Trivializable sub-Riemannian structures on spheres

Bull. Sci. math. 137 (2013) 361–385 www.elsevier.com/locate/bulsci

Trivializable sub-Riemannian structures on spheres W. Bauer a,∗,1 , K. Furutani b,2 , C. Iwasaki c,3 a Mathematisches Institut, Bunsen-Strasse 3-5, Georg-August-Universität Göttingen, Germany b Department of Mathematics, Tokyo University of Science, Japan c Department of Mathematics, University of Hyogo, Japan

Available online 24 September 2012

Abstract We classify the trivializable sub-Riemannian structures on odd-dimensional spheres SN that are induced by a Clifford module structure of RN+1 . The underlying bracket generating distribution is of step two and spanned by a set of global linear vector fields X1 , . . . , Xm . As a result we show that such structures only exist in the cases where N = 3, 7, 15. The corresponding hypo-elliptic sub-Laplacians sub are defined as the (negative) sum of squares of the vector fields Xj . In the case of a trivializable rank four distribution on S7 and a trivializable rank eight distribution on S15 we obtain a part of the spectrum of sub . We also remark that in both cases there is a relation between the eigenfunctions and Jacobi polynomials. © 2012 Elsevier Masson SAS. All rights reserved. MSC: 53C17; 35P20 Keywords: Sub-Laplacian; Clifford algebra; Spectrum; Jacobi polynomials

1. Introduction Let M be a smooth connected and oriented manifold endowed with a sub-bundle D of the tangent bundle T M. If ·,·x where x ∈ M denotes a family of smoothly varying inner products on the fibers Dx , then (M, D, ·,·) is called a sub-Riemannian manifold if the distribution D is * Corresponding author. Tel.: +49(0)551 397749; fax: +49(0)551 3922985.

E-mail addresses: [email protected] (W. Bauer), [email protected] (K. Furutani), [email protected] (C. Iwasaki). 1 Supported by the DFG (Deutsche Forschungsgemeinschaft). 2 Supported by the “FY 2011 Researcher Exchange Program between JSPS and DAAD”. 3 Partially supported by the Grant-in-aid for Scientific Research (C) No. 20540218 of JSPS (Japan Society for the Promotion of Science). 0007-4497/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.bulsci.2012.09.004

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bracket generating. Recall that D being bracket generating means that all linear combinations of vector fields on M taking values in D together with a finite number of their Lie brackets span the tangent space T Mx at any point x ∈ M. The sub-Riemannian structure on M is called in the strong sense or trivializable if D is spanned by a family {X1 , . . . , Xm } of globally defined vector fields Xj on M where m = dim Dx (which is independent of x ∈ M). Now let M = SN be the N -dimensional Euclidean sphere in RN+1 . As is well-known odddimensional spheres always carry a contact structure which is bracket generating but not trivializable in general. The present paper is a continuation of [5] where we have started the spectral analysis of two sub-Laplace operators (cf. (1.1)) induced by a co-rank one trivializable subRiemannian structure (shortly: TSR-structure) on S3 and S7 . Therein it was remarked that due to topological obstructions the condition N ∈ {1, 3, 7, 15, 23, 31, 63} is necessary for the existence of a TSR-structure on SN induced by a Clifford module structure of RN+1 . First we give an exact classification of such type of sub-Riemannian structures on spheres and we show that they only exist for the dimensions N = 3, 7, 15 (cf. Theorem 4.4). Whereas in the case of N = 3 and N = 15 a trivializable bracket generating distribution D  T SN of the above kind must have rank two and rank eight, respectively, we show that there are three essentially different TSRstructures on S7 of rank 4, 5 and 6. It also follows that among the non-parallelizable spheres only S15 admits a TSR-structure. Let us recall the construction of the linear vector fields that span the distribution D. Fix a set of (N + 1) × (N + 1) skew-symmetric real matrices A1 , . . . , Ar that fulfill the anti-commuting relations Ai Ak + Ak Ai = −2δik , where i, k = 1, . . . , r. Then a collection of r linear vector fields on SN that are orthonormal in each point of the sphere can be defined by X(Aα ) :=

N+1  N+1 

(Aα )i,j xj

i=1 j =1

∂ , ∂xi

α = 1, . . . , r.

Recall that the maximal number r = γ (N ) of such vector fields coincides with the maximal dimension of a trivial sub-bundle of T SN due to a famous theorem by J.F. Adams (see Theorem 2.1 below and [1]). It was remarked in [5] that D := span{X(Aα ): α = 1, . . . , r} is of step two and only for particular choices of N and r the distribution D can be bracket generating. Let ·,·x denote the restriction of the standard metric on SN to Dx and assume that D is bracket generating. To the trivializable sub-Riemannian manifold (SN , D, ·,·) one assigns a (positive) sub-Laplace operator sub r (shortly: sub-Laplacian) which is defined as the (negative) sum of squares of the vector fields Xα := X(Aα ) sub r := −

r 

X(Aα )2 .

(1.1)

α=1

In general sub r is not elliptic but sub-elliptic due to a classical theorem of L. Hörmander, cf. [13] sub sub and in particular, sub r has only discrete spectrum. The full spectrum of 2 and 6 acting on 3 7 S and S had been calculated explicitly in [5] and by using these data we have analyzed the spectral zeta functions of both operators. In the case of a distribution of co-rank larger than sub one, the analysis is more complicated. For two sub-Laplace operators sub 4 and 8 acting 7 15 on S and S , respectively, we show in the present paper that the spectra σ4 = σ (sub 4 ) and sub σ8 = σ (8 ) are independent of the choice of the matrices Aα above (see Corollary 5.4 and sub Corollary 6.1). Then we determine certain series of eigenvalues of sub 4 and 8 together with their eigenfunctions which are among the spherical harmonics and we calculate the eigenspace

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dimensions. However, the complete spectra remain unknown and shall be further studied in the future in order to explicitly express the spectral zeta functions and to analyze the asymptotic expansions of the heat trace of these operators. With d ∈ {4, 8} and via a map Φ : M := Sd−1 × Sd−1 × (0, 1) → S2d−1 we pull back the sub2d−1 to an operator D sub on M. We express D sub in an explicit form Laplace operator sub d on S d d containing the Laplace–Beltrami operator on Sd−1 and a second order differential operator Ld on the interval (0, 1). By determining the spectrum and the eigenfunctions of Ld which are given by the Jacobi polynomials we derive a series of eigenfunctions of sub d and their eigenvalues that are different from the ones obtained in Sections 5 and 6. In Section 2 we recall the construction of TSR-structures on SN induced by a Clifford module structure of RN+1 . In Section 3 we classify the TSR-structures on S7 and S15 that can be obtained by this method. By excluding the spheres of dimension N = 23, 31, 63 as candidates for such a structure we give a complete classification in Theorem 4.4 of Section 4. Sections 5 and 6 contain sub 7 our results on the spectral analysis of the sub-Laplace operators sub 4 and 8 acting on S 15 and S , respectively. Finally a relation between the sub-Laplacian and the Jacobi polynomials is explained in Section 7 and some further eigenvalues are calculated. In the last Section 8 we mention various open problems in this area. 2. Adams theorem and step two trivializable distributions on spheres Let N ∈ N and by SN denote the N -dimensional unit sphere in RN+1 with tangent bundle T SN . We start by recalling a classical theorem by J.F. Adams which gives the maximal dimension of a trivial sub-bundle in T SN . We then explain an explicit construction of a set of linear vector fields on SN (where N is odd) that are orthonormal in each point of SN . Let D0 = D ⊂ T SN be the distribution spanned by such vector fields and inductively put Dn+1 := Dn + [D, Dn ], where n ∈ N0 and [D, Dn ]x is the space spanned by all Lie brackets of vector fields at a point x ∈ SN having value in D and Dn . It is shown that D1 = D2 = · · · = Dn holds whenever n > 1 and by using this fact we recall that the equality D1 = T SN implies that N ∈ {1, 3, 7, 15, 23, 31, 63} (see [5]). Theorem 2.1. (See J.F. Adams, 1962 [1].) The maximal dimension γ (N ) of a trivial sub-bundle in T SN is given by γ (N ) = 2a + 8b − 1 where the integers 0  a < 4 and 0  b can be obtained through the relation: N + 1 = 2a+4b × [odd number]. A set {Xi : i = 1, . . . , γ (N)} of linear vector fields such that {Xi (x): i = 1, . . . , γ (N)} is orthonormal in each point x ∈ SN and gives a trivialization of a γ (N )-dimensional sub-bundle in T Sn is constructed via a Clifford module structure of RN+1 (cf. [1]). We call it a set of canonical vector fields and recall the construction here. Let C (V , Q) be a Clifford algebra on a k-dimensional real vector space V with a positive definite quadratic form Q. More precisely, if T (V ) :=

