LOCAL SYMMETRY OF HARMONIC SPACES AS DETERMINED BY THE SPECTRA OF SMALL GEODESIC SPHERES

TERESA ARIAS-MARCO AND DOROTHEE SCHUETH

Abstract. We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm |∇R| of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants |R|^{2} and |Ric|^{2} of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó.

1. Introduction

For a compact closed Riemannian manifold S the spectrum of S is the eigenvalue spectrum, including multiplicities, of the associated (positive semi-definite) Laplace operator Δ acting on smooth functions. A central question of inverse spectral geometry asks to which extent the geometry of S is determined by its spectrum. The so-called heat invariants a_{k}(S) of S are examples of geometric invariants which are determined by the spectrum of S; indeed, they are the coefficients in the famous asymptotic expansion by Minakshisundaram-Pleijel,

Tr(exp(−tΔ)) ∼ (4πt)^{− dim(S)/2} ∑_{k=0}^{∞} a_{k}(S)t^{k}

for t ↓ 0. The first few of these coefficients are given by

a_{0}(S) = vol(S), a_{1}(S) = \frac{1}{6} ∫_{S} scal dvol_{S} , a_{2}(S) = \frac{1}{360} ∫_{S}(5scal^{2} − 2|Ric|^{2} + 2|R|^{2})dvol_{S} ,

where scal, Ric, and R denote the scalar curvature, the Ricci operator, and the Riemannian curvature operator of S, respectively. In general, each a_{k}(S) is the integral over S of certain curvature invariants; see [9] for more information.

Nevertheless, there exist many examples of pairs or families of isospectral Riemannian manifolds (i.e., sharing the same spectrum) which are not isometric, sometimes not even locally isometric; see, for example, the survey article [10]. Still, many questions remain open; for example, it is not

2000 Mathematics Subject Classification. 53C25, 53C20, 58J50, 58J53, 53C30, 22E25. Key words and phrases. Harmonic space, curvature invariants, second fundamental form, Ledger’s recursion

formula, geodesic spheres, geodesic balls, heat invariants, isospectral manifolds, Damek-Ricci spaces.

The authors were partially supported by DFG Sonderforschungsbereich 647. The first author’s work has also been supported by D.G.I. (Spain) and FEDER Project MTM2010-15444, by Junta de Extremadura and FEDER funds, and the program “Estancias de movilidad en el extranjero ‘José Castillejo’ para jóvenes doctores” of the Ministry of Education (Spain).

Dedication. Dorothee Schueth would like to dedicate this article to her former high school teacher, Martin Berg. It was he who first made her see the beauty of Mathematics.

known whether a locally symmetric compact closed Riemannian manifold can be isospectral to a locally nonsymmetric Riemannian manifold.

On the other hand, the geometry of geodesic spheres plays an interesting role in Riemannian geometry. Chen and Vanhecke [4] formulated the following general question: To what extent do the properties of small geodesic spheres determine the Riemannian geometry of the ambient space? For example, Gray and Vanhecke [12] studied the information contained in the volume function of small geodesic spheres and investigated the question whether a Riemannian manifold whose geodesic spheres have the same volumes as spheres in euclidean space must necessarily be flat (answering this question in the positive under various choices of additional assumptions).

In the context of inverse spectral geometry, an interesting special version of the above question is: To what extent do the spectra of small geodesic spheres in a (possibly noncompact) Riemannian manifold M determine the geometry of M ? For example, Theorem 6.18 in [4] uses the information contained in the heat invariants a_{0} and a_{1} of small geodesic spheres (viewed as functions of the radius) and concludes local isometry of manifolds with adapted holonomy to certain model spaces under the assumption that all small geodesic spheres around each point are isospectral to the corresponding geodesic spheres in those model spaces.

In order to arrive at such and similar results, one uses radial power series expansions of curvature invariants, both of the ambient space and of the geodesic spheres. In general, even the first few coefficients of such expansions become very complicated; see, for example, the various formulas in [12] or [4]. One setting in which a quite restrictive geometric assumption on the ambient space makes the calculations considerably easier is the setting of harmonic ambient spaces.

A manifold is called harmonic if the volume density function of the geodesic exponential map is radial around each point. The notion of harmonicity was first introduced by Copson and Ruse [5] and intensively studied by Lichnerowicz [13]; see also [16]. Chapter 6 of the book by Besse [2] gives a useful survey of properties of harmonic spaces. One of the important facts about harmonic spaces is that they are Einstein [2] and hence analytic [7] (the latter result was not yet known when [2] was written). A locally symmetric manifold is harmonic if and only if it is flat or of rank one; the famous Lichnerowicz conjecture postulated that, conversely, each harmonic space is locally symmetric; i.e., satisfies ∇R = 0 (this condition is classically known to be equivalent to the condition that the local geodesic symmetries around each point be isometries). For the case of compact manifolds with finite fundamental group the Lichnerowicz conjecture was proved by Szabó [18]; however, Damek and Ricci gave examples of noncompact homogeneous harmonic manifolds which are not locally symmetric in infinitely many dimensions greater or equal to seven [6]. These spaces are usually referred to as Damek-Ricci spaces; see [1] for more information.

Specializing the above question about the information contained in the spectra of small geodesic spheres to the setting of harmonic spaces, we are able to prove in the present paper that the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric (see Corollary 1.2 below). More precisely, we obtain:

Main Theorem 1.1. Let M_{1} and M_{2} be harmonic spaces, and let p_{1} ∈ M_{1}, p_{2} ∈ M_{2}. If there exists ε > 0 such that for each r ∈ (0, ε) the geodesic spheres S_{r}(p_{1}) and S_{r}(p_{2}) are isospectral, then |∇R|_{p_{1}}^{2} = |∇R|_{p_{2}}^{2}.

Corollary 1.2. Let M_{1} and M_{2} be harmonic spaces. Assume that the hypothesis of Theorem 1.1 is satisfied for each pair of points p_{1} ∈ M_{1}, p_{2} ∈ M_{2}. Then M_{1} is locally symmetric if and only if M_{2} is locally symmetric.

LOCAL SYMMETRY OF HARMONIC SPACES AS DETERMINED BY SPECTRA OF GEODESIC SPHERES 3

In particular, note that in the case of locally homogeneous harmonic spaces M_{1} and M_{2}, the hypothesis of Theorem 1.1 implies that M_{1} is locally symmetric if and only if M_{2} is locally symmetric. Actually, all known examples of harmonic spaces are locally homogeneous; it is an open question whether there exist harmonic spaces which are not locally homogeneous.

Interestingly, our result implies that certain pairs of geodesic spheres which were claimed to be isospectral by Szabó in [20], [21] are actually not isospectral. In fact, Szabó considered (as the featured examples in a more general construction) geodesic spheres in certain symmetric spaces M_{1} (namely, quaternionic hyperbolic space of real dimension 4m ≥ 12) and in certain associated locally nonsymmetric Damek-Ricci spaces M_{2} of the same dimension (see also Remark 2.4). He stated that every pair of geodesic spheres S_{r}(p_{1}) ⊂ M_{1} and S_{r}(p_{2}) ⊂ M_{2} of the same radius was isospectral. Since these ambient manifolds M_{1} and M_{2} are harmonic and homogeneous, and M_{1} is locally symmetric while M_{2} is not, Corollary 1.2 immediately implies that Szabó’s result cannot be correct. Note that it was Fürstenau [8] who first discovered that actually there was a gap in Szabó’s isospectrality argument. The question of whether that proof could be repaired or not had since remained open; our result settles this question in the negative.

