Riemannian submersions and eigenforms of the Witten Laplacian

Transcription

1 Proceedings of The Sixteenth International Workshop on Diff. Geom. 16(2012) Riemannian submersions and eigenforms of the Witten Laplacian Hyunsuk Kang Department of Mathematics, Korea Institute for Advanced Study, Seoul , Korea h.kang@kias.re.kr (2010 Mathematics Subject Classification : 53B20, 58J50.) Abstract. It is known that a Riemannian submersion and the Laplacian commute so that the eigenforms are preserved with the identical eigenvalue if and only if the fiber of the submersion is minimal and the horizontal distribution is integrable. We generalize this for the case of the Witten Laplacian, a perturbed Laplacian, and show that there is an obstruction to have the volume form on the 2-sphere as an eigenform for the Hopf fibration. 1 Introduction Riemannian submersions have received much attention particularly since the publication of O Neill s celebrated formula [4] which relates the curvatures of the base manifold and the total manifold where basic theories can be found in [1]. Later in 1990 s, the spectral geometry of Riemannian submersion was developed by Gilkey, Leahy and Park ([2], [3]). We extend their work to the case of the Witten Laplacian showing that results such as the rigidity of eigenvalues still hold. 2 The spectral geometry of Riemannian submersions Let π : Z Y be a submersion where Y and Z are closed Riemannian manifolds, that is, π : T z Z T π(z) Y is surjective for all z Z. Then the vertical distribution V z is defined as ker(π ) and the horizontal distribution H z is defined as its orthogonal complement V z so that T Z = V H. Let V and H be the dual codistributions of the cotangent bundle. Definition. If π : Z Y is a submersion, and π (z) : H z T π(z) Y is a linear Key words and phrases: Eigenvalues, Laplacian. 143

2 144 Hyunsuk Kang isometry for all z Z, then π is called a Riemannian submersion. For indices, we use a, b, and c to index local orthonormal frames {f a }, {f a }, {F a } and {F a } for H, H, T Y and T Y, and i, j, and k to index local orthonormal frames {e i } and {e i } for V and V. We adopt the Einstein convention and sum over repeated indices. Throughout the paper, we assume that π : Z Y is a Riemannian submersion. The following metric dependent tensors determine the spectral rigidity through Riemannian submersions. Definition. The unnormalised mean curvature covector of the fibres of π is defined by θ := g Z ([e i, f a ], e i )f a = Γ iia f a C (H), and the tensor ω := ω abi = 1 2 g Z(e i, [f a, f b ]) = 1 2 (Γ abi Γ bai ) is called the integrability curvature tensor. Remark 2.1. Each fibre is a minimal submanifold of the total space if and only if θ = 0. Eells and Sampson [5] showed that π is a harmonic map if and only if the fibres are minimal. Moreover the distribution H is integrable if and only if ω = 0. Let d Y and d Z be the exterior derivatives on Y and Z respectively and let δ Y and δ Z be their adjoint operators on the spaces of smooth forms. For C (Y ), d = e de is called the twisted exterior derivative, and δ = e δe is its adjoint operator. The Witten Laplacian is defined by := d δ + δ d, and a W-eigenform will mean an eigenform of the Witten Laplacian. For a covector σ, we use left exterior multiplication by σ to define the linear operator ext(σ) on the bundle of exterior differential forms Λ Y : ext(σ)α := σ α. Let int(σ) be the adjoint operation of interior multiplication; these two operations are related by the duality equation: < ext(σ)α, β > Y =< α, int(σ)β > Y, and satisfy the following commutation rules: ext(σ a )ext(σ b ) = ext(σ b )ext(σ a ), int(σ a )int(σ b ) = int(σ b )int(σ a ), ext(σ a )int(σ b ) + int(σ b )ext(σ a ) = δ ab. Since π commutes with d, it also holds for the twisted exterior derivative: π d Y = π (d Y + ext(d Y )) = d Z π + ext(d Z π )π = d Z π π. But, in general, π does not commute with δ as we shall see. In [3], the authors introduced the operators Ω and Ξ defined by Ω := w abi ext Z (e i )int Z (f a )int Z (f b ), and Ξ := int Z (θ) + Ω. The following lemma is a key ingredient to determine the rigidity and the change of eigenvalues under Riemannian submersions.