∞  i (⊗ V ) i=0

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denotes the tensor algebra of V , then C (V , Q) is defined as the quotient algebra of T (V ) divided by the two sided ideal which is generated by elements of the form x ⊗ x + Q(x) with x ∈ V . In the case of V = Rk equipped with the Euclidean inner product ·,· we can choose Q(x) := x, x = x 2 and we shortly write C k := C (Rk , Q) for the corresponding Clifford algebra. By H we denote the quaternion numbers. Let K ∈ {R, C, H} and with n ∈ N let K(n) be the n × n-matrix algebra over K. As is well-known the following isomorphisms exist (see [2,16]): C 1 ∼ C 2 ∼ C 3 ∼ C 4 ∼ = C, = H, = H ⊕ H, = H(2), ∼ ∼ ∼ C 6 = R(8), C 7 = R(8) ⊕ R(8), C 8 ∼ C 5 = C(4), = R(16). Canonical vector fields on SN are linear and explicitly given in the form: Xα =

N+1  N+1 

aijα xj

i=1 j =1

∂ , ∂xi

(2.1)

where α = 1, . . . , γ (N). The coefficients (aijα ) = Aα ∈ R(N + 1) in (2.1) come from generators {eα ∈ V : α = 1, . . . , γ (N)} of a Clifford algebra C (V , Q) where γ (N ) = dim V . The matrices Aα , Aβ are skew-symmetric and satisfy the anti-commuting relations Aα Aβ + Aβ Aα = −2δαβ . In the case of the dimensions N = 1, 3, 7 it holds γ (N ) = N and it can be verified that the operator L := −

γ (N)

Xα2

α=1

coincides with the Laplace–Beltrami operator on SN with respect to the standard Riemannian ∂f metric. By grad f = ( ∂x , . . . , ∂x∂f ) we denote the gradient of f . Then we define a map X 1 N+1 from R(N + 1) into the space of linear vector fields on RN+1 by X(A) = (Ax)t · gradt , where A ∈ R(N + 1). Let U ∈ R(N + 1) and define the operator TU on C ∞ (RN+1 ) by composition of functions with U , i.e. TU f := f ◦ U . Then we have   X(A)TU f (x) = (Ax)t · U t (grad f )t (U x) = (U Ax)t (grad f )t (U x). In particular it follows: Lemma 2.2. Let U ∈ GL(N + 1, R), then X(U −1 AU ) = TU ◦ X(A) ◦ TU −1 . Let h ∈ C ∞ (SN ), then the vector field X(A) can be also expressed by d h(exp tA · x)|t=0 . dt The canonical vector fields (2.1) can be shortly written in the form X(A)h(x) =

Xα = X(Aα ) = (Aα x)t · gradt ,

(2.2)

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where α = 1, . . . , γ (N). In the following we use the same notation [·,·] for the Lie bracket of vector fields and the commutator of matrices. We write H [f ] = (∂ 2 f/∂xi ∂xj )i,j for the Hessian of f ∈ C 2 (RN+1 ). Then X(A)X(B)f = (Ax)t H [f ]Bx + X(BA)f.

(2.3)

In particular, if we choose skew-symmetric matrices Aα with α = 1, . . . , γ (N) fulfilling the anticommuting relations Aα Aβ + Aβ Aα = −2δαβ , then we obtain: (i) Let α = β, then [Xα , Xβ ] = −X([Aα , Aβ ]) = −2X(Aα Aβ ). (ii) Let α, β, γ be pairwise distinct, then   Aα , [Aβ , Aγ ] = 2(Aα Aβ Aγ − Aβ Aγ Aα ) = 2 (Aα Aβ + Aβ Aα ) Aγ = 0.   =0

It follows that [Xα , [Xβ , Xγ ]] = X([Aα , [Aβ , Aγ ]]) = 0. (iii) Let α = β, then   Aα , [Aα , Aβ ] = 2[Aα , Aα Aβ ] = 2A2α Aβ − 2Aα Aβ Aα = −4Aβ . In this case we have [Xα , [Xα , Xβ ]] = −4X(Aβ ) = −4Xβ . Hence from these relations one finds: Lemma 2.3. Let Xα with α = 1, . . . , γ (N) be a set of canonical vector fields. Then all higher Lie brackets [Xα [Xβ , [Xγ . . .]]] are contained in 

X := span Xi , [Xj , Xk ]  i, j, k = 1, . . . γ (N) . Hence a necessary condition for the distribution in T SN spanned by {Xα : α = 1, . . . , γ (N)} to fulfill the bracket generating condition is given by

 γ (N ) (2a + 8b)(2a + 8b − 1) =: ρ(N). (2.4) N  γ (N ) + = 2 2 Here a and b are the integers in Theorem 2.1. A straightforward calculation shows that (2.4) holds in dimension N = 1, 3, 7, 15, 23, 31, 63: N

1

3

7

15

23

31

63

γ (N ) ρ(N )

1 1

3 6

7 28

8 36

7 28

9 45

11 66

Note that the dimensions N in the first row which fulfill ρ(N)  N are also indicated by the proof of the following lemma. Lemma 2.4. Let N > 63, then ρ(N) < N .

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Proof. From Theorem 2.1 we see that the condition ρ(N )  N means that there are integers 0  a < 4 and 0  b such that N + 1 = 2a+4b × [odd number]  ρ(N) + 1 =

(2a + 8b)(2a + 8b − 1) + 1. 2

In the case of a = 0 we conclude that 24b+1 × [odd number]  (1 + 8b)8b + 2. It is easy to check that this inequality is only fulfilled in the following case: (a) [odd number] = 1 and b = 1. This means that N = 15. In the case of a = 1 we have 22+4b × [odd number]  (2 + 8b)(1 + 8b) + 2. This can only be realized in the cases (b) [odd number] = 1 and b = 0, 1. This means that N ∈ {1, 31}. In case of a = 2 we have 23+4b × [odd number]  (4 + 8b)(3 + 8b) + 2. The only solutions for integers b  0 are (c) [odd number] = 1 and b = 0, 1. This means that N ∈ {3, 63}. Finally, in the case of a = 3 we have 24+4b × [odd number]  (8 + 8b)(7 + 8b) + 2. This inequality is realized in the cases (d) [odd number] = 1, 3 and b = 0. This means that N ∈ {7, 23}. Since the maximal dimension N appearing in this list is N = 63 the assertion of the lemma follows. 2 Corollary 2.5. (See [5].) Let SN be a sphere equipped with a trivializable sub-Riemannian structure that is induced by a Clifford module structure on RN+1 , then N ∈ {1, 3, 7, 15, 23, 31, 63}. In the following we will see that the spheres of dimension N = 3, 7, 15 in fact carry a trivializable sub-Riemannian structure. The case N = 3 is well-known (cf. [5,8,11,12]): Theorem 2.6. Let Xi with i = 1, 2 be canonical vector fields on S3 , then the distribution H23 := span{X1 , X2 } is bracket generating of step two. 3. Trivializable sub-Riemannian structures on S7 and S15 Since γ (7) = 23 + 8 × 0 − 1 = 7 one recovers the well-known fact that the tangent bundle T S7 is trivializable. Consider any set of skew-symmetric matrices {Ai }7i=1 ⊂ R(8) which fulfill the anti-commuting relations Ai Aj + Aj Ai = −2δij ,

i, j ∈ {1, . . . , 7}.

(3.1)

As described before we assign to each matrix Ai a linear vector field X(Ai ) for i = 1, . . . , 7 such that the distribution H77 := span{X(Ai ) | i = 1, . . . , 7} gives a trivialization of the tangent bundle T S7 . Consider now a distribution of the form:

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Hr7 := span X(Ai )  i = 1, . . . , r , where r ∈ {1, . . . , 7} is fixed. The following result was already proved in [8] by different methods but here we give a simpler argument. Proposition 3.1. (See [8].) Let r ∈ {4, 5, 6}, then the trivial sub-bundle Hr7 of T S7 is bracket generating of step two. Proof. It is sufficient to prove that H47 is bracket generating. We have     X(Ai ), X(Aj ) = −X [Ai , Aj ] = −2X(Ai Aj ). By expressing the tangent bundle T S7 in the form 

T S7 = (x, ξ ) ∈ R8 × R8  x = 1, x, ξ  = 0 we can identify X(Ai )(x) with (x, Ai x) and [X(Ai ), X(Aj )](x) with (x, −2Ai Aj x), respectively. We fix an arbitrary point x ∈ S7 and assume that there is (x, y) ∈ Tx S7 with 0 = y such that y is orthogonal to all the vectors {Ai x}i=1,...,4 and {Ai Ak x}1i
y, Ai x = 0 and y, Ai Ak x = 0.

Since the matrices Ai are skew-symmetric we see that: x, y = 0,

Ai y, x = 0 and Ai y, Ak x = 0,

which implies that the 5-dimensional spaces spanned by the vectors {x, Ai x}i=1,...,4 and {y, Ai y}i=1,...,4 , respectively, are orthogonal in R8 . Hence y cannot be non-trivial which implies the bracket generating property of the sub-bundle H47 . 2 Finally, note that 3 +

3 2

= 6 < 7 and therefore H37 cannot be bracket generating.

Theorem 3.2. (See [5,8,18].) On the sphere S7 there are trivializable sub-Riemannian structures of rank four, five and six, which are defined via sets of canonical vector fields. We will see that S15 carries a trivializable sub-Riemannian structure and the co-rank of the defining distribution is seven. Since γ (15) = 8 there is a Clifford module structure of R16 realized by the standard irreducible representation of the Clifford algebra C 8 ∼ = R(16). Choose a collection of skewsymmetric matrices Bj ∈ R(16) with the properties: Bi Bk + Bk Bi = −2δik . For r = 1, . . . , 8 consider the distribution fibers Hr15 (x).

(3.2) Hr15

:= span{X(Bi ) | i = 1, . . . , r} in

T S15

with the

Theorem 3.3. The trivial sub-bundle H815 of T S15 is bracket generating on S15 of step two. Proof. Assume that H815 is not bracket generating at a fixed point x ∈ S15 . Then we can use the same argument as in Proposition 3.1 to construct two 9-dimensional spaces spanned by {x, Bi x: i = 1, . . . , 8} and {y, Bi y: i = 1, . . . , 8} with y ∈ R16 that are orthogonal in R16 . Since this is not possible, the assertion follows. 2

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Proposition 3.4. The distributions Hr15 in T S15 with r ∈ {1, . . . , 7} are not bracket generating. More precisely, there are points x ∈ S15 such that  

dim H715 (x) + H715 , H715 x = 7. Proof. Let x ∈ R16 and consider the space Wx := span{x, Bi x, Bi Bj x: 1  i < j  7} which is contained in the orbit of the Clifford algebra C 7 ∼ = R(8) ⊕ R(8) ⊂ R(16). Due to the structure of C 7 there is x ∈ R16 such that Wx ∼ = R8 and the assertion follows. 2 4. The remaining cases N = 23, 31, 63 With Corollary 2.5 in mind we still have to decide whether there are trivializable bracket generating distributions in T SN defined as above via skew-symmetric anti-commuting matrices in R(N + 1) where N ∈ {23, 31, 63}. Note that we have shown earlier in [5] that such a distribution of T S63 does not exist. Here we present an essentially simplified approach. We start with N = 23. Since γ (23) = 7 we can choose a set of skew-symmetric matrices {Ai }i=1,...,7 ⊂ R(24) such that the anti-commuting relations Ai Ak + Ak Ai = −2δik

(4.1)

are fulfilled for i, k = 1, . . . , 7. Let r ∈ {1, . . . , 7} and define an r-dimensional trivial sub-bundle Hr23 ⊂ T S23 by Hr23 := span{X(Ai ): i = 1, . . . , r}. The same argument we have used in the proof of Proposition 3.4 shows that there is a subspace Wx,24 ⊂ R24 with dim Wx,24 = 8 and invariant under A1 , . . . , A7 . It follows: Lemma 4.1. Let r ∈ {1, . . . 7}, then Hr23 cannot be bracket generating in T S23 . Now, consider the case N = 31. It holds γ (31) = 9 and hence there is a set of skew-symmetric matrices {Ai ∈ R(32): i = 1, . . . , 9} with (4.1). With x ∈ R32 consider Wx,31 := span{x, Ai x, Ai Aj x: 1  i < j  8}. Since C 8 ∼ = R(16), we have dim Wx,31  16 for some x ∈ R32 and it follows that dim span{x, Ai x, Ai Aj x: 1  i < j  9}  dim Wx,31 + 9  25 < 31. Put Hr31 = span{X(Ai ): i = 1, . . . , r} where r ∈ {1, . . . , 9} then we have shown: Lemma 4.2. Let r ∈ {1, . . . , 9}, then Hr31 cannot be bracket generating in T S31 . In the case of S63 we can use a similar argument. Now we have γ (63) = 11 and therefore one has a set {Ai }i=1,...,11 ⊂ R(64) of skew-symmetric matrices which fulfill (4.1). With some x ∈ R64 and Wx,63 := span{x, Ai x, Ai Aj x: 1  i < j  8} we have dim Wx,63  16. Hence dim span{x, Ai x, Ai Aj x: 1  i < j  11}  ρ(63) − ρ(7) + 16 = 54 < 63. Put

Hr63

(4.2)

= span{X(Ai ): i = 1, . . . , r} with r ∈ {1, . . . , 11}, then it follows from (4.2):

Lemma 4.3. Let r ∈ {1, . . . , 11}, then Hr63 cannot be bracket generating in T S63 . Finally, we summarize our results on the classification of TSR-structures. As before let HrN with r  γ (N ) be a distribution of T SN spanned by a family of canonical vector fields

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{X(Ai ): i = 1, . . . , r} where the matrices Ai ∈ R(N + 1) are skew-symmetric and fulfill the anti-commuting relations (4.1) with i, k = 1, . . . , r. Then: Theorem 4.4. The distribution HrN is bracket generating if and only if one of the cases (i)–(iv) is fulfilled: (i) (ii) (iii) (iv)

N N N N

= 1 and r = 1, = 3 and r = 2, 3, = 7 and r = 4, 5, 6, 7, = 15 and r = 8.

Moreover, (as is well-known) one has H11 = T S1 , H33 = T S3 and H77 = T S7 . 5. Spectral analysis of a sub-Laplacian on S 7 In the following we consider the standard Riemannian metric on S7 and its restriction to the sub-bundles Hj7 where j ∈ {4, 5, 6}. Let dVS7 denote the associated volume form on S7 . Assume that the matrices A ∈ R(8) with = 1, . . . , 7 are skew-symmetric and fulfill the anti-commuting relations (3.1). Each vector field X(Aj ) is skew-symmetric with respect to dVS7 . As was pointed out in Theorem 4.4 of the previous section the distribution Hj7 is bracket generating for j = 4, 5, 6. According to a theorem by L. Hörmander in [13] we can assign to each distribution Hj7 for j = 4, 5, 6 a sub-elliptic sub-Laplacian sub acting on C ∞ (S7 ). More precisely, sub is j j positive and defined as a (negative) sum of squares of canonical vector fields sub j := −

j 

X(A )2 .

(5.1)

=1

The sub-Laplacian (5.1) is essentially self-adjoint on C ∞ (S7 ) and in general depends on the particular choice of the matrices A . However, we will not indicate this dependence. Moreover, ∞ 7 in our notation we do not distinguish between the operator sub j on C (S ) and its self-adjoint 7 extension to L2 (S7 ). Note that sub 7 coincides with the Laplace–Beltrami operator S7 on S with respect to the standard metric. Corollary 5.1. The vector fields X(A ) with = 1, . . . , 7 commute with sub 7 . Proof. We can assume that = 7 and put X := X(A7 ). Since sub 7 coincides with the Laplace– Beltrami operator on S7 it commutes with the family ϕt = etA7 : S7 → S7 of isometries where t ∈ R. Since ϕt generates X the assertion follows. 2 As a consequence of the preceding corollary we see that sub commutes with the Laplaj cian S7 for j = 4, 5, 6 and therefore the spaces H of -homogeneous harmonic polynomials on R8 where ∈ N0 are invariant under sub j . Hence, in principle, one can calculate the full discrete spectrum of the hypo-elliptic operators sub j with j = 4, 5, 6 from their matrix representations on H . Moreover, note that for all j = 4, 5, 6 the following commutativity relations hold:   7    sub   sub = j , S7 = 0. X(A )2 = sub sub j , j , S7 − j =j +1

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The sum of squares sub 7 = S7 is independent of the choice of the matrices A ∈ R(8). In the case of j ∈ {4, 5, 6} the sub-Laplacians sub j in fact depend on the particular generators A ∈ R(8) of a Clifford algebra. However, as we will see next, the spectrum σ (sub j ) is independent of this choice. We only deal with the case where j = 4. (k) Let Ak := {Aj : j = 1, . . . , 4} ⊂ R(8) for k = 1, 2 be any two families of skew-symmetric matrices that are anti-commuting for i, j = 1, . . . , 4 and k ∈ {1, 2}, i.e. (k) (k) (k) A(k) i Aj + Aj Ai = −2δij .

(5.2)

(k) (k) Put Ak := span{A(k) i1 Ai2 · · · Ai : 1  i1 < i2 < · · · < i  4} where k = 1, 2. It can be verified that dim Ak = 16 and therefore Ak for k = 1, 2 form realizations of C 4 . In the following we call (1) (2) two families A1 and A2 equivalent if there is C ∈ GL(8, R) such that Ai = C −1 Ai C for all i = 1, . . . , 4. Now, we have

Lemma 5.2. All families Ak ⊂ R(8) of skew-symmetric matrices that fulfill the relations (5.2) for k = 1, 2 are equivalent. Moreover, the matrix C can be chosen to be orthogonal. Proof. Note that each choice of skew-symmetric matrices Ak , k = 1, 2 with (5.2) induces an irreducible representation of C 4 on R8 (cf. Theorem 2.1). Since all irreducible representations of C 4 are equivalent (see Theorem 5.6 in [16]) we find an isomorphism C ∈ GL(8, R) such that (1) (2) Aj = C −1 Aj C for j = 1, . . . , 4. Without loss of generality we can assume that C = 1 and it remains to show that C is orthogonal. From the assumptions on A(k) j with k = 1, 2 it follows that Aj = C t Aj (C −1 )t and therefore:  t −1 (1) t  t −1 (2) A(2) Aj C = CC Aj CC t , j = C (1)

(2)

where j = 1, . . . , 4. Hence the self-adjoint matrix Q := CC t is invertible and all its eigenspaces (2) are invariant under the left multiplication by Aj where j = 1, . . . , 4. Hence Q = γ I where γ > 0. From Q = 1 it follows that CC t = Q = I and the assertion is proved. 2 We give an easy counter example to Lemma 5.2 in the case of three skew-symmetric matrices {A1 , A2 , A3 } that fulfill (5.2). Of course, this reflects the fact that C 3 ∼ = H ⊕ H admits two inequivalent irreducible representations. Example 5.3. Let {Aj : j = 1, . . . , 4} ⊂ R(8) be skew-symmetric with (5.2). Consider the two sets of matrices A1 := {A1 , A2 , A3 } and A2 := {A1 , A2 , A1 A2 }. Note that the elements of A2 are skew-symmetric and fulfill the anti-commuting relations. Moreover, there is no isometry U ∈ O(8) such that A1 = U −1 A2 U . Otherwise the set {A1 , A2 , A1 A2 , U A4 U −1 } was equivalent 4 := to {A1 , A2 , A3 , A4 } and therefore would fulfill the anti-commuting relations. However, if A U A4 U −1 anti-commutes with A1 and A2 , then it easily can be seen that it commutes with A1 A2 which gives a contradiction. With j = 1, . . . , 4 and the above matrices A(k) j for k = 1, 2 and C ∈ O(8) we have  (1)    (2)  (2)  X Aj = X C −1 Aj C = TC X Aj TC −1 , where the operator TC acts on C ∞ (R8 ) (or L2 (S7 ), respectively) by TC f = f ◦ C.

(5.3)

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Corollary 5.4. The two sub-Laplacians sub,k := − 4 i.e.

4

(k) 2 j =1 X(Aj )

371

with k = 1, 2 are equivalent,

sub,1 = TC ◦ sub,2 ◦ TC −1 . 4 4 In particular, the spectrum of the sub-Laplacian (5.1) is independent of the choice of skewsymmetric anti-commuting matrices Aj ∈ R(8). Next, we wish to determine infinite sub-series of the (discrete) spectrum of sub 4 . We start with a particular form of the matrices Aj and first we choose a set of skew-symmetric matrices 2 , A 3 , A 4 } ⊂ R(4) such that A i A j A j + A i = −2δij for i, j ∈ {2, 3, 4}. For example one can {A obtain these by using the isomorphism H ⊗ H ∼ = R(4). Then we form four matrices on R8 as follows 



j 0 I 0 A ∈ R(8), (5.4) ∈ R(8) and Aj :=  A1 := −I 0 Aj 0 where j = 2, 3, 4. It is easy to check that Ai Aj + Aj Ai = −2δij with i, j ∈ {1, . . . , 4}. We decompose the variables of R8 in the form (u, v) ∈ R4 × R4 . With f ∈ C ∞ (R8 ) and j = 2, 3, 4 we obtain   j v)t , (A j u)t (grad f )t . X(Aj )f (u, v) = (A  For any w, z ∈ C4 let us write w, z := 4i=1 wi zi . Fix z ∈ C4 and for a given integer ∈ N0 consider the functions: + (u) := z, u gz,

− and gz, (v) := z, v .

In the case of j = 2, 3, 4 we calculate + j z, vg + (u), X(Aj )gz, (u, v) = − A z, −1 + + (u, v) = z, vgz, −1 (u). X(A1 )gz,

Since Aj is skew-symmetric we have + j z, v2 z, u −2 , (u, v) = − z, u + ( − 1)A X(Aj )2 gz, + X(A1 )2 gz, (u, v) = − z, u + ( − 1)z, v2 z, u −2 . − Similarly, one has in the case of gz, (v) and with j = 2, 3, 4: − j z, ug − (v), (u, v) = − A X(Aj )gz, z, −1 − − (u, v) = − z, ugz, −1 (v). X(A1 )gz,

After applying these vector fields a second time we obtain − j z, u2 z, v −2 , X(Aj )2 gz, (u, v) = − z, v + ( − 1)A − X(A1 )2 gz, (u, v) = − z, v + ( − 1)z, u2 z, v −2 . + − In order to calculate the sum of squares of the vector fields X(Aj ) applied to gz, and gz, we need the following relation:

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Lemma 5.5. Let z ∈ C4 be fixed such that z2 = above. Then it follows for all u ∈ R4 that

4

2 k=1 zk

j for j = 2, 3, 4 as = 0 and choose A

4  j z, u2 = −z, u2 . A

(5.5)

j =2

 Proof. On C4 consider the hermitian inner product (z, w) := 4j =1 zj wj = z, w. Let u ∈ R4 j with j = 2, 3, 4 are skewand without restriction assume that u = 1. Since the matrices A i A j A j + A i = −2δij it is straightforward to check that symmetric with A 2 u, A 3 u, A 4 u} ⊂ R4 ⊂ C4 {u, A 1 := Id and fix z = forms an orthonormal basis of (C4 , (·,·)). Put A Then we have z2 = z, z =

4 

i u)(z, A j u)A i u, A j u = (z, A

i,j =1

4





j =1 (z, Aj u)Aj u

∈ C4 .

4  i u)2 . (z, A i=1

i z, u the assertion follows from z2 = 0. i u) = −A Since (z, A

2

Now, let z ∈ C4 be as in Lemma 5.5, then it follows: + sub 4 gz, (u, v) = −

4 

+ + X(Aj )2 gz, (u, v) − X(A1 )2 gz, (u, v)

j =2

= 3 z, u − ( − 1)z, u −2

4  j z, v2 + z, u A j =2

− ( − 1)z, v z, u 2

−2

= 4 z, u + = 4 gz, (u).

(5.6)

In the third equality we have used the relation (5.5). In the same way it follows that − − sub 4 gz, (v) = 4 gz, (v).

(5.7)

Now let ∈ N0 and choose k ∈ {0, 1, . . . , }. With z ∈ C4 we then define the function − + (v)gz,k (u) = z, v −k z, uk . Gz,k, (u, v) := gz, −k

Note that in the case of z2 = 0 the function Gz,k, is harmonic on R8 . Now, we can show  Proposition 5.6. Let z ∈ C4 with z2 = 4j =1 zj2 = 0 and k ∈ {0, 1, . . . , }. Then the restriction of the function Gz,k, to S7 defines an eigenfunction of sub 4 corresponding to the eigenvalue λk, := 4 + 4k( − k). Proof. With our notation above we obtain + − sub − sub + sub 4 Gz,k, = gz,k 4 gz, −k + gz, −k 4 gz,k − 2

4   j =1

 − +  X(Aj )gz, −k X(Aj )gz,k .

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373

+ j z, vz, uk−1 and the relations Applying (5.6), (5.7) together with X(Aj )gz,k (u, v) = −kA − −k−1  X(Aj )gz, −k (u, v) = −( − k)Aj z, uz, v where j = 2, 3, 4 shows that



    − + sub 4 − 4 Gz,k, (u, v) = −2 X(A1 )gz, −k (u, v) X(A1 )gz,k (u, v) − 2k( − k)z, uk−1 z, v −k−1

4 

j z, vA j z, u. A

j =2

By using the assumption z2 = 0 and Lemma 5.5 one obtains with x, y ∈ R4 : 4 

j z, x − yA j z, x + y = A

j =2

j z, x2 − A j z, y2 A



j =2

Choosing x = 

4  

u+v 2

and y =

v−u 2

2  = − z, x2 − z, y2 = −z, x − yz, x + y.  j z, uA j z, v = −z, uz, v. Moreover, yields 4j =2 A

  − + X(A1 )gz, −k (u, v) X(A1 )gz,k (u, v) = −k( − k)z, uk z, v −k .

Hence, we obtain sub 4 Gz,k, = 4( + k( − k))Gz,k, which proves the proposition.

2

Let S (Rn ) denote the space of harmonic polynomials on Rn that are homogeneous of degree ∈ N0 . If n = 4 then one knows that   Γ ( + 2)(2 + 2) = ( + 1)2 . dim S R4 = Γ ( + 1) · Γ (3) In the following we write P (X3 ) for the space of -homogeneous polynomials on C4 restricted to the quadric  X3 := z ∈ C : 4

4 

 zj2

= 0, z = 0 .

j =1

Let ΩS be the Liouville volume form on the cotangent bundle T0∗ (S3 ) (without the zero section). Recall that both spaces T0∗ (S3 ) and X3 can be identified via the correspondence (x, y) ↔ z := √

y x + −1y and hence we can consider ΩS as a volume form on X3 (for details see [6,19]). Let ∈ N0 and consider the integral transform   4 T : P (X3 ) → S R : [T q](x) := q(z)¯z, x e− z ΩS (z). X3

As is well-known T defines an isomorphism of (finite dimensional) vector spaces [19]. With any number k ∈ {0, . . . , } we set   8 Tk : P (X3 ) → S R : [Tk q](u, v) := q(z)¯z, v −k ¯z, uk e− z ΩS (z). X3

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Note that the operator Tk is still one-to-one. Indeed, assume that [Tk q](u, v) ≡ 0, then we can choose u = v and one obtains 0 ≡ [Tk q](v, v) = 3k [T q](v). Since T is one-to-one we conclude that q ≡ 0. Define a family of + 1 subspaces of the harmonic functions on R8 by   (5.8) W (k) := Tk P (X3 ) ⊂ S R8 , where k = 0, . . . , . The result in Proposition 5.6 can be extended as follows: Theorem 5.7. Let ∈ N0 be fixed and k ∈ {0, . . . , }. Then: (i) W (k) is an eigenspace of sub 4 with respect to the eigenvalue λk = 4 + 4k( − k). (ii) Independently of k it holds dim W (k) = ( + 1)2 . (iii) Consider the spaces  W (k) + W ( − k) if k < 2 , V (k) := W (k) if k = 2 . Then V (k1 ) is orthogonal to V (k2 ) in L2 (S7 ) in the case where k1 = k2 . Moreover,  2( + 1)2 if k < 2 , dim V (k) = ( + 1)2 if k = 2 .

(5.9)

Proof. (i): Follows directly from Proposition 5.6 and the definition of the operator Tk . (ii): Since Tk is one-to-one we have dim W (k) = dim P (X3 ) = dim S (R4 ) = ( + 1)2 . (iii): Since it holds λs = λk if and only if s ∈ {k, − k} we see that V (k) and V (s) with s∈ / {k, − k} are eigenspaces of sub 4 with respect to different eigenvalues. The sub-Laplace operator sub is symmetric and therefore V (k) and V (s) must either coincide or be orthogonal 4 2 7 in L (S ). In order to show (5.9) it is sufficient to prove that W (k) ∩ W ( − k) = {0}. Let k < 2 and by t we denote a real parameter. Assume that Tk p = T −k q ∈ W (k) ∩ W ( − k), then we obtain for any v ∈ R4 : t k [Tp](v) = [Tk p](tv, v) = [T −k q](tv, v) = t −k [T q](v). Since t ∈ R was chosen arbitrary it follows that Tp = T q = 0 and since T is one-to-one we have p = q = 0. Hence Tk p = T −k q = 0. 2 Recall that the eigenspace S (R8 ) of the Laplace–Beltrami operator on S7 is invariant under sub 4 and contains V (k). Moreover, from Theorem 5.7 (iii) we find    ( + 5)!( + 3) V (k) = − ( + 1)3 . dim S R8 / !360 k=0

6. Spectral analysis of a sub-Laplacian on S 15 Let {Aj : j = 1, . . . , 8} ⊂ R(16) be a set of skew-symmetric matrices that fulfill the anticommuting relations Ai Aj + Aj Ai = −2δij where i, j = 1, . . . , 8, cf. [14]. As was shown in

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375

Theorem 4.4 the corresponding distribution H815 := span{X(Ai ): i = 1, . . . , 8} is bracket generating of step two and we can assign to it a positive hypo-elliptic sub-Laplacian on S15 as a (negative) sum of squares of vector fields: sub 8 := −

8 

X(Aj )2 .

(6.1)

j =1

First, we discuss the dependence of the sub-Laplacian (6.1) on the particular choice of vector fields X(Aj ) in (6.1). Let {Bj : j = 1, . . . , 8} ⊂ R(16) be a second families of skew-symmetric matrices that fulfill the anti-commuting relations Bi Bj + Bj Bi = −2δij

(6.2) ∼ for i, j = 1, . . . , 8. Note that up to equivalence C 8 = R(16) admits only one irreducible representation. Hence the sets {Ai : i = 1, . . . , 8} and {Bi : i = 1, . . . , 8} in R(16) are equivalent, i.e. there is C ∈ GL(16, R) such that Ai = C −1 Bi C for all i = 1, . . . , 8 and similar to the proof of Lemma 5.2 we conclude: Corollary 6.1. The sub-Laplacians induced by {Ai : i = 1, . . . , 8} and {Bi : i = 1, . . . , 8} are equivalent and the above matrix C can be chosen to be orthogonal. In particular, the spectrum of (6.1) is independent of the choice of Aj . In order to calculate a sub-series of eigenvalues in the spectrum of the sub-Laplacian (6.1) and to estimate their multiplicities we proceed as in Section 5 and we choose the above matrices i ∈ R(8) with i = 2, . . . , 8 and A i A j A j + A i = −2δij . Ai ⊂ R(16) in a particular form. Let A Then we can define 



j 0 I 0 A ∈ R(16). (6.3) ∈ R(16) and Aj :=  A1 := −I 0 Aj 0 One easily verifies that Ai Aj + Aj Ai = −2δij for i, j ∈ {1, . . . , 8}. We decompose the coordinates of R16 in the form (u, v) ∈ R8 × R8 and as before we write 8 w, z := j =1 wj zj where (w, z) ∈ C8 × C8 . Let k ∈ {0, 1, . . . , } and with z ∈ C8 consider the function: Fz,k, (u, v) := z, v −k z, uk .

(6.4)

Under the additional assumption that = 0 it can be checked that the functions Fz,k, (u, v) are harmonic on R16 . Completely analogous to the proof of Proposition 5.6 one can show: z2

Proposition 6.2. Let z ∈ C8 with z2 = 0 and k ∈ {0, 1, . . . , } where ∈ N0 . Then the restriction of the function Fz,k, to S15 defines an eigenfunctions of sub 8 corresponding to the eigenvalue γk, = 8 + 4k( − k). Let ΩS be the Liouville volume form on the cotangent bundle T0∗ (S7 ) (without the zero section) and put   8  8 2 zj = 0, z = 0 . X7 := z ∈ C : j =1

By identifying X7 and T0∗ (S7 ) one can consider ΩS as a volume form on X7 . Let P (X7 ) be the space of -homogeneous polynomials on C8 restricted to X7 and consider the operators

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Tk : P (X7 ) → S R

16



 : [Tk q](u, v) :=

q(z)¯z, v −k ¯z, uk e− z ΩS (z)

X7

where k = 0, . . . , . Note that Tk is one-to-one for all k (cf. [19]) and define   W (k) := Tk P (X7 ) ⊂ S R16 . Recall that the dimension of the space S (R8 ) of all -homogeneous harmonic polynomials on R8 is given by   Γ ( + 6)(2 + 6) ( + 5)!( + 3) = . dim S R8 = Γ ( + 1)Γ (7) !360 Moreover, we have dim P (X7 ) = dim S (R8 ). Similar to Theorem 5.7 we can now prove: Theorem 6.3. Let ∈ N0 be fixed and k ∈ {0, 1, . . . , }. (i) W (k) is an eigenspace of sub 8 with respect to the eigenvalue γk, = 8 + 4k( − k). (ii) Independently of k is holds dim W (k) = ( +5)!( +3) . !360 (iii) Consider the spaces  W (k) + W ( − k), if k < 2 , V (k) := W (k), if k = 2 . In the case where k1 = k2 we have the orthogonality V (k1 ) ⊥ V (k2 ) in L2 (S15 ). Moreover,  ( +5)!( +3) , if k < 2 , !180 dim V (k) = ( +5)!( +3) , if k = 2 . !360 In particular,    ( + 13)!(2 + 14) ( + 1)( + 3)( + 5)! V (k) = − . dim S R16 / !15! !360 k

7. Sub-Laplacians and the Jacobi equation In the following section we will find a series of eigenfunctions and their eigenvalues of the sub 7 15 sub-Laplacians sub 4 and 8 acting on S and S , respectively, that are different from the ones obtained before. We will treat both cases simultaneously. j : j = 2, . . . , d} of skew-symmetric matrices Let d ∈ {4, 8} be fixed and consider a set {A j A j + A i = −2δij where i, j ∈ {2, . . . , d}. i A in R(d) that fulfill the anti-commuting relations A As earlier we put



 j 0 I 0 A ∈ R(2d) and Aj :=  ∈ R(2d). (7.1) A1 := −I 0 Aj 0 Consider the map Φ : Rd × Rd × (0, 1) → Rd × Rd defined by √ √ Φ(x, y, t) = ( tx, 1 − ty). Note that Φ restricts to a map from Sd−1 × Sd−1 × (0, 1) → S2d−1 which is one-to-one with an open dense range R := {(u, v) ∈ S2d−1 : u = 0 and v = 0} in S2d−1 . With d ∈ {4, 8} let us write

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377

(u, v) = (u1 , . . . , ud , v1 , . . . , vd ) ∈ R2d ∼ = Rd × Rd and consider the volume form on S2d−1 given by dVS2d−1 (u, v) =

d  i ∧ · · · ∧ dud ∧ dv1 ∧ · · · ∧ dvd (−1)i−1 ui · du1 ∧ · · · ∧ du i=1

+ d

d  j ∧ · · · ∧ dvd . (−1)j −1 vj · du1 ∧ · · · ∧ dud ∧ dv1 ∧ · · · ∧ dv j =1

i ∧ · · · ∧ dxd be the volume form associLet dVSd−1 (x) = i=1 (−1)i−1 xi dx1 ∧ · · · ∧ dx ated with the standard Riemannian metric on the unit sphere Sd−1 . Pulling back dVS2d−1 to the manifold Sd−1 × Sd−1 × (0, 1) via the map Φ gives: Proposition 7.1. Let Φ be as above and d ∈ {4, 8}, then: t 2 −1 (1 − t) 2 −1 Φ dVS2d−1 (u, v) = dVSd−1 (x) ∧ dVSd−1 (y) ∧ dt. 2 ∗



d



d

Proof. From the definition of Φ we obtain d √ √ √ √    (−1)i−1 txi · d( tx1 ) ∧ · · · ∧ d( txi ) ∧ · · · ∧ d( txd ) Φ ∗ dVS2d−1 (u, v) = i=1

√ √ ∧ d( 1 − ty1 ) ∧ · · · ∧ d( 1 − tyd )

+

d  √ √ √ (−1)j −1 1 − tyj · d( tx1 ) ∧ · · · ∧ d( txd ) j =1

√ √ √ ∧ d( 1 − ty1 ) ∧ · · · ∧ d(  1 − tyj ) ∧ · · · ∧ d( 1 − tyd ) = I1 + I2 .

(7.2)

A straightforward calculation shows that the first sum I1 and the second sum I2 on the right hand side of (7.2) have the form: d

d

t 2 (1 − t) 2 −1 dVSd−1 (x) ∧ dVSd−1 (y) ∧ dt, 2 d d t 2 −1 (1 − t) 2 dVSd−1 (x) ∧ dVSd−1 (y) ∧ dt. I2 = 2

I1 =

d

d

d

d

d

d

From t 2 (1 − t) 2 −1 + t 2 −1 (1 − t) 2 = t 2 −1 (1 − t) 2 −1 and (7.2) we obtain the assertion.

2

 denote the tensor product of Hilbert spaces, then we have Corollary 7.2. Let ⊗     L2 S2d−1 , dVS2d−1 ∼ = L2 Sd−1 × Sd−1 , dVSd−1 ∧ dVSd−1   d d  L2 (0, 1), 2−1 t 2 −1 (1 − t) 2 −1 dt . ⊗ Let X(Aj ) with j = 1, . . . , d denote the linear vector field on S2d−1 induced by Aj , then we define Yj acting on C ∞ (Rd × Rd × (0, 1)) by

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  Yj Φ ∗ f := X(Aj )f, where j = 1, . . . , d. First we want to express Yj in global coordinates. ∂ ∂  + bj , ∂y  + cj · ∂t∂ where Proposition 7.3. For 1  j  d it holds Yj = aj , ∂x   1−t

t

y − x, yx , b1 = −x + x, yy , a1 = t 1−t  c1 = 2 (1 − t)tx, y. √ j y, x and In the case of 2  j  d we have cj = 2 (1 − t)tA   1 − t  t   j y, xy . bj = Aj y − Aj y, xx , Aj x + A aj = t 1−t

Now consider the second order differential operators Yj2 =

d 

(aj )k (aj )m

k,m=1

d d   ∂2 ∂2 ∂2 +2 (aj )k (bj )m + (bj )k (bj )m ∂xk ∂xm ∂xk ∂ym ∂yk ∂ym k,m=1

k,m=1

d 

∂2 ∂2 (bj )k cj +2 ∂xk ∂t ∂yk ∂t k=1 k=1     ∂ ∂ ∂ + Yj (aj ), + Yj (bj ), + Yj (cj ) . ∂x ∂y ∂t

+ cj2

∂2 +2 ∂t 2

d 

(aj )k cj

(7.3)

We wish to calculate the coefficients of the terms in Yj2 that involve a derivative with respect to the variable t . The following relations are essential for the calculation: i A j A j ∈ R(d): j = 2, . . . , d} with A j + A i = −2δij and At = −Aj . Lemma 7.4. Let {A i  j xA j x + y, xx, (i) y x 2 = dj =2 y, A d 2  Aj y, x + y, x2 = y 2 x 2 , (ii) jd =2 2   (iii) j =2 (Aj x)α (Aj x)β + xα xβ = δαβ x where α, β ∈ {1, . . . , d}. d x} forms 2 x, . . . , A Proof. In the proof of (i)–(iii) we can assume that x = 1. Then {x, A an orthonormal basis of Rd . From this observation (i) directly follows and (ii) is the Parseval d x] ∈ R(d). 2 x, . . . , A identity. In order to prove (iii) consider the orthogonal matrix T = [x, A 1 := Id, then Let α, β ∈ {1, 2, 3, 4} and put A d d d        m x)α (A m x)β . (T )α,m T t m,β = (T )α,m (T )β,m = (A δαβ = T t T α,β = m=1

Now, we calculate the sum of squares

m=1

d

2 j =1 Yj

2

m=1

by using the identities in Proposition 7.5.

Proposition 7.5. Let (x, y, t) ∈ Sd−1 × Sd−1 × (0, 1). With the notation in (7.3) we have the following relations:

W. Bauer et al. / Bull. Sci. math. 137 (2013) 361–385

(1) (2) (3) (4) (5) (6) (7) (8) (9)

d

2 j =1 cj

d

jd =1 jd =1 jd =1

jd =1 jd =1 jd =1

379

= 4t (1 − t),

Yj (cj ) = 2d(1 − 2t), Yj (aj ) = Yj (bj ) =

t−1 t (d t t−1 (d

− 1)x, − 1)y,

cj aj = 0, cj bj = 0, (aj )k (aj )m =

− ym yk ), d   j =1 (aj )k (bj )m = xk ym − yk xm + j =2 (Aj y)k (Aj x)m .

jd =1

(bj )k (bj )m =

− xm xk ),

1−t t (δkm t 1−t (δkm

Proof. We only show the identities (1), (2), (5) and (6) which in the following are needed to calculate the second order differential operator Ld in (7.5) below. According to Proposition 7.3 and Lemma 7.4 (ii) we have  d  d d    j y, x2 + y, x2 = 4(1 − t)t cj2 = cj2 + c12 = 4(1 − t)t A j =1

j =2

j =2

and (1) follows. To prove (2) note that d 

Yj (cj ) =

j =1

d 

Yj (cj ) + Y1 (c1 )

j =2

=

d   1−t

 j y, xx · ∂ + j y − A A ∂x

t

j =2



 t  j y, xy · ∂ Aj x + A 1−t ∂y

    j y, x ∂ 2 (1 − t)tA j y, x + 2 (1 − t)tA ∂t    ∂  ∂ 1−t t  + y − x, yx · + −x + x, yy · t ∂x 1−t ∂y   ∂   + 2 (1 − t)tx, y 2 (1 − t)tx, y ∂t   d  2 2 2 j y, x − x, y = 2(1 − t) d y − A j =2



d  j y, x2 + x, y2 + 2t −d x 2 + A

 + 2(1 − 2t)

j =2 d 





j y, x + x, y A 2

2

j =2

= 2d (1 − t) y 2 − t x 2 = 2d(1 − 2t). j y = y 2 and Lemma 7.4 (ii). Next we prove (5). By Proposij y, A Here we have used A tion 7.3 together with Lemma 7.4 (i) and (ii) we have

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W. Bauer et al. / Bull. Sci. math. 137 (2013) 361–385 d 

aj cj =

j =1

d 

a j c j + a1 c 1

j =2



d  j y, xA j y + x, yy − = 2(1 − t) A j =2



!

d 

"  j y, x2 + x, y2 x A

j =2



= 2(1 − t) y 1 − x x = 0. 2

2

Similarly, one has d 

bj cj =

j =1

d 

c j b j + c1 b 1

j =2



d  j x, yA j x − x, yx + = 2t − A j =2



and (6) is shown.

"  d  2 2  Aj y, x + x, y y j =2

 = 2t x y − 1 y = 0 2

!

2

2

By summing (7.3) over j = 1, . . . , d and plugging in the relations (1)–(9) in Proposition 7.5 one obtains  d   d  1−t  ∂2 ∂ 2 Yj = (δkm − xm xk ) − (d − 1) x, t ∂xm ∂xk ∂x j =1 k,m=1  d    t ∂2 ∂ + (δkm − ym yk ) − (d − 1) y, 1−t ∂ym ∂yk ∂y k,m=1   d d   ∂2   +2 (Aj y)k (Aj x)m (xk ym − yk xm ) + ∂xk ∂ym j =2

k,m=1

∂2

∂ + 2d(1 − 2t) . (7.4) ∂t ∂t 2 We denote the operator in the last line which only depends on the parameter t ∈ (0, 1) by + 4t (1 − t)

∂2 ∂ + 2d(1 − 2t) . (7.5) ∂t ∂t 2 1 := Id. Note that in the cases Let B = {ej : j = 1, . . . , d} be the standard basis of Rd and put A j ∈ R(d) that fulfill the anti-commuting of d ∈ {4, 8} we can choose skew-symmetric matrices A i A j A j + A i = −2δij for j = 2, . . . , d and that leaves the set B ± := {±ej : j = relations A 1, . . . , d} invariant. More precisely, for each pair (i, k) with i, k ∈ {1, . . . , d} exactly one of the j ei = ek or A j ei = −ek can be solved for some index j = j (i, k) ∈ {1, . . . , 4}. equations A d 2f Let x = i=1 xi ei ∈ Rd and in the following we write H [f ] = ( ∂x∂i ∂x )i,j =1,...,d for the Hesj Ld := 4t (1 − t)

sian where f ∈ C ∞ (Rd ). Then we have

d d d    j x)t H [f ]A j x = j ei )t H [f ]A j em = (∗). (A xi xm (A j =1

i,m=1

j =1

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381

j1 ei = A j2 em . If i = m, then we can choose for any j1 ∈ {1, . . . , d} some index j2 = j1 such that A      Therefore, in the case where j1 , j2 = 1 we have Aj2 ei = −Aj2 Aj1 Aj2 em = −Aj1 em . It follows that j1 ei )t H [f ]A j1 em = −(A j2 em )t H [f ]A j2 ei = −(A j2 ei )t H [f ]A j2 em (A j1 ei )t H [f ]A j1 em = −et H [f ]em in the case where j2 = 1. Moreover, if i = m we have and (A i d 

j ei )t H [f ]A j ei = (A

j =1

d  ∂ 2f j =1

∂xj2

= f.

Hence we can simplify the above summation (∗) to d  j x)t H [f ]A j x = x 2 f. (A x H [f ]x + t

(7.6)

j =2

Let Sd−1 denote the Laplace–Beltrami operator on the sphere Sd−1 . Then it follows from (2.3) that: Sd−1 f = −

d 

j )2 f = − X(A

j =2

d 

j x)t H [f ]A j x + (d − 1)X(A 1 )f (A

j =2

  ∂f = x H [f ]x − x f + (d − 1) x, . ∂x In the last line we have used (7.6). Hence we can write (7.4) in the simpler form 2

t

d−1 

t −1 t Sd−1 ,x −  d−1 t t − 1 S ,y   d d   j y)k (A j x)m +2 (A (xk ym − yk xm ) +

Yj2 = −

j =1

j =2

k,m=1

∂2 + Ld . ∂xk ∂ym

sub Next, we comment on a relation between the two sub-Laplace operators sub 4 and 8 acting 7 15 on S and S and Jacobi polynomials. Moreover, we give a sequence of eigenfunctions of both operators that is different from the one in Proposition 5.6 and Theorem 6.3, respectively. Let F (a, b; c; t) be a hypergeometric function

F (a, b; c; t) =

∞  Γ (a + k) Γ (b + k) k=0

Γ (a)

Γ (b)

Γ (c) t k , Γ (c + k) k!

where Γ (s) denotes the usual Gamma function. Moreover, for n = 0, 1, . . . put Gn (α, γ ; t) = F (−n, α + n; γ ; t)  Γ (γ ) d n  γ +n−1 = t 1−γ (1 − t)γ −α t (1 − t)α+n−γ . n Γ (γ + n) dt The functions Gn (α, γ ; t) are called Jacobi polynomials and satisfy the following differential equation (Jacobi equation): t (1 − t)

 d d2 Gn + γ − (α + 1)t Gn + n(α + n)Gn = 0. dt dt 2

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Moreover, it is well-known that: 1

Gn (α, γ ; t)Gm (α, γ ; t)t γ −1 (1 − t)α−γ dt =

0

n!Γ (α + n − γ + 1) δn,m . (α + 2n)Γ (α + n)Γ (γ + n)

(7.7)

Put α = d − 1 and γ = d2 , then we have: d

d

Proposition 7.6. The operator Ld acting on L2 ((0, 1), 2−1 t 2 −1 (1 − t) 2 −1 dt) has the eigenfunctions: 



d d Gn (t) := Gn d − 1, ; t ≡ F −n, d − 1 + n; ; t , n ∈ N0 2 2 with the eigenvalues λn = −4(d − 1 + n)n of multiplicity one. In particular, one has the following inclusion {−λn = 4(d − 1 + n)n: n ∈ N0 } ⊂ σ (sub d ). The functions Gn can be seen as eigenfunctions of the sub-Laplacian sub d invariant under the action of the group Spin(d). Example 7.7. We express the first few eigenfunctions in terms of the original variable (u, v) n = 0:

G0 (t) ≡ 1

n = 1:

G1 (t) = 1 − 2t = v 2 − u 2

n = 2:

G2 (t) = (1 − t)2 − 2(1 + 2/d)t (1 − t) + t 2

n = 3:

G3 (t) = (1 − t)3 − 3(1 + 4/d)t (1 − t)2 + 3(1 + 4/d)t 2 (1 − t) − t 3 = ( v 2 − u 2 )[( u 2 + v 2 )2 − (4 + 12/d) u 2 v 2 ]

= ( u 2 + v 2 )2 − 4(1 + d −1 ) u 2 v 2

Finally, we want to remark that in the case of n +  1 the eigenfunctions Gn of sub d defined in the present section are orthogonal to the eigenspaces W (k) of Theorem 5.7 and Theorem 6.3, respectively. Let w = (u, v) ∈ Rd × Rd and recall that functions f ∈ W (k) with k = 0, . . . , can be expressed in the form  f (w) = q(z)¯z, uk ¯z, v −k e− z ΩS (z), Xd−1

where q ∈ P (Xd−1 ). Hence it is sufficient to show that 1 0= 2



 1

√ √ d d ¯z, tuk ¯z, 1 − tv −k Gn (t) dVSd−1 (u) dVSd−1 (v)t 2 −1 (1 − t) 2 −1 dt

Sd−1 Sd−1 0

whenever n +  1. In the case of = k = 0 we have n  1 and this identity follows from 1 0

d

d

Gn (t)t 2 −1 (1 − t) 2 −1 dt = 0.

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383

Hence we can assume that  1 and therefore it holds − k  1 or k  1. Without loss of generality let k  1 and by rewriting the integral as a product of three integrals over Sd−1 and (0, 1) we see that it is enough to show that under the assumption z2 = 0 one has  ¯z, uk dVSd−1 (u) = 0. (7.8) Sd−1

Note that z2 = 0 implies that the integrand in (7.8) forms a k-homogeneous harmonic polynomials which proves the assertion. 8. Open problems As is well-known, S3 and S7 are the only spheres that have a trivial tangent bundle. However, a product of spheres is parallelizable if one of them is odd-dimensional, cf. [15,20]. On the one hand the product S3 × S1 carries a TSR-structure of co-rank one and the spectral analysis of the corresponding sub-Laplacian was studied in [4]. On the other hand it can be shown that the product S2 × S1 admits no bracket generating sub-bundle D  T (S2 × S1 ). Also S3 × S3 has a co-rank three TSR-structure (cf. [10]). The spectral analysis of the sub-Laplacians on the spheres S3 , S7 and S15 which had been started in [5] and is continued in the present paper is still not complete. Below we pose some open problems that shall be studied in a future work. A. Determine the trivializable sub-Riemannian structures on Sn × Sk where k is odd. More generally, one may ask to classify all trivializable sub-Riemannian structures on finite products of spheres M := Sn1 × · · · × Snk . Let  be the Laplace–Beltrami operator on a closed Riemannian manifold M with (discrete) spectrum σ () = {0 = λ0 < λ1 < λ2 < · · ·} where λj denote the distinct eigenvalues with multiplicities mj . Then it is well-known that the corresponding spectral zeta function ζM (s) :=

∞  mj j =1

λsj

converges uniformly compact for Re(s) > dim2 M = sM and extends to a meromorphic function on the complex plane with only simple poles located on the real axis. Moreover, ζM (s) is holomorphic for s in a zero neighborhood. The number sM coincides with the order of the leading singularity in the asymptotic expansion of the heat trace of  as time tends to zero. In [4,5,7,8] the spectral zeta function ζ sub (s) of a sub-Laplace operator on S3 and S7 , on U (2) ∼ = S3 × S1 , on Heisenberg manifolds, and even on arbitrary compact nilmanifolds equipped with a trivializable sub-Riemannian structure of step two was studied (see also [9,10]). Let (N, D, ·,·) be one of these examples with corresponding distribution D of rank d and let sN ∈ R again denote the location of the largest pole of ζ sub (s). Then the following relation was shown in all the cases above: d (8.1) sN = dim N − . 2 Indeed (8.1) follows from the complete spectral data of the sub-Laplacian or from an explicit form of its heat kernel. To our knowledge the validation of (8.1) in a more general framework

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seems to be unknown. In particular, a relation of the type (8.1) is not clear in the case of the sub-Laplace operators studied in this paper. Hence a solution of the following problem would be interesting. sub 7 B. Let sub 4 and 5 be the sub-Laplace operators on S , which are induced by a rank four sub and a rank five distribution, respectively. Determine the full spectrum of sub 4 and 5 . sub 15 Similarly, determine the full spectrum of 8 on S . Analyze the meromorphic extension to the complex plane of the corresponding spectral zeta functions (in the case of existence). In particular, calculate the numbers sS3 and sS7 in all three cases and compare the result with (8.1).

Similar to [4,5,7,8] an explicit form of the heat kernel for the sub-Laplacians in B can be used to calculate their spectrum. Hence, the problems B and C below are closely related. C. Express the heat kernel of the sub-Laplacians in B in an explicit form (cf. [3,9,10,17]). Acknowledgement We wish the thank Prof. Ines Kath for many helpful discussions. In particular she pointed our various simplifications of our former proofs. References [1] J.F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962) 603–632. [2] M.F. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology 3 (Suppl. I) (1964) 3–38. [3] F. Baudoin, M. Bonnefort, The subelliptic heat kernel on SU(2), representations, asymptotics and gradient bounds, Math. Z. 263 (3) (2009) 647–672; F. Baudoin, J. Wang, The subelliptic heat kernel on the CR sphere, arXiv:1112.3084v1, 2011. [4] W. Bauer, K. Furutani, Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zetaregularized determinant, J. Geom. Phys. 60 (2009) 1209–1234. [5] W. Bauer, K. Furutani, Spectral analysis and geometry of a sub-Riemannian structure on S 3 and S 7 , J. Geom. Phys. 58 (2008) 1693–1738. [6] W. Bauer, K. Furutani, Quantization operators on quadrics, Kyushu J. Math. 62 (1) (2008) 221–258. [7] W. Bauer, K. Furutani, C. Iwasaki, Spectral zeta function of the sub-Laplacian on two step nilmanifolds, J. Math. Pures Appl. (9) 97 (3) (2012) 242–261. [8] W. Bauer, K. Furutani, C. Iwasaki, Spectral analysis and geometry of a sub-Laplacian and related Grushin type operators, in: Partial Differential Equations and Spectral Theory, in: Oper. Theory Adv. Appl., vol. 211, Springer, 2011, pp. 183–287. [9] R. Beals, B. Gaveau, P. Greiner, The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemannian complexes, Adv. Math. 121 (1996) 288–345. [10] R. Beals, B. Gaveau, P. Greiner, Hamilton–Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl. 79 (7) (2000) 633–689. [11] O. Calin, D.C. Chang, Sub-Riemannian geometry on the sphere S 3 , Canad. J. Math. 61 (4) (2009) 721–739. [12] D.C. Chang, I. Markina, A. Vasiliev, Sub-Riemannian geodesics on the 3-D sphere, Complex Anal. Oper. Theory 3 (2) (2009) 361–377. [13] L. Hörmander, Hypo-elliptic second order differential equations, Acta Math. 119 (1967) 147–171. [14] I.M. James, The Topology of Stiefel Manifolds, London Math. Soc. Lecture Note Ser., vol. 24, Cambridge Univ. Press, 1977. [15] M.A. Kervaire, Courbure intégrale généralisée et homotopie, Math. Ann. 131 (1956) 219–252. [16] H.B. Lawson, M.-L. Michelson, Spin Geometry, Princeton Univ. Press, 1989. [17] I. Markina, M.G. Molina, Sub-Riemannian geodesics and heat operator on odd dimensional spheres, Anal. Math. Phys. 2 (2) (2012) 123–147, arXiv:1008.5265v1.

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