The incorrect examples of geodesic spheres mentioned above had the notable property that one is homogeneous and the other not. While it remains unknown whether a homogeneous metric on a sphere can be isospectral to a non-homogeneous one, Szabó in an earlier article [19] did construct a pair of isospectral metrics, only one of which is homogeneous, on the product of a sphere and a torus (those results are not affected by the error in the later papers).

We obtain analogs of our above results for geodesic balls endowed with either Dirichlet or Neumann boundary conditions; see Theorem 5.1 and Corollary 5.2. Similarly as above, this implies that Szabó’s examples in [20], [21] of isospectral geodesic balls (of any given radius) in quaternionic hyperbolic space of real dimension at least 12 and in certain associated locally nonsymmetric Damek-Ricci spaces were erroneous.

Note that nevertheless there do exist isospectral pairs and even continuous families of isospectral metrics on spheres and balls; the first such examples were due to Gordon [11].

In order to prove Theorem 1.1 we use the heat invariants a_{0}(S_{r}(p)) and a_{2}(S_{r}(p)) of geodesic spheres in harmonic spaces. In particular, we study the coefficients of r^{2} in the radial power series expansions of \frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} |Ric^{S}|^{2}dvol_{S_{r}(p)} and \frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} |R^{S}|^{2}dvol_{S_{r}(p)}, where Ric^{S} and R^{S} denote the Ricci operator and the Riemannian curvature operator of S_{r}(p). From the form of these coefficients (see Proposition 3.2 and its mean value version Proposition 4.2), we are able to conlude that the heat invariants a_{0} and a_{2} of S_{r}(p), viewed as functions of r, together determine the value of |∇R|^{2} at the midpoint p. Note that the same is not true for a_{0}(S_{r}(p)) alone; see Remark 2.4. Moreover, in the harmonic setting, the function r ↦ a_{1}(S_{r}(p)) = \frac{1}{6} ∫_{S_{r}(p)} scal^{S} dvol_{S_{r}(p)} does actually not contain more information than r ↦ a_{0}(S_{r}(p)) = vol(S_{r}(p)); see Remark 2.3. So it is indeed necessary for our purpose to consider a_{2}(S_{r}(p)).

Our computations rely heavily on the harmonicity of the ambient space. Note that they are related to certain more general computations in [12] and [4]; for example, Theorem 8.1 of [4] actually includes a kind of analog to our Proposition 3.2, and this even for general, not only for harmonic manifolds; however, that theorem contains information only on the coefficients of r^{j} with j ≤ 0, while we need the coefficients of r^{2}. In fact, in the harmonic case, the lower order coefficients turn out to be determined already by the function r ↦ a_{0}(S_{r}(p)) = vol(S_{r}(p)); see Proposition 3.2 and Remark 2.3(i).

This paper is organized as follows:

In Section 2 we gather the necessary background on harmonic spaces, mostly following [2]; in particular, we recall Ledger’s recursion formula for the power series expansion of the second fundamental form of geodesic spheres, and the resulting curvature identities in harmonic spaces.

In Section 3, we study the coefficients of r^{j} for j ≤ 2 in |Ric^{S_{r}(p)}|_{exp(ru)}^{2} and |R^{S_{r}(p)}|_{exp(ru)}^{2} for unit tangent vectors u of harmonic spaces, using the power series expansion of the second fundamental form and its radial covariant derivative, as well as the Taylor series expansion of the Riemannian curvature tensor. Proposition 3.2 is the main result of this section.

Section 4 is devoted to the proof of the Main Theorem 1.1. In preparation for this, we first derive a mean value version of Proposition 3.2; see Proposition 4.2.

Finally, in Section 5, we prove the analog of Theorem 1.1 for geodesic balls. We consider the heat coefficients of geodesic balls in harmonic spaces and show that the functions r ↦ a_{0}(B_{r}(p)) and r ↦ a_{2}(B_{r}(p)) (either for Dirichlet or for Neumann boundary conditions) together determine the value of |∇R|^{2} at the midpoint p of the balls. More precisely, we show that the coefficient of r^{3} in the radial power series expansion of the quotient a_{2}(B_{r}(p))/a_{0}(B_{r}(p)) is a sum of a nonzero multiple of |∇R|_{p}^{2} and of terms determined by the function r ↦ a_{0}(B_{r}(p)).

2. Preliminaries

2.1. Volume density and the shape operator of geodesic spheres. In the following, let M be a complete, connected, n-dimensional Riemannian manifold. For p ∈ M, let exp_{p} = exp |_{T_{p}M} : T_{p}M → M denote the associated geodesic exponential map. For a vector v ∈ T_{p}M we denote by γ_{v} the geodesic with initial velocity v. Identifying T_{v}(T_{p}M) with T_{p}M, we regard the differential d(exp_{p})_{v} as a linear map from T_{p}M to T_{exp v}M. We denote parallel translation along γ_{v} by P_{γ_{v}}^{s,t} : T_{γ_{v}(s)}M → T_{γ_{v}(t)}M. Given any unit vector u ∈ S_{1}(0_{p}) := {u ∈ T_{p}M | |u| = 1} and r ∈ ℝ, we consider the volume density

θ_{u}(r) := det(P_{γ_{u}}^{r,0} ∘ d(exp_{p})_{ru}).

Note that θ_{u}(r) is the infinitesimal volume distortion of the map exp_{p} at the point ru ∈ T_{p}M. Recall the Gauss lemma: The vector d(exp_{p})_{ru}u is a unit vector perpendicular to each d(exp_{p})_{ru}w with w ⊥ u. Thus, for each r ∈ (0, i(p)), where i(p) denotes the injectivity radius of M at p,

(1)  v_{u}(r) := r^{n−1}θ_{u}(r)

is the infinitesimal volume distortion at u of the map

S_{1}(0_{p}) ∋ u ↦ γ_{u}(r) = exp(ru) ∈ S_{r}(p),

where S_{r}(p) ⊂ M denotes the geodesic sphere of radius r around p. Let σ_{u}(r) denote the shape operator of S_{r}(p) at exp(ru); that is,

σ_{u}(r) := (∇ν)|_{T_{exp(ru)}M} ,

where ∇ is the Levi-Civita connection of M and ν denotes the outward pointing unit normal vector field on the geodesic ball B_{i(p)}(p) \ {p}. In particular, ν ∘ γ_{u} = γ̇_{u}, σ_{u}ν = 0, and the image of σ_{u}(r) is contained in T_{γ_{u}(r)}S_{r}(p). It is well-known that for all r ∈ (0, i(p)),

(2)  v′_{u}(r)/v_{u}(r) = Tr(σ_{u}(r)),

and that the covariant derivative σ′_{u} of the endomorphism field σ_{u} along γ_{u}|_{(0,i(p))} satisfies the so-called Riccati equation

