr −1

j

c r −1 c r −1

1 Tr −2 jrr − Aβ ir −1 Aα i δαβ δirr −1 jr j

i

ir i

jr −1

j

j

1 −1 α Tr −2 jrr − Aα irr − A i −1 jir 1

i

iir

j

j

1 Tr −2 jr jrr − Aα irr Aα i Aβ ir −1 Aβ k + cr Tr(Tr ) −1 j

ir i

i

j

k

jr −1

j

 j  j i k i i = −Tr jrr −−11 j + δjk Tr jrr −−11 − δjkr −1 Tr jr −1 Aβ irr −−11 Aβ k + cr (n − r )Sr = −Tr jrr ++11 j Aβ irr ++11 Aβ k + Tr(Tr Aβ ) Tr(Aβ ) − Tr(Tr Aβ Aβ ) + cr (n − r )Sr i

k

j

j

= −(r + 1)(r + 2)Sr +2 + Tr(Tr Aβ ) Tr(Aβ ) − Tr(Tr Aβ Aβ ) + cr (n − r )Sr .

K. Andrzejewski, P.G. Walczak / Journal of Geometry and Physics 60 (2010) 708–713

713

By the use of (6) and Lemma 1, we see that the second term on the right-hand side of (5) is of the form Tr ij eα (Aα i ) = Tr ij (Aα Aα )i + clTr ii = Tr(Tr Aα Aα ) + cl Tr(Tr ) j

j

= Tr(Tr Aα Aα ) + cl(n − r )Sr .

(7)

Hence (5) is of the form eα (Tr(Tr Aα )) = −(r + 1)(r + 2)Sr +2 + Tr(Tr Aα ) Tr(Aα ) + c (r + l)(n − r )Sr . Putting (8) into (4) we complete the proof of theorem.

(8)



Corollary 1. Let D be a distribution on a closed Riemannian manifold M with constant sectional curvature c ≥ 0 and (SrT ) Sr its (total) rth mean curvature. Let us assume that F is a totally geodesic distribution orthogonal to D. Then

Z

S r +2 Ω =

c (n − r )(l + r )

Z

M

M

(r + 1)(r + 2)

Sr Ω ,

equivalently SrT+2 =

c (n − r )(l + r )

(r + 1)(r + 2)

SrT .

(9)

Finally, note that we can use Corollary 1 to prove Theorem 2. Proof of Theorem 2. For even n using (9) and induction one gets SrT as in Theorem 2. When n is odd, then c must be zero, because there is no totally geodesic foliation on a closed Riemannian manifold of constant positive curvature. Indeed, without loss of generality, we may assume that M = S m . For the existence of foliations the sphere should have odd dimension. Since n is odd the foliation should be even dimensional and should not contain any compact spherical leaf. Otherwise, we might pull back the Euler class of the foliation to this spherical even dimensional leaf, proving that it has Euler number zero. On the other hand, totally geodesic foliations on round spheres should have spheres as leaves — contradiction. Consequently c = 0 and using again (9) we end the proof of Theorem 2.  Since Lemma 2 holds for arbitrary Riemannian manifolds, it seems to be reasonable to search for formulae analogous to the ones from Theorem 2 in the more general case (see also [5]). Furthermore, for r = 0 are known applications of Theorem 3 in different areas of differential geometry, analysis and mathematical physics; see, for example [14–16]. The reader is warmly invited to find them for other r. Acknowledgement We are grateful to Fabiano Brito for helpful e-mail discussion. References [1] F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom. 18 (2000) 371–383. [2] D. Asimow, Average gaussian curvature of leaves of foliations, Bull. Amer. Math. Soc. 84 (1978) 131–133. [3] F. Brito, R. Langevin, H. Rosenberg, Intégrales de courbure sur des variétés feuilletées, J. Differential Geom. 16 (1981) 19–50. [4] A. Ranjan, Structural equations and an integral formula for foliated manifolds, Geom. Dedicata 20 (1986) 85–91. [5] V. Rovenski, P. Walczak, Integral formulae for foliations on Riemannian manifolds, in: Diff. Geom. and Appl., Proc. of Conf., Olomouc, World Scientific, Singapore, 2008, pp. 203–214. [6] P. Walczak, An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math. 58 (1990) 243–252. [7] L.J. Alías, S. de Lira, J.M. Malacarne, Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J Inst. Math. Jussieu 5 (4) (2006) 527–562. [8] J.L.M. Barbosa, A.G. Colares, Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997) 277–297. [9] L. Cao, H. Li, r-Minimal submanifolds in space forms, Ann. Global Anal. Geom. 32 (2007) 311–341. [10] J.F. Grosjean, Upper bounds for the first eigenvalue of the laplacian on compact submanifolds, Pacific J. Math. 206 (2002) 93–11. [11] H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993) 211–239. [12] K. Andrzejewski, P.G. Walczak, The Newton transformation and new integral formulae for foliated manifolds, Ann. Global Anal. Geom. 37 (2010) 103–111. [13] A. Zeghib, Feuilletages géodésiques des variétés localement symétriques, Topology 36 (4) (1997) 805–828. [14] V. Brinzanescu, R. Slobodeanu, Holomorphicity and Walczak formula on Sasakian manifolds, J. Geom. Phys. 57 (2006) 193–207. [15] F. Brito, P.G. Walczak, On the energy of unit vector fields with isolated singularities, Ann. Polon. Math. 73 (2000) 269–274. [16] M. Svensson, Holomorphic foliations, harmonic morphisms and the Walczak formula, J. London Math. Soc. 68 (2003) 781–794.