3 Riemannian submersions and eigenforms of the Witten Laplacian 145 Lemma 2.2. Let Ω and Ξ be defined as above. Then (i) δ Z π π π δ Y = Ξπ, (ii) p,z π π π p,y = d Z π Ξπ + Ξd Z π π. Proof. The case = 0 was shown in [3] using the facts that for a smooth p-from ω, d Z ω = ext Z (e i ) e iω + ext Z (f a ) f aω, δ Z ω = int Z (e i ) e iω int Z (f a ) f aω. Suppose that is non-trivial. For a smooth p-form ω on Z, δ Z π ω = int Z (e i ) e iω int Z (f a ) f aω + int Z (e i ) ( ( e iπ )ω ) +int Z (f a ) ( ( f aπ )ω ) = δ Z ω + int Z (d Z π )ω. For a smooth p-form Φ on Y, using the fact that int is natural under the pullback since ext is, δ Z π π Φ = (δ Z + int Z (d Z π ))π Φ = π δ Y Φ + Ξπ Φ + int Z (d Z π )π Φ = π δ Y Φ + Ξπ Φ. The assertion (ii) follows from (i) and the definition of the Witten Laplacian: p,z π π π p,y = d Z π (δ Z π π π δ Y ) + (δ Z π π π δ Y )d Y = d Z π Ξπ + Ξd Z π π. Having established Lemma 2.2, the analogous results regarding the rigidity of eigenvalues follow in a similar manner to the proofs of theorems in [2] and [3]. Let E λ (, p,y ) be the eigenspace corresponding to the eigenvalue λ of the p- form valued Witten Laplacians on Y. We say that an eigenform is preserved if its pull-back is also an eigenform. For convenience, we denote the pullback to Z of a smooth function f on Y by f. Theorem 2.3. Let π : Z Y be a Riemannian submersion. If f E(λ, 0,Y ) is non-trivial and if f := π f E(µ, 0,Z ), then λ = µ.

4 146 Hyunsuk Kang Proof. Suppose that f E(λ, 0,Y ) and f E(µ, 0,Z ). Note that Ω = 0 on functions and 1-forms, and hence Ξ = 0 on functions. By Lemma 2.2, we have (µ λ) f = ( 0,Z π π 0,Y )f = ΞdZ π f = int Z (θ)d Z f. Choose a point y 0 Y so that e f (y 0 ) 0 is maximal and choose z 0 in the fibre π 1 (y 0 ). Then f(z 0 ) 0 and (d Y f)(y 0) = 0. This implies that (d Z f)(z 0 ) = 0 and we have (µ λ) f(z 0 ) = 0. One can also easily obtain the result on the commuting the Laplacian and the pullback of the Riemannian submersion. Theorem 2.4. The following conditions are equivalent. (i) 0,Z π = π 0,Y, (ii) For any λ, π E(λ, 0,Y ) E(µ, 0,Z) for some µ R, (iii) The fibres of π are minimal submanifolds, i.e. θ = 0. So far we have observed that the spectral rigidity of the Laplacian on functions extends to the Witten Laplacian on functions. A natural question that one can ask is whether the same works for the form-valued Witten Laplacian. To discuss this, we introduce fibre products and define the horizontal space and the vertical space on them. Definition. Let π i : U i Y be a Riemannian submersion for i = 1, 2. Then the fibre product of the two Riemannian submersions is W (U 1, U 2 ) := {w = (u 1, u 2 ) U 1 U 2 π 1 (u 1 ) = π 2 (u 2 )}. Define smooth submersions π W : W Y and σ i : W U i by π W (w) := π 1 (u 1 ) = π 2 (u 2 ) σ i (w) := u i for w = (u 1, u 2 ) W. Let H i and V i be the horizontal space and the vertical space of π i respectively. The vertical space V W (w) of π W at w is V 1 (u 1 ) V 2 (u 2 ) T U 1 T U 2 and H W (w) := {(ξ 1, ξ 2 ) H 1 (u 1 ) H 1 (u 1 ) (π 1 ) (ξ 1 ) = (π 2 ) (ξ 2 )} defines a complementary orthogonal summand. Let the metrics on V i be induced from the metrics on U i. Then we define a metric on W so that H W V i, V 1 V 2, and π W is a Riemannian submersion.

5 Riemannian submersions and eigenforms of the Witten Laplacian 147 Lemma 2.5. Let W be a fibre product as above. Let θ W, ω W, θ i,and ω i be the mean curvature vectors and horizontal integrability tensors of π W and π i respectively. Then we have θ W = σ 1θ 1 + σ 2θ 2, ω W = σ 1ω 1 + σ 2ω 2. Proof. Use the fact that Riemannian submersions pass through the Lie brackets and compute the θ and ω in terms of horizontal lifts of submersions σ i. See Lemma in [2]. Lemma 2.6. Let W be a fibre product as above. If Φ E(λ, p,y ) and if π i Φ E(λ + ɛ i, p,ui πi ) for i = 1, 2, then π W Φ E(λ + ɛ 1 + ɛ 2, p,w πw ). Proof. From Lemma 2.5, we have Ξ W π W = int W (θ W )π W + Ω W π W = σ 1Ξ 1 π 1 + σ 2Ξ 2 π 2, since π W = σ i π i. Then by Lemma 2.2, Lemma 2.5, and the fact π i dy = dui π i π i, we have ( p,w π W π W π W p,y )Φ = (Ξ W π W d Y + d W π W Ξ W π W )Φ = σ 1(Ξ 1 π 1d Y + d U1 π 1 Ξ 1π 1)Φ + σ 2(Ξ 2 π 2d Y + d U2 π 2 Ξ 2π 2)Φ = σ1( p,u1 π1 π 1 π1 p,u1 )Φ + σ2( p,u2 π2 π 2 π2 p,u2 )Φ = σ 1ɛ 1 π 1Φ + σ 2ɛ 2 π 2Φ = (ɛ 1 + ɛ 2 )π W Φ. Theorem 2.7. Let π : Z Y be a Riemannian submersion. If Φ E(λ, p,y ) and if π Φ E(µ, p,z π ), then λ µ. Proof. Let ɛ := µ λ. Then π Φ E(λ + ɛ, p,y ). Suppose that ɛ < 0. Define Z(0) := Z and let Z(n) := W (Z(n 1), Z(n 1)) with Riemannian submersions π n : Z(n) Y where π 0 = π. π1φ E(λ + 2ɛ, p,z(1) π1 ) and induction implies that By Lemma 2.6, the first step n = 1 gives that π nφ E(λ + 2 n ɛ, p,z(n) π n ).

6 148 Hyunsuk Kang Since the Witten Laplacian on Z(n) is a nonnegative operator, λ + 2 n ɛ 0 for all n N. Hence ɛ 0. Having obtained the necessary and sufficient conditions for the rigidity of eigenvalues of the Witten Laplacian on functions (Theorem 2.4), we now show the analogous rigidity theorem for the Witten Laplacian on forms. Let V (y) be the volume of the fibre π 1 (y) over y given by V (y) = π 1 (y) e1 e r, where r = dim(π 1 (y)). We perturb the metric ds 2 Z on Z with the horizontal distribution preserved. Let ds 2 Z (t) := V 2t ds 2 V + ds2 H be a conformal deformation of the fibre metric. We compute the mean curvature covector θ(t) and the integrability tensor ω(t) of the perturbed metric. Lemma 2.8 ([2]). Let r be the dimension of the fibre X. Suppose that Z is equipped with the perturbed metric ds 2 Z (t). Then (i) θ(t) = θ rtπ d Y (ln(v )), (ii) ω(t) = V 2t ω, (iii) π θ = d Y ln(v ). Now we are in a position to give the rigidity theorem for eigenvalues of W- eigenforms. In view of Lemma 2.2, we have the following theorem analogous to the classical case: Theorem 2.9. Let π : Z Y be a Riemannian submersion. Let = π be the pullback of the dilaton. For a fixed p such that 1 p dim(y ), the following conditions are equivalent. (i) p,z π = π p,y. (ii) For all λ, π E(λ, p,y ) E(λ, p,z ). (iii) For all λ, there exists µ = µ(λ) such that π E(λ, p,y ) E(µ, p,z ). (iv) The fibres of the submersion are minimal and the horizontal distribution of Z is integrable, i.e. θ = 0 and w = 0. For smooth 1-forms, if one has a Riemannian submersion with minimal fibres, the rigidity of the eigenvalues follows straight from Lemma 2.2. It is still an open question whether a smooth eigen 1-form of the Laplacian can change through a Riemannian submersion.

7 Riemannian submersions and eigenforms of the Witten Laplacian 149 Theorem Let π : Z Y be a Riemannian submersion with minimal fibres. Assume we are given a non-zero Φ in E(λ, 1,Y ) such that π Φ E(µ, 1,Z ). Then λ = µ. Proof. Suppose that θ = 0. By Lemma 2.2, (µ λ)π Φ = (d Z Ξ + Ξd Z )π Φ = d Z Ωπ Φ + Ωd Z π Φ. By the definition of Ω, Ω introduces a vertical component in the first term of the second equality and the second term vanishes. Since the left hand side has no vertical dependence, it must vanish. 3 Obstructions In the previous section, we have obtained the necessary and sufficient conditions for the spectral rigidity of Riemannian submersions and one can ask whether the eigenvalues can change through Riemannian submersions. In [2], the authors showed that eigenvalues of the real Laplacian on p-forms can change if p 2; in Hopf fibration, the volume form for S 2 is harmonic and its pullback to S 3 is an eigenform with corresponding eigenvalue 4. But, as we shall show below, even this relatively simple case does not give such a change of eigenvalues for the Witten Laplacian. This is understandable in the sense that the eigenvalues of Schrödinger operators are more difficult to compute than those of the Laplacian. For details of the Hopf fibration, see section 2.6 in [2]. Lemma 3.1 ([2]). Let π : S(L) Y be a principal circle bundle. Then (i) the fibres of π are totally geodesic so that θ = 0, (ii) the normalised curvature two form F := d Y A is invariantly defined where A is the normalised curvature 1-form, and (iii) de 1 = π F and Ω = ext S (e 1 )π int Y (F) where e 1 is the covector spanning the vertical codistribution V. For the Hopf fibration, ν 2 = 1 2 F and its pull back is the product of the two covectors spanning the horizontal distribution. We shall show that the volume form is not an eigenform of the Witten Laplacian.

8 150 Hyunsuk Kang Let e k := ext(w k ) where {w k } is a basis for the tangent space of the manifold M m, and let i k := int(w k ) be its adjoint. We denote the covariant differentiation k := wk with respect to the Levi-Civita connection by ; k. Then one can obtain = + ;j ;j + ;kj (e j i k i j e k ) on C ΛM. Since e j i k i j e k = δ jk i k e j i j e k = e j i k + e k i j δ jk, the Witten Laplacian can be written as 0 = on C (M m ), m = m + 2 on C (Λ m M m ). Note that we have taken the sign of the Laplacian to be = ;ii. Proposition 3.2. Let M m be a closed manifold. Then the volume form ν on M m is a W-eigenform, moreover W-harmonic if and only if is constant. Proof. This can be shown by the max-min principle. Suppose that ν has an eigenvalue λ. Then we have m ν = ( 2 )ν = λν. At the maximum (resp. minimum) of, = 0 and 0 (resp. 0). If = 0 at the maximum (resp. minimum) of, the only possible value for λ is zero. Then, integrating the equation above, one has M 2 = 0, which implies that is constant. If 0 at the maximum and the minimum, then this forces to be constant. The result follows from the fact that the volume form is harmonic. Note that the same argument works for the Witten Laplacian on smooth functions. Proposition 3.3. For a non-constant dilaton, no 2-form can be a W-eigenform for the standard Hopf fibration. Proof. Let Φ = fν 2, where ν 2 is the volume form on S 2 and f C (S 2 ). Note that ν 2 is the eigenform in the classical case. Suppose that Φ E λ ( 2,S2 ) and Φ E µ ( 2,S3 ). Using the facts that π ν 2 = ζ 2 ζ 3, ω 231 = ω 321 = 1, and Ω = 2ext(ζ 1 )int(ζ 3 )int(ζ 2 ), we have that (µ λ)π Φ = (Ξd + d Ξ)π fν 2 = d fωπ ν 2 = 2(d + ext(d )) fζ 1 = 2d f ζ fdζ fd ζ 1 = 2π (df + fd) ζ 1 + 4π Φ.

9 Riemannian submersions and eigenforms of the Witten Laplacian 151 Hence we need to choose such that given a smooth function f, df + fd = 0, i.e. df f = d for nonvanishing f. For f C (S 2 ) and f > 0 on S 2, = lnf + constant. In other words, f = ke, where k is a constant. Without loss of generality, we assume that k = 1. Then f = ( ;i e ) ;i = ;i ;i e + ;ii e = ( 2 + )f. also this choice of f should satisfy 2,S2 fν 2 = λfν 2. That is, λφ = Φ = fν 2 + ( 2 )fν 2 = 2( )Φ. Integrating the both sides, one obtains λ = 0 by Stokes theorem. But = 0 implies that is constant. The following theorem and its corollary show that for certain fibre bundles, the eigenvalue can change through Riemannian submersions under rather strong restrictions. Theorem 3.4. Let be a unitary connection on a complex line bundle over Y with associated curvature two form F and associated principal circle bundle S = S(L). Assume that Φ E(λ, p,y ), and ext Y (F)int Y (F)Φ = ɛφ, where ɛ is a constant. Also suppose that Φ satisfies one of the following conditions: (i) d Y Φ = 0, d Y int Y (F)Φ = 0, and {ext Y (d Y )int Y (F) int Y (F)ext Y (d Y )}Φ = 0, (ii) d Y Φ = 0, d Y int Y (F)Φ = 0, and int Y (F)ext(d Y )Φ = 0, (iii) d Y Φ = 0, dy int Y (F)Φ = 0, and ext Y (d Y )int Y (F)Φ = 0, (iv) d Y Φ = 0, and dy int Y (F)Φ = 0. Then π Φ is in E(λ + ɛ, p,s ).

10 152 Hyunsuk Kang Proof. For convenience, we omit the subscripts S and Y. Recall that θ = 0, F = da, de 1 = π F and Ω = ext S (e 1 )π int Y (F) from Lemma 3.1. We use Lemma 2.2 to compute the difference between the eigenvalues. p,s π Φ π p,y Φ = (d + ext(d ))(ext(e 1 )π {int Y (F)Φ}) (ext(e 1 )π {int Y (F)})π d Φ = ext(π F)π {int Y (F)Φ} + ext(e 1 )dπ {int Y (F)Φ} +ext(e 1 )ext(π d)π {int Y (F)Φ} ext(e 1 )π {int Y (F)d Φ} = π {ext(f)int(f)φ} + ext(e 1 )π {d(int(f)φ) +ext(d)int(f)φ int(f)d Φ} = ɛπ Φ + ext(e 1 )π {d(int(f)φ) + ext(d)int(f)φ int(f)(dφ + ext(d)φ)}. The hypotheses in (i) imply that all the summands of the second factor of the second term vanish. Hence we have p,s π Φ = (λ + ɛ)π Φ. The proofs for the rest the cases follow similarly, noting that d Φ = 0 implies d π Φ = 0. Corollary 3.5. Let be a unitary connection on a complex line bundle over Y with associated curvature two form F and associated principal circle bundle S = S(L). Suppose that µ := int(f)f = F 2 is constant. (i) If Φ 1 := F E(λ, 2,Y ), then π Φ 1 E(λ + µ, 2,S ), (ii) If Φ 2 := e F E(λ, 2,Y ), then π Φ 2 E(λ + µ, 2,S ), (iii) If Φ 3 := F A d E(λ, 4,Y ), and A d is a closed 2-form, then π Φ 3 E(λ + µ, 4,S ), (iv) If Φ 4 := e F A d E(λ, 4,Y ), and A d is a closed 2-form, then π Φ 4 E(λ + µ, 4,S ).

11 Riemannian submersions and eigenforms of the Witten Laplacian 153 Proof. (i) Since dφ 1 = df = d 2 A = 0, d(int(f)φ 1 ) = dµ = 0, and {ext(d)int(f) int(f)ext(d)}φ 1 = µd int(f)(f d) = 0. Hence the result follows from Theorem 3.4 (i). (ii) This follows from Theorem 3.4 (iv) with d Φ 2 = e df = 0 and d int(f)φ 2 = e d(int(f)f) = 0. (iii) The fact that F = da implies dφ 3 = 0. Since A d is closed, we have d(int(f)φ 3 ) = µd(a d) = 0. Then the result follows from Theorem 3.4 (i). (iv) Similarly, this follows from Theorem 3.4 (iv). References [1] A.L. Besse, Einstein manifolds, Springer (1987). [2] P.B. Gilkey, J.V. Leahy and J.H. Park, Spectral Geometry, Riemannian submersions and the Gromov-Lawson Conjecture, Chapman & Hall/CRC (1999). [3] P.B. Gilkey and J.H. Park, Riemannian submersions which preserves the eigenforms of the Laplacian, Illinois J. Math., 40 (1996), no.2, [4] B. O Neill, The fundamental equation of a submersion, Mich. Math. J., 13 (1966), [5] J. Eells and A.J. Sampson, Harmonic mapping of Riemannian manifolds, Springer (1987).