(3)  σ′_{u} = −σ_{u}^{2} − Rγ̇_{u},

where R is the Riemannian curvature tensor of M, given by R(x, y)z = −∇_{x}∇_{y}z+∇_{y}∇_{x}z+∇_{[x,y]}z, and where R_{ν} := R(ν, .)ν. (Note that here we use the same sign for R as Besse [2].) Let C_{u}(r) := rσ_{u}(r). This endomorphism field along γ_{u}|_{(0,i(p))} is smoothly extendable to r = 0 by C_{u}(0) := I_{u}, where I_{u} is defined by I_{u}(u) = 0 and I_{u}|_{{u}^{⊥}} = Id_{{u}^{⊥}}. Moreover, from (3) one can derive Ledger’s recursion formula for the covariant derivatives of C_{u} at r = 0 (see, e.g., [4]):

(k − 1)C_{u}^{(k)}(0) = −k(k − 1)R_{u}^{(k−2)} − ∑_{ℓ=0}^{k} \binom{k}{ℓ}C_{u}^{(ℓ)}(0)C_{u}^{(k−ℓ)}(0)

for all k ∈ ℕ, where R_{u}^{(k)} is the k-th covariant derivative of the endomorphism field Rγ̇_{u} along γ_{u} at r = 0. This formula allows one to successively compute the C_{u}^{(k)}(0) in terms of the endomorphisms R_{u}^{(k)} of T_{p}M. Forming the Taylor series of C_{u} and dividing by r, one obtains (see, e.g., [2], [4]):

\begin{aligned}P_{γ_{u}}^{r,0} ∘ σ_{u}(r) ∘ P_{γ_{u}}^{0,r} = & \frac{1}{r}I_{u} − \frac{r}{3}R_{u} − \frac{r^{2}}{4} R′_{u} − (\frac{1}{10}R′′_{u} + \frac{1}{45}R_{u}R_{u})r^{3} \\ & − (\frac{1}{36}R′′′_{u} + \frac{1}{72}R_{u}R′_{u} + \frac{1}{72}R′_{u}R_{u})r^{4} \\ & − (\frac{1}{168}R_{u}^{(4)} + \frac{1}{210}R_{u}R′′_{u} + \frac{1}{210}R′′_{u}R_{u} + \frac{1}{112}R′_{u}R′_{u} + \frac{2}{945}R_{u}R_{u}R_{u})r^{5} \\ & + O(r^{6}). \end{aligned} ; (4)

2.2. Curvature identities in harmonic spaces. The manifold M is called a harmonic space if for every p ∈ M the above function θ_{u} does not depend on u ∈ S_{1}(0_{p}). An equivalent condition is that for all r ∈ (0, i(p)), the geodesic spheres S_{r}(p) have constant mean curvature (recall equations (1), (2)). For more information on harmonic spaces see [16] or [2]. If M is harmonic then the function θ_{u} does in fact not even depend on p; that is, there exists θ : [0, ∞) → ℝ such that

θ_{u}(r) = θ(r)

for all u ∈ T M with |u| = 1. Moreover, even the local or infinitesimal versions of the above condition imply that the manifold is Einstein [2] and therefore analytic [7]. Hence, the local or infinitesimal versions of the above conditions are equivalent to the global versions. Since θ_{u}(r) depends only on r, so does v_{u}(r) and hence Tr(σ_{u}(r)). From this one can successively derive, using the expansion (4):

Proposition 2.1 (see [2], Chapter 6). If M is harmonic then there exist constants C, H, L ∈ ℝ such that for all p ∈ M and all u ∈ T_{p}M with |u| = 1:

(i) Tr(R_{u}) = C; in particular:

(ii) Tr(R_{u}^{(k)}) = 0 for all k ∈ ℕ.

(iii) Tr(R_{u}R_{u}) = H; in particular:

(iv) Tr(R_{u}R′_{u}) = 0 and

(v) Tr(R_{u}R′′_{u}) = −Tr(R′_{u}R′_{u}).

(vi) Tr(32R_{u}R_{u}R_{u} − 9R′_{u}R′_{u}) = L.

In fact, taking traces in (4), one has in the harmonic case:

(5)  Tr(σ_{u}(r)) = (n − 1)\frac{1}{r} − \frac{1}{3}Cr − \frac{1}{45}Hr^{3} − \frac{1}{15120}Lr^{5} + O(r^{7})

for r ↓ 0 and all u ∈ T M with |u| = 1. Note that Proposition 2.1(i) just says that the Einstein constant of M is C; that is, Ric = CId on each T_{p}M. Recall that the Ricci operator is defined by ⟨Ric(x), y⟩ = Tr(R(x, .)y) for all x, y ∈ T_{p}M and all p ∈ M. From Proposition 2.1 one can further derive:

Proposition 2.2 (see [2], Chapter 6). If M is harmonic, then for the above constants C, H, L and each p ∈ M:

(i) ⟨R(x, .) . , R(y, .) . ⟩ = \frac{2}{3}((n + 2)H − C^{2})⟨x, y⟩ for all x, y ∈ T_{p}M; in particular:

(ii) |R|_{p}^{2} = \frac{2}{3}n((n + 2)H − C^{2}).

(iii) 32(nC^{3} + \frac{9}{2}C|R|_{p}^{2} + \frac{7}{2} R̂(p) − R^{∘}(p)) − 27|∇R|_{p}^{2} = n(n + 2)(n + 4)L.

Here, the functions R̂, R^{∘} ∈ C^{∞}(M) are certain curvature invariants of order six which are defined as follows: If {e_{1}, …, e_{n}} is an orthonormal basis of T_{p}M and R_{ijkℓ} := ⟨R(e_{i}, e_{j})e_{k}, e_{ℓ}⟩, then

R^{∘}(p) := ∑_{i,j,k,ℓ,a,b} R_{ijkℓ}R_{jaℓb}R_{aibk}, R̂(p) := ∑_{i,j,k,ℓ,a,b} R_{ijkℓ}R_{kℓab}R_{abij}.

Note that the term nC^{3} in Proposition 2.2(iii) reads nC^{2} in the corresponding equation 6.67 in [2], but this was obviously a misprint (note that curvature terms of different order cannot occur here); see also formula (3.1) in [22].

Proposition 2.2(iii) will be used in Section 4, together with the following formula which actually holds in any Einstein manifold; see formula (6-7) in [14] or formula (11.3) in [12]:

(6)  −\frac{1}{2}Δ(|R|^{2}) = 2C|R|^{2} − R̂ − 4R^{∘} + |∇R|^{2},

where Δ denotes the Laplace operator on functions, that is, Δf = − ∑_{i}(e_{i}(e_{i}f) − (∇_{e_{i}}e_{i})f) for local orthonormal frames {e_{1}, …, e_{n}}. (Again, there is a misprint in two of the coefficients in the corresponding formula 6.65 in [2].) If M is harmonic, then the left hand side of (6) is zero by Proposition 2.2(ii). Finally, we recall the following well-known observations which will be used in Section 4:

Remark 2.3. Let M be an n-dimensional harmonic space with volume density function θ as above.

(i) For any p ∈ M, the volume of the geodesic sphere S_{r}(p) with 0 < r < i(p) equals the volume ω_{n−1} of the standard unit sphere S^{n−1} in ℝ^{n} multiplied by the factor

v(r) := r^{n−1}θ(r)

Note that v(r) = v_{u}(r) for each unit vector u ∈ T M, where v_{u} is the function defined in (1). The function v determines the volume growth function v′/v of the geodesic spheres, and thus it determines, by (2), the function Tr(σ_{u}(r)) (which is independent of u). By (5), the function which associates to small values of r the volume of geodesic spheres of radius r in a given harmonic space M determines the constants C, H, L (and of course n) associated with M.

(ii) Let scal = nC denote the scalar curvature of M. Let p ∈ M, fix some r ∈ (0, i(p)), and let scal^{S} denote the scalar curvature function of S_{r}(p). A routine calculation using the Gauss equation shows that for each unit vector u ∈ T_{p}M we have

scal^{S}(exp(ru)) = scal − 2⟨Ric(γ̇_{u}(r)), γ̇_{u}(r)⟩ + (Tr(σ_{u}(r)))^{2} − Tr(σ_{u}(r)^{2})

which by the Einstein condition and equations (2) and (3) implies

scal^{S}(exp(ru)) = (n − 2)C + (v′(r)/v(r))^{2} + Tr(σ′_{u}(r)) + Tr(R_{γ̇_{u}(r)})
= (n − 2)C + (v′(r)/v(r))^{2} + (v′/v)′(r) + C = (n − 1)C + v′′(r)/v(r).

Therefore, geodesic spheres in M have constant scalar curvature, and the respective constant depends only on the radius, not on the midpoint. Finally, using (i) one concludes that the function which associates to small values of r the scalar curvature of geodesic spheres of radius r is determined already by the function which associates to small values of r the volume of geodesic spheres of radius r.

Remark 2.4. As mentioned in the Introduction, the aim of this paper is to show that in harmonic spaces, the heat invariants a_{0}(S_{r}(p)) = vol(S_{r}(p)) and a_{2}(S_{r}(p)), viewed as functions of r, together determine |∇R|_{p}^{2} . This is not the case for a_{0} alone, as manifested by certain pairs of Damek-Ricci spaces. A Damek-Ricci space AN is a certain type of solvable Lie groups with left invariant metric, namely, the standard 1-dimensional solvable extension of a simply connected Riemannian nilmanifold N of Heisenberg type. The volume density function of AN is radial and depends only on the dimensions of N and its center [6]; see also the book [1]. Within the class of Damek-Ricci spaces, there exist pairs of symmetric spaces AN and locally nonsymmetric spaces AN′ where N and N′ have the same dimension and so do their centers. (In fact, certain such pairs AN and AN′ were the ambient manifolds used by Szabó in [20], [21]; recall the Introduction.) In particular, geodesic spheres of the same radius in AN and AN′ have the same volume. This shows that in harmonic spaces, the function r ↦ vol(S_{r}(p)) alone does not determine |∇R|_{p}^{2} . In turn, Remark 2.3 shows that in harmonic spaces, the function r ↦ a_{0}(S_{r}(p)) = vol(S_{r}(p)) already determines the function r ↦ a_{1}(S_{r}(p)) = \frac{1}{6} ∫_{S_{r}(p)} scal^{S} dvol_{S_{r}(p)} . Therefore, we need to consider a_{2}(S_{r}(p)). The next section gives some necessary preparations for this.

3. Radial expansions of |Ric|^{2} and |R|^{2} for geodesic spheres in harmonic spaces

In this section we will describe a certain coefficient in the radial power series expansions of the curvature invariants |Ric|^{2} and |R|^{2} of geodesic spheres in harmonic spaces. First we need the following lemma.

Lemma 3.1. Let M be an n-dimensional harmonic space, and let C and H be the constants from Proposition 2.1. Let p ∈ M, and let S := S_{r}(p) be a geodesic sphere around p with radius r ∈ (0, i(p)), endowed with the induced Riemannian metric. Let u be a unit vector in T_{p}M, let σ := σ_{u}(r) be as in Section 2, and write σ′ := σ′_{u}(r). Let R^{S} and Ric^{S} denote the curvature tensor, resp. the Ricci operator, of S. Then in the point q := exp(ru) ∈ S we have:

(i) |Ric^{S}|_{q}^{2} = (n − 1)C^{2} + 2C(Tr(σ))^{2} + (Tr(σ))^{2}Tr(σ^{2}) + 2CTr(σ′) + 2Tr(σ)Tr(σσ′) + Tr(σ′σ′),
(ii) |R^{S}|_{q}^{2} = \frac{2}{3}(n − 4)((n + 2)H − C^{2}) + 4H + 2(Tr(σ^{2}))^{2} − 2Tr(σ^{4}) + 4 ∑_{i=1}^{n} Tr(σ ∘ R(e_{i}, .)σe_{i}),
where {e_{1}, …, e_{n}} is an orthonormal basis of T_{q}M.

Proof. (i) Let ν be the outward pointing radial unit vector field as in Section 2. From the Gauss equation one easily derives the following formula whose analog is valid for submanifolds of codimension one in arbitrary Riemannian manifolds:

Ric_{q}^{S} = (Ric − R_{ν_{q}} + Tr(σ)σ − σ^{2})|_{T_{q}S}

Using the Einstein condition and the Riccati equation (3), this formula becomes in our situation:

Ric_{q}^{S} = (CId + Tr(σ)σ + σ′)|_{T_{q}S}

(see also [15], p. 67). Now one obtains the desired formula immediately, keeping in mind that both σ and σ′ are symmetric and annihilate ν_{q}.

(ii) Choose an orthonormal basis {e_{1}, …, e_{n}} of T_{q}M such that e_{1} = ν_{q}. For all i, j, k, ℓ ∈ {2, …, n} we have by the Gauss equation (recall our sign convention for R):

⟨R^{S}(e_{i}, e_{j})e_{k}, e_{ℓ}⟩ = ⟨R(e_{i}, e_{j})e_{k}, e_{ℓ}⟩ + ⟨σe_{i}, e_{k}⟩⟨σe_{j}, e_{ℓ}⟩ − ⟨σe_{j}, e_{k}⟩⟨σe_{i}, e_{ℓ}⟩.

Squaring both sides and forming the sum over i, j, k, ℓ, while recalling that σ is symmetric and annihilates e_{1}, we get

|R^{S}|_{q}^{2} = ∑_{i,j,k,ℓ=2}^{n}⟨R(e_{i}, e_{j})e_{k}, e_{ℓ}⟩^{2} + |σ|^{2}|σ|^{2} + |σ|^{2}|σ|^{2}

− 2|σ^{2}|^{2} + 2 ∑_{i,j=1}^{n}⟨R(e_{i}, e_{j})σe_{i}, σe_{j}⟩ − 2 ∑_{i,j=1}^{n}⟨R(e_{i}, e_{j})σe_{j}, σe_{i}⟩

= ∑_{i,j,k,ℓ=2}^{n}⟨R(e_{i}, e_{j})e_{k}, e_{ℓ}⟩^{2} + 2(Tr(σ^{2}))^{2} − 2Tr(σ^{4}) + 4 ∑_{i=1}^{n} Tr(σ ∘ R(e_{i}, .)σe_{i}).

The desired formula now follows from the fact that the first sum on the right hand side is equal to |R|_{q}^{2} − 4|R(e_{1}, .) . |^{2} + 4|R_{e_{1}}|^{2} which by Proposition 2.2(i), (ii) and Proposition 2.1(iii) becomes \frac{2}{3}(n − 4)((n + 2)H − C^{2}) + 4H. □

Using the radial power series expansion of σ together with the previous lemma, we will make conclusions concerning the first few coefficients of the radial expansions of |Ric^{S}|^{2} and |R^{S}|^{2}. The following proposition will be the key of the proof of the Main Theorem 1.1. Actually we will use only the statements about α_{2} and β_{2} in this proposition.

Proposition 3.2. Let M be an n-dimensional harmonic space, and let C, H, and L be the constants from Proposition 2.1. Let p ∈ M, and let u be a unit vector in T_{p}M. Then

|Ric^{S_{r}(p)}|_{exp(ru)}^{2} = α_{−4}r^{−4} + α_{−2}r^{−2} + α_{0} + α_{2}(u)r^{2} + O(r^{3}) and
|R^{S_{r}(p)}|_{exp(ru)}^{2} = β_{−4}r^{−4} + β_{−2}r^{−2} + β_{0} + β_{2}(u)r^{2} + O(r^{3})

for r ↓ 0, where the coefficients α_{i} and β_{i} for i ∈ {−4, −2, 0} are constants depending only on n, C, and H. Moreover,

α_{2}(u) = α̂_{2} + \frac{1}{16}Tr(R′_{u}R′_{u}) and
β_{2}(u) = β̂_{2} + \frac{4}{9} ∑_{i=1}^{n} Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i}),

where α̂_{2} and β̂_{2} are constants depending only on n, C, H, and L, and where {e_{1}, …, e_{n}} is an orthonormal basis of T_{p}M.

Proof. We use Lemma 3.1 together with the power series expansions (4), (5) of σ := σ_{u}(r) and Tr(σ). Let us first consider |Ric^{S_{r}(p)}|_{exp(ru)}^{2} and the individual contributions of the nonconstant terms in Lemma 3.1(i) to its expansion. By (5) we have

(Tr(σ))^{2} = ((n − 1)\frac{1}{r} − \frac{1}{3}Cr − \frac{1}{45}Hr^{3} − \frac{1}{15120}Lr^{5})^{2} + O(r^{6})

for r ↓ 0. Moreover, from the expansion (4) and Proposition 2.1 one gets

(7)  Tr(σ^{2}) = (n − 1)\frac{1}{r^{2}} − \frac{2}{3}C + \frac{1}{15}Hr^{2} + \frac{1}{3024}Lr^{4} + O(r^{5}).

Further,

Tr(σ′) = \frac{d}{dr} Tr(σ) = −(n − 1)\frac{1}{r^{2}} − \frac{1}{3}C − \frac{1}{15}Hr^{2} + O(r^{4})

by (5), and

2Tr(σσ′) = \frac{d}{dr} Tr(σ^{2}) = −2(n − 1)\frac{1}{r^{3}} + \frac{2}{15}Hr + \frac{1}{756}Lr^{3} + O(r^{4})

by (7). Using these expansions and (5), one easily checks that each of the the four individual terms 2C(Tr(σ))^{2}, (Tr(σ))^{2}Tr(σ^{2}), 2CTr(σ′), and 2Tr(σ)Tr(σσ′) appearing on the right hand side of Lemma 3.1(i) has the property that the corresponding coefficients of r^{−4}, r^{−2}, r^{0} depend only on n, C, H, the coefficient of r^{2} depends only on n, C, H, L, and the coefficients of r^{−3}, r^{−1}, r vanish.

It remains to consider the term Tr(σ′σ′) in Lemma 3.1(i). From (4) we get

P_{γ_{u}}^{r,0} ∘ σ′ ∘ P_{γ_{u}}^{0,r} = − \frac{1}{r^{2}} I_{u} − \frac{1}{3}R_{u} − \frac{r}{2}R′_{u} − (\frac{3}{10}R′′_{u} + \frac{1}{15}R_{u}R_{u})r^{2}

− (\frac{1}{9}R′′′_{u} + \frac{1}{18}R_{u}R′_{u} + \frac{1}{18}R′_{u}R_{u})r^{3}

− (\frac{5}{168}R_{u}^{(4)} + \frac{1}{42}R_{u}R′′_{u} + \frac{1}{42}R′′_{u}R_{u} + \frac{5}{112}R′_{u}R′_{u} + \frac{2}{189}R_{u}R_{u}R_{u})r^{4}

+ O(r^{5})

and thereby, using Proposition 2.1:

\begin{aligned}Tr(σ′σ′) = & (n − 1)\frac{1}{r^{4}} + \frac{2}{3} \frac{C}{r^{2}} + \frac{11}{45}H \\ & + ((−\frac{2}{21} + \frac{5}{56} − \frac{1}{5} + \frac{1}{4})Tr(R′_{u}R′_{u}) + (\frac{4}{189} + \frac{2}{45})Tr(R_{u}R_{u}R_{u}))r^{2} + O(r^{3}). \end{aligned}

The coefficient of r^{2} in the latter expansion is

\frac{37}{840}Tr(R′_{u}R′_{u}) + \frac{62}{945}Tr(R_{u}R_{u}R_{u})

which by Proposition 2.1(vi) turns out to be

\frac{62}{32 · 945}L + (\frac{37}{840} + \frac{9 · 62}{32 · 945})Tr(R′_{u}R′_{u}) = \frac{31}{15120}L + \frac{1}{16}Tr(R′_{u}R′_{u}).

This concludes the proof of the statements concerning the expansion of |Ric^{S_{r}(p)}|_{exp(ru)}^{2} .

We now turn to |R^{S_{r}(p)}|_{exp(ru)}^{2} and study the individual contributions of the nonconstant terms in Lemma 3.1(ii) to its expansion. Squaring (7), we see that in the expansion of the term 2(Tr(σ^{2}))^{2} the coefficients of r^{−4}, r^{−2}, r^{0} depend only on n, C, H, the coefficient of r^{2} depends only on n, C, H, L, and the coefficients of r^{−3}, r^{−1}, r vanish.

Regarding the term −2Tr(σ^{4}) we obtain from (4):

P_{γ_{u}}^{r,0} ∘ σ^{4} ∘ P_{γ_{u}}^{0,r} = \frac{1}{r^{4}} I_{u} − \frac{4}{3r^{2}} R_{u} − \frac{1}{r}R′_{u} + (−\frac{2}{5}R′′_{u} + \frac{26}{45}R_{u}^{2}) + (−\frac{1}{9}R′′′_{u} + \frac{4}{9}(R_{u}R′_{u} + R′_{u}R_{u}))r

+ (−\frac{1}{42}R_{u}^{(4)} + (−\frac{2}{105} + \frac{1}{5})(R′′_{u}R_{u} + R_{u}R′′_{u}) + (−\frac{1}{28} + \frac{3}{8})R′_{u}R′_{u} + (−\frac{8}{945} + \frac{4}{45} − \frac{4}{27})R_{u}^{3})r^{2}

+ O(r^{3})

for r ↓ 0. Using Proposition 2.1 we get

−2Tr(σ^{4}) = − 2(n − 1)\frac{1}{r^{4}} + \frac{8}{3} \frac{C}{r^{2}} − \frac{52}{45}H

+ ((−\frac{8}{105} + \frac{4}{5} + \frac{1}{14} − \frac{3}{4})Tr(R′_{u}R′_{u}) + (\frac{16}{945} − \frac{8}{45} + \frac{8}{27})Tr(R_{u}R_{u}R_{u}))r^{2} + O(r^{3}).

The coefficient of r^{2} in the latter expansion is

\frac{19}{420}Tr(R′_{u}R′_{u}) + \frac{128}{945}Tr(R_{u}R_{u}R_{u})

which by Proposition 2.1(vi) equals

(8)  \frac{128}{32 · 945}L + (\frac{19}{420} + \frac{9 · 128}{32 · 945})Tr(R′_{u}R′_{u}) = \frac{4}{945}L + \frac{1}{12}Tr(R′_{u}R′_{u}).

It remains to consider the term 4 ∑_{i=1}^{n} Tr(σ ∘ R(e_{i}, .)σe_{i}) in Lemma 3.1(ii). We make some preliminary observations. For k ∈ ℕ_{0}, let R^{(k)}, resp. Ric^{(k)} denote the k-th covariant derivative of the curvature tensor, resp. the Ricci operator, along γ_{u} at r = 0. We will use the the Taylor series expansion of the Riemannian curvature tensor along γ_{u} (recall that M is analytic):

(9)  P_{γ_{u}}^{r,0} ∘ R_{γ_{u}(r)} ∘ P_{γ_{u}}^{0,r} = ∑_{k=0}^{∞} \frac{r^{k}}{k!} R^{(k)}

Moreover, Ric^{(k)} = 0 for k ≥ 1 since M is Einstein. Note that Ric|_{T_{p}M} = ∑_{i=1}^{n} R_{e_{i}} and similarly on each T_{γ_{u}(r)}M if we extend {e_{1}, …, e_{n}} parallelly along γ_{u}. For any k ∈ ℕ_{0} we have, using Proposition 2.1:

∑_{i=1}^{n} Tr(I_{u} ∘ R^{(k)}(e_{i}, .)I_{u}e_{i}) = Tr(I_{u} ∘ Ric^{(k)}) − Tr(I_{u} ∘ R_{u}^{(k)})

= Tr(Ric^{(k)}) − ⟨Ric^{(k)}u, u⟩ − Tr(R_{u}^{(k)}) = \begin{cases}(n − 2)C, & k = 0, \\ 0, & k ≥ 1. \end{cases}

(10)

Moreover,

∑_{i=1}^{n} Tr(R_{u} ∘ R^{(k)}(e_{i}, .)I_{u}e_{i}) = Tr(R_{u} ∘ Ric^{(k)}) − Tr(R_{u}R_{u}^{(k)})

= \begin{cases}C^{2} − H, & k = 0, \\ −Tr(R_{u}R_{u}^{(k)}), & k ≥ 1, \end{cases}

(11)

∑_{i=1}^{n} Tr(R′_{u} ∘ R^{(k)}(e_{i}, .)I_{u}e_{i}) = Tr(R′_{u} ∘ Ric^{(k)}) − Tr(R′_{u}R_{u}^{(k)})

= \begin{cases}0, & k = 0, \\ −Tr(R′_{u}R_{u}^{(k)}), & k ≥ 1, \end{cases}

(12)

(13)  ∑_{i=1}^{n} Tr(R′′_{u} ∘ R(e_{i}, .)I_{u}e_{i}) = Tr(R′′_{u} ∘ Ric) − Tr(R′′_{u}R_{u}) = 0 + Tr(R′_{u}R′_{u}),

(14) ∑_{i=1}^{n} Tr(R_{u}R_{u} ∘ R(e_{i}, .)I_{u}e_{i}) = Tr(R_{u}R_{u} ∘ Ric) − Tr(R_{u}R_{u}R_{u}) = CH − Tr(R_{u}R_{u}R_{u}).

Note that for any pair of symmetric endomorphisms F, G of T_{p}M we have

(15)  ∑_{i=1}^{n} Tr(F ∘ R^{(k)}(e_{i}, .)Ge_{i}) = ∑_{i=1}^{n} Tr(G ∘ R^{(k)}(e_{i}, .)F e_{i})

by the symmetries of the curvature operator. Keeping the expansions (4) and (9) in mind, we see that the expression in (10) contributes only to the coefficient of r^{−2} in the expansion of ∑_{i=1}^{n} Tr(σ∘ R(e_{i}, .)σe_{i}), the expression in (11) contributes to the coefficients of r^{0} and r^{2} (and higher order), the expressions in (12), (13), (14) contribute to the coefficient of r^{2} (and higher order). The only additional contribution to the coefficient of r^{2} is given by the sum of Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i}). Recalling (15) (and multiplying R^{(k)} by 1/k!), we obtain from (4), 2.1(iv), (9), and the above observations:

4 ∑_{i=1}^{n} Tr(σ ∘ R(e_{i}, .)σe_{i}) = 4((n − 2)\frac{C}{r^{2}} − \frac{2}{3}(C^{2} − H)

+ [\frac{2}{3 · 2!}Tr(R_{u}R′′_{u}) + (\frac{2}{4} − \frac{2}{10})Tr(R′_{u}R′_{u}) − \frac{2}{45}CH + \frac{2}{45}Tr(R_{u}R_{u}R_{u})

+ \frac{1}{9} ∑_{i=1}^{n} Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i})]r^{2}) + O(r^{3}).

By Proposition 2.1(v), the coefficient of r^{2} in the latter expansion is

−\frac{8}{45}CH − \frac{2}{15}Tr(R′_{u}R′_{u}) + \frac{8}{45}Tr(R_{u}R_{u}R_{u}) + \frac{4}{9} ∑_{i=1}^{n} Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i})

By Proposition 2.1(vi), the two terms involving Tr(R′_{u}R′_{u}) and Tr(R_{u}R_{u}R_{u}) become

\frac{8}{32 · 45}L + (−\frac{2}{15} + \frac{9 · 8}{32 · 45})Tr(R′_{u}R′_{u}) = \frac{1}{180}L − \frac{1}{12}Tr(R′_{u}R′_{u}).

Combining this with the result for the r^{2}-coefficient of −2Tr(σ^{4}) from (8), we conclude that the terms involving Tr(R′_{u}R′_{u}) in the coefficient of r^{2} in the power series expansion of |R^{S_{r}(p)}|_{exp(ru)}^{2} cancel each other, and the only remaining term apart from those which depend solely on n, C, H, L is \frac{4}{9} ∑_{i=1}^{n} Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i}), as claimed. □

Remark 3.3. For the purpose of the proof of the Main Theorem 1.1 in Section 4, which we will perform using the heat invariants a_{0}(S_{r}(p)) = vol(S_{r}(p)) and a_{2}(S_{r}(p)) = \frac{1}{360} ∫_{S_{r}(p)}(5(scal^{S})^{2} − 2|Ric^{S}|^{2} +2|R^{S}|^{2})dvol_{S_{r}(p)}, we would actually not have needed the exact statement of the previous proposition – which might, however, be interesting in its own right. Rather, we could have restricted our attention to the term Tr(σ′σ′) in the expression of |Ric^{S}|_{γ_{u}(r)}^{2} in Lemma 3.1(i), and to the last two terms in the expression of |R^{S}|_{γ_{u}(r)}^{2} in Lemma 3.1(ii). In fact, even without the explicit calculation of the expansion of the other terms, one easily sees that those are determined by the volume function r ↦ a_{0}(S_{r}(p)) = vol(S_{r}(p)) of the geodesic spheres (which is just the function v multiplied by the volume of the standard unit sphere, see Remark 4.1 below). More precisely, in the spirit of Remark 2.3 we obtain

2C(Tr(σ))^{2} = 2C(v′/v)^{2},
(Tr(σ))^{2}Tr(σ^{2}) = (v′/v)^{2}(−(v′/v)′ − C),
2CTr(σ′) = 2CTr(σ)′ = 2C(v′/v)′,
2Tr(σ)Tr(σσ′) = 2v′/v · \frac{1}{2}(Tr(σ^{2}))′ = v′/v · (−(v′/v)′′),
2(Tr(σ^{2}))^{2} = 2(−(v′/v)′ − C)^{2}.

4. Proof of the Main Theorem

In this section we will first derive an integrated version of Proposition 3.2. Using this and the heat invariants a_{0}, a_{1}, a_{2} of geodesic spheres in harmonic spaces we will then prove our Main Theorem 1.1. We need the following general remark on mean values.

Remark 4.1. In any harmonic space M, the average (or mean value) of a smooth function f on a geodesic sphere S_{r}(p) (with 0 < r < i(p)) is the same as the average of f(exp(r .)) over the unit sphere S_{1}(0_{p}) in T_{p}M. More explicitly: Let ω_{n−1} denote the volume of the (n − 1)- dimensional standard sphere. In particular, ω_{n−1} is the volume of S_{1}(0_{p}). Recall from Section 2 that θ(r) = θ_{u}(r) is independent of u (and even of p) by harmonicity. We have

vol(S_{r}(p)) = r^{n−1}θ(r)ω_{n−1} = v(r)ω_{n−1} ,

and for any smooth function f on S_{r}(p),

\frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} f dvol_{S_{r}(p)} = \frac{1}{v(r)ω_{n−1}} ∫_{S_{1}(0_{p})} f(exp(ru))v(r) du

= \frac{1}{ω_{n−1}} ∫_{S_{1}(0_{p})} f(exp(ru)) du.

Now we can give an “integrated” version of Proposition 3.2.

Proposition 4.2. Let M be an n-dimensional harmonic space, and let C, H, and L be the constants from Proposition 2.1. Let p ∈ M. Then

\frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} |Ric^{S_{r}(p)}|^{2}dvol_{S_{r}(p)} = α_{−4}r^{−4} + α_{−2}r^{−2} + α_{0} + \overline{α}_{2}r^{2} + O(r^{3}) and
\frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} |R^{S_{r}(p)}|^{2}dvol_{S_{r}(p)} = β_{−4}r^{−4} + β_{−2}r^{−2} + β_{0} + \overline{β}_{2}r^{2} + O(r^{3})

for r ↓ 0, where the coefficients α_{i} and β_{i} for i ∈ {−4, −2, 0} are the constants from Proposition 3.2 depending only on n, C, and H. Moreover,

\overline{α}_{2} = α̃_{2} + \frac{3}{16n(n + 2)(n + 4)}|∇R|_{p}^{2} and
\overline{β}_{2} = β̃_{2} + \frac{1}{8n(n + 2)}|∇R|_{p}^{2},

where α̃_{2} and β̃_{2} are constants depending only on n, C, H, and L.

Proof. For any unit vector u in T_{p}M, let α_{2}(u) and β_{2}(u) be the coefficients from Proposition 3.2. Using that proposition and Remark 4.1, we only need to show that

\overline{α}_{2} := \frac{1}{ω_{n−1}} ∫_{S_{1}(0_{p})} α_{2}(u) du = α̂_{2} + \frac{1}{ω_{n−1}} ∫_{S_{1}(0_{p})} \frac{1}{16}Tr(R′_{u}R′_{u}) du and

\overline{β}_{2} := \frac{1}{ω_{n−1}} ∫_{S_{1}(0_{p})} β_{2}(u) du = β̂_{2} + \frac{1}{ω_{n−1}} ∫_{S_{1}(0_{p})} \frac{4}{9} ∑_{i=1}^{n} Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i}) du

are of the claimed form, where α̂_{2}, β̂_{2} are as in Proposition 3.2. For \overline{α}_{2} this follows immediately (with α̃_{2} := α̂_{2}) from the following formula (see the proof of Theorem 5.7 of [15]; details of the computation can be found on p. 170 of [12]):

(16)  ∫_{S_{1}(0_{p})} Tr(R′_{u}R′_{u}) du = \frac{3ω_{n−1}}{n(n + 2)(n + 4)}|∇R|_{p}^{2}

This confirms the statement concerning \overline{α}_{2}.

We now consider \overline{β}_{2}. Writing u = ∑_{i=1}^{n} u_{i}e_{i} and R_{ijkℓ} = ⟨R(e_{i}, e_{j})e_{k}, e_{ℓ}⟩ we have

\begin{aligned}∑_{i=1}^{n} Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i}) & = ∑_{i,j,k,ℓ=1}^{n}⟨R(e_{i}, e_{j})e_{k}, e_{ℓ}⟩⟨R_{u}e_{i}, e_{k}⟩⟨R_{u}e_{j}, e_{ℓ}⟩ \\ & = ∑_{a,b,c,d=1}^{n}[ ∑_{i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aibk}R_{cjdℓ}]u_{a}u_{b}u_{c}u_{d}. \end{aligned} ; (17)

Note that the integral of u_{a}u_{b}u_{c}u_{d} over S_{1}(0_{p}) is zero whenever {a, b, c, d} contains at least three different elements. Abbreviating A_{abcd} := ∑_{i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aibk}R_{cjdℓ} we have, using the Einstein

condition and recalling the definition of R̂ and R^{∘} from Section 2:

∑_{a,b=1}^{n} A_{aabb} = ∑_{a,b,i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aiak}R_{bjbℓ} = C^{2} ∑_{i,j,k,ℓ=1}^{n} R_{ijkℓ}δ_{ik}δ_{jℓ} = C^{2} ∑_{i,j=1}^{n} R_{ijij} = nC^{3},

∑_{a,b=1}^{n} A_{abab} = ∑_{a,b,i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aibk}R_{ajbℓ} = ∑_{a,b,i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aibk}R_{jaℓb} = R^{∘}(p),

∑_{a,b=1}^{n} A_{abba} = ∑_{a,b,i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aibk}R_{bjaℓ} = ∑_{a,b,i,j,k,ℓ=1}^{n} R_{ijkℓ}R_{aibk}R_{jbℓa} = R^{∘}(p) − \frac{1}{4}R̂(p),

where for the last equality we have used formula (2.7)(vi) of [17]; see also formula (2.15) of [12]. Let S^{n−1} ⊂ R^{n} denote the (n−1)-dimensional standard sphere. Note that ∫_{S^{n−1}} u_{1}^{2}u_{2}^{2} du = \frac{ω_{n−1}}{n(n+2)} and ∫_{S^{n−1}} u_{1}^{4} du = \frac{3ω_{n−1}}{n(n+2)}. From the above equations and (17) we thus obtain

∫_{S_{1}(0_{p})} ∑_{i=1}^{n}Tr(R_{u} ∘ R(e_{i}, .)R_{u}e_{i}) du

= ∑_{{a,b∈\{1,...,n\} \atop a≠b}}[A_{aabb} + A_{abab} + A_{abba}] ∫_{S^{n−1}} u_{1}^{2}u_{2}^{2} du + ∑_{a=1}^{n} A_{aaaa} ∫_{S^{n−1}} u_{1}^{4} du

= ∑_{a,b=1}^{n}[A_{aabb} + A_{abab} + A_{abba}]\frac{ω_{n−1}}{n(n + 2)} = (nC^{3} + 2R^{∘}(p) − \frac{1}{4}R̂(p))\frac{ω_{n−1}}{n(n + 2)}

Hence,

(18) \overline{β}_{2} = β̂_{2} + \frac{4}{9}(nC^{3} + 2R^{∘}(p) − \frac{1}{4}R̂(p))\frac{1}{n(n + 2)} = β̂_{2} + \frac{4C^{3}}{9(n + 2)} + (\frac{8}{9}R^{∘}(p) − \frac{1}{9}R̂(p))\frac{1}{n(n + 2)}

Recall from Proposition 2.2(ii), (iii) and equation (6) that

112R̂(p) − 32R^{∘}(p) = 27|∇R|_{p}^{2} + some constant depending only on n, C, H, L and
R̂(p) + 4R^{∘}(p) = |∇R|_{p}^{2} + some constant depending only on n, C, H,

using which one easily computes that

−\frac{1}{9}R̂(p) + \frac{8}{9}R^{∘}(p) = \frac{1}{8}|∇R|_{p}^{2} + some constant depending only on n, C, H, L.

Thus we conclude from (18):

\overline{β}_{2} = β̃_{2} + \frac{1}{8n(n + 2)}|∇R|_{p}^{2} ,

where β̃_{2} is a constant depending only on n, C, H, L. □

Proof of the Main Theorem 1.1: Let M_{1}, M_{2} be harmonic spaces, p_{1} ∈ M_{1}, p_{2} ∈ M_{2}, and assume there exists ε in the interval (0, min{i(p_{1}), i(p_{2})}) such that for each 0 < r < ε the geodesic spheres S_{r}(p_{1}) and S_{r}(p_{2}) are isospectral. Then dim M_{1} =: n = dim M_{2}, and the heat invariants of the geodesic spheres coincide:

a_{k}(S_{r}(p_{1})) = a_{k}(S_{r}(p_{2}))

for each r ∈ (0, ε) and all k ∈ ℕ_{0}. We want to deduce that |∇R|_{p_{1}}^{2} = |∇R|_{p_{2}}^{2}. Actually this will follow using just a_{0} and a_{2}.

Reformulating the problem, let M be an n-dimensional harmonic space and p ∈ M. We want to show that for any ε ∈ (0, i(p)), the two functions

φ_{k} : (0, ε) ∋ r ↦ a_{k}(S_{r}(p)) ∈ ℝ

with k ∈ {0, 2} together determine the value of |∇R|_{p}^{2} . By Remark 2.3(i), the function

φ_{0} : r ↦ a_{0}(S_{r}(p)) = vol(S_{r}(p)) = v(r)ω_{n−1}

determines the constants C, H, L associated with M (see Section 2). Recall that the scalar curvature scal^{S} =: scal^{S_{r}} of S_{r}(p) is constant on the manifold S_{r}(p), and that the function v : (0, ε) → ℝ determines, by Remark 2.3(ii), the function (0, ε) ∋ r ↦ scal^{S_{r}} ∈ ℝ. In particular, the function φ_{0} = vω_{n−1} also determines the function (0, ε) ∋ r ↦ ∫_{S_{r}(p)}(scal^{S})^{2}dvol_{S_{r}(p)} = φ_{0}(r)·(scal^{S_{r}})^{2} ∈ ℝ. By

φ_{2}(r) = a_{2}(S_{r}(p)) = \frac{1}{360} ∫_{S_{r}(p)}(5(scal^{S})^{2} − 2|Ric^{S}|^{2} + 2|R^{S}|^{2})dvol_{S_{r}(p)}

it follows that the functions φ_{0} and φ_{2} together determine the function

(0, ε) ∋ r ↦ \frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)}(|R^{S}|^{2} − |Ric^{S}|^{2})dvol_{S_{r}(p)} ∈ ℝ.

By Proposition 4.2, the r^{2}-coefficient in the power series expansion of this function is the sum of the term

(\frac{1}{8n(n + 2)} − \frac{3}{16n(n + 2)(n + 4)})|∇R|_{p}^{2} = \frac{2n + 5}{16n(n + 2)(n + 4)}|∇R|_{p}^{2}

and β̃_{2} − α̃_{2}. Recall that the latter is a constant depending only on n, C, H, L, and is thus determined by φ_{0}. We conclude that the functions φ_{0} and φ_{2} together determine |∇R|_{p}^{2} , as claimed. □

5. Geodesic balls

In this section we will prove the following version of the Main Theorem 1.1 for geodesic balls:

Theorem 5.1. Let M_{1} and M_{2} be harmonic spaces, and let p_{1} ∈ M_{1}, p_{2} ∈ M_{2}. If there exists ε > 0 such that for each r ∈ (0, ε) the geodesic balls B_{r}(p_{1}) and B_{r}(p_{2}) are Dirichlet isospectral, then |∇R|_{p_{1}}^{2} = |∇R|_{p_{2}}^{2}. The same holds if the assumption of Dirichlet isospectrality is replaced by the assumption of Neumann isospectrality.

This theorem implies the corresponding analog of our Main Corollary 1.2:

Corollary 5.2. Let M_{1} and M_{2} be harmonic spaces. Assume that the Dirichlet isospectrality hypothesis of Theorem 5.1 is satisfied for each pair of points p_{1} ∈ M_{1}, p_{2} ∈ M_{2}. Then M_{1} is locally symmetric if and only if M_{2} is locally symmetric. The same holds if the assumption of Dirichlet isospectrality is replaced by the assumption of Neumann isospectrality.

For the proof of Theorem 5.1 we will use the heat invariants for manifolds with boundary. Let M be an n-dimensional Riemannian manifold, and let B ⊂ M be a compact domain with smooth boundary. If Δ denotes the Laplace operator on B with Dirichlet boundary conditions then there is an asymptotic expansion

Tr(exp(−tΔ)) ∼ (4πt)^{−n/2} ∑_{k} a_{k}^{D}(B)t^{k}

for t ↓ 0, where k = 0, 0.5, 1, 1.5, … ranges over the nonnegative half integers (see [3]). For the Laplace operator on B with Neumann boundary conditions the analog of this formula holds with certain coefficients a_{k}^{N}(B). The coefficients a_{k}^{D}(B) (resp. a_{k}^{N}(B)) are given by certain curvature integrