Let F be a foliation of dimension p and codimension q on a smooth manifold of dimension n.

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1 Trends in Mathematics Information Center for Mathematical Sciences Volume 5, Number 2,December 2002, Pages VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS KYOUNG HEE HAN AND HOBUM KIM Abstract. We obtain a second variation formula for the energy functional for a harmonic Riemannian foliation in terms of the second and third fundamental tensors, integrability tensor and curvature regarding the foliation. From this formula, we conclude that if the second fundamental tensor and the integrability tensor of a harmonic Riemannian foliation is small compared with the partial Ricci tensor, then it is stable. 1. Introduction. A foliation F on a Riemannian manifold (M, ) is called Riemannian and bundle-like if all the leaves are locally equi-distant to each other. Such a foliation is characterized by the property that a geodesic orthogonal to the foliation at one point is orthogonal everywhere. F is said to be harmonic if all the leaves of F are minimal submanifolds([4]). F. Kamber and Ph. Tondeur([4],[5],[6]) showed that a Riemannian foliation is harmonic if and only if it is critical for the energy functional under an appropriate class of so-called special variations defined by sections of the normal bundle([4]). They also derived a second variational formula for the energy functional and considered the stability of a harmonic Riemannian foliations ([5],[6]). In this note, exploiting the second variation formula by Kamber and Tondeur([5]), we obtain a second variation formula for the energy functional for a harmonic Riemannian foliation in terms of the second fundamental tensor, integrability tensor and curvature regarding the foliation. As an immediate consequence it will follow that if the second fundamental tensor and the integrability tensor of a harmonic Riemannian foliation is small compaired with the partial Ricci tensor, then it is stable. This result manifests the fact that the second fundamental tensor, integrability tensor and curvature reflects fundamental aspects of geometry of foliations. 2. Energy of a foliation Let F be a foliation of dimension p and codimension q on a smooth manifold of dimension n Mathematics Subject Classification. 53C12, 58E30. Key words and phrases. Riemannian foliations, harmonic foliations, energy functional, stability. Received October 1, c 2002 Information Center for Mathematical Sciences

2 60 KYOUNG HEE HAN AND HOBUM KIM The tangent bundle L of a foliation F is the subbundle of T M, consisting of all vectors tangent to the leaves of F. The normal bundle Q of codimension-q foliation F on M is the quotient bundle Q = T M/L. The tangent bundle L of F is integrable or involutive and Q appears in exact sequence of vector bundles (2.1) 0 L T M π Q 0. Consider a Riemannian metric on M. Then T M splits orthogonally as T M = L L with σ : Q L T M splitting the sequence (2.1). The metric on T M is then a direct sum = g. With g = L Q σ g L, the splitting map σ : (Q, ) (L, g L ) is a metric isomorphism. Let denote the (partial) Bott connection in Q defined by (2.2) X s = π[x, Y s ] for X ΓL, Y s ΓT M with π(y s ) = s. We observe that the RHS in (2.2) is independent of the choice of Y s. Let M denote the Levi-Civita connection associated to the Riemannian metric on M. A connection in Q is defined by { π[x, Ys ] for X ΓL (2.3) X s = π( M Y X s) for X ΓσQ, where s ΓQ and Y s = σ(s) ΓσQ. The first condition says that is an adapted connection on Q, i.e. a connection extending the (partial) Bott connection alon. A foliation F is Riemannian (or an R-foliation), if the normal bundle is equipped with a holonomy invariant fiber metric. This condition is expressed in terms of the Bott connection by X = 0 for X ΓL. A Riemannian metric on M is bundle-like with respect to the foliation F, if the fiber metric induced on Q turns the foliation into an R-foliation. An R-foliation admits a bundle-like metric. Let F be an R-foliation with a metric on the normal bundle Q. For r 0, We denote by Ω r (M, Q) the space of all smooth Q-valued r-forms on M. For a connection on Q, we consider the exterior derivative d : Ω r (M, Q) Ω r+1 (M, Q), r 0, given by for ω Ω r (M, Q), (2.4) r+1 (d ω)(x 1,, X r+1 ) = ( 1) i+1 Xi ω(x 1,, X i,, X r+1 ) + i<j( 1) i+j ω([x i, X j ], X 1,, X i,, X j,, X r+1 ),

3 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS 61 where the X i s are vector fields on M. The covariant derivative X : Ω r (M, Q) Ω r (M, Q), X ΓT M, r 0, is given by (2.5) ( X ω)(x 1,, X r ) = X ω(x 1,, X r ) r ω(x 1,, M X X i,, X r ), where ω Ω r (M, Q) and X i ΓT M. Since M is torsion-free, d ω can be expressed as (2.6) r+1 (d ω)(x 1,, X r+1 ) = ( 1) i+1 ( Xi ω)(x 1,, X i 1, X i+1,, X r+1 ). For the case r = 1, this formula reads (2.7) (d ω)(x, Y ) = ( X ω)(y ) ( Y ω)(x). The Q-valued bilinear form on M α(x, Y ) = ( π)(x, Y ) = π( M X Y ) X π(y ) for X, Y ΓT M is called the second fundamental form of the foliation. For each ν ΓQ the map W (ν) : L L defined by (W (ν)u, V ) = (α(u, V ), ν) for U, V ΓL is called theweingarten operator. F is said to be harmonic if Trace W(ν) = 0 for each ν ΓQ. The harmonicity of the foliation is related with the Laplacian (2.8) = d d + d d where the star operator on M extends to Q-valued forms : Ω r (M, Q) Ω n r (M, Q) and the codifferential d : Ωr (M, Q) Ω r 1 (M, Q), r > 0 of the exterior differential d is given in terms of the star operator by (2.9) d ω = ( 1)n(r+1)+1 d ω, ω Ω r (M, Q).

4 62 KYOUNG HEE HAN AND HOBUM KIM The codifferential d becomes the formal adjoint of d with respect to the naturally induced scalar product (2.10) µ, ν = M (µ, ν) η M of sections µ, ν ΓQ on Q-valued forms over a compact oriented manifold M. Here η M is the volume form associated to the metric on M. The kernel of the Laplacian coincides precisely with the forms which are both d -closed and d -closed. A foliation is harmonic if and only if π = 0([4]). The energy of the foliation F is (2.11) E(F) = 1 2 π 2 where π Ω 1 (M, Q) is the canonical projection π : T M Q and the norm is given by the scalar product, on Ω r (M, Q), r 1 such that (2.12) ω, ω = M (ω ω ). 3. Second variation formula and stability of a harmonic Riemannian foliation. Assume to be bundle-like. A section ν ΓQ defines a special variation F t of F 0 = F through Riemannian foliations by patching the local data (3.1) Φ α t (x) = exp f α (x)(tν α (x)), where f α is a local submersion defining F in an open set U α. Φ α t is then the local submersion defining F t for t ɛ, where ɛ > 0 is sufficiently small. The RHS in (3.1) denotes the endpoint of the geodesic segment starting at f α (x) and determined by tν α (x). This is the construction of Eells-Sampson. Clearly (3.2) ν α (x) = d dt Φ α t (x). t=0 The following facts are due to Kamber and Tondeur([4],[5]). Lemma 3.1. Let M be a compact oriented manifold with a Riemannian foliation F and bundle-like metric. Then F is harmonic if and only if it is an extremal of the energy functional for special variations of F.

5 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS 63 Lemma 3.2(Second variation formula). Let M be a compact oriented manifold and F a harmonic Riemannian foliation with respect to a bundle-like metric. Consider the 2-parameter family F s,t of special variations of F = F 0,0 defined by two sections µ, ν of the normal bundle Q. Then for the second derivative of the energy we have (3.3) ( 2 / s t)e(f s,t ) s=0,t=0 = ( ρ )µ, ν, where ρ is the Ricci operator given by (ρ µ) x = n γ=p+1 R (µ, e γ)e γ orthonormal basis {e p+1,, e n } of Q x. for an Theorem 3.3. Let F be a harmonic Riemannian foliation of dimension p with normal bundle Q on a compact oriented Riemannian manifold M of dimension n. Let M be the Levi-Civita connection on M and the connection on Q defined by (2.3). Consider the family F t of special variations of F = F 0 defined by ν ΓQ. Then for the second derivative of the energy we have (3.4) d 2 dt 2 E(F t ) t=0 = ( M ν, ν) W (ν) 2 A(ν) 2 (ρ ν, ν), M where W is the Weingarten operator and A is the integrability tensor given by (3.6) A E F = π ( M π(e) π(f )) + π( M π(e) π (F )) for E, F ΓT M. Proof. Consider the family F t of special variations of F = F 0 defined by the section ν ΓQ. Let be a bundle-like metric and E 1,, E n an orthonormal local frame of T M on a neighborhood of x M such that E 1,, E p ΓL and E p+1,, E n ΓQ. From (3.3), we have (3.7) d 2 dt 2 E(F t ) = ( ρ )ν, ν t=0 = ( EA ν, EA ν) (ρ ν, ν). M A=1

6 64 KYOUNG HEE HAN AND HOBUM KIM Using the integrability tensor A and the Weingarten operator W, we find that ( EA ν, EA ν) A=1 = = ( Ei ν, Ei ν) + ( M ν, M ν) E i E i ( E α ν, Eα ν) (π ( M ν), π ( M ν)) E i E i (π( M E ν i), π( M E ν i)) + = ( M ν, ν) (π ( E α ν), π ( E α ν)) (W (ν)e i, W (ν)e i ) (A ν E i, A ν E i ) = ( M ν, ν) W (ν) 2 A(ν) 2. ( M Eα ν, M Eα ν) (A Eα ν, A Eα ν) In the above caculations, we have used the fact that W (ν)e α = 0 and A Eα ν = A ν E α for α = p + 1,, n. If there is an ɛ > 0 such that for all t < ɛ, E(F t ) E(F), then the foliation F is said to be a stable. Corollary 3.4. Let F be a harmonic Riemannian foliation on a compact oriented Riemannian manifold M with a bundle-like metric. If W 2 + A 2 ρ(z) for any z L, where ρ(z) is the partial Ricci curvature in the direction of z, then F is stable. Acknowledgements. The present studies were supported by the Basic Science Research Institute Program, Ministry of Education, BSRI References 1. James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), Franz W. Kamber and Philppe Tondeur, Harmonic foliations, Sprin- ger lecture notes in mathematics 949 (1982), Franz W. Kamber and Philppe Tondeur, The index of harmonic foliations on spheres, Transactions of the American Mathematical Society 279; 1 (1983), Franz W. Kamber and Philppe Tondeur, Infinitesimal automorphisms and second variarion of the energy for harmonic foliations, Tôhoku Math. Journ. 34 (1982), Barrett O Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), Philppe Tondeur, Foliations on Riemannian manifold, Lecture note (1986). Department of Mathematics, Yonsei University, Seoul , Korea address: kimhb@yonsei.ac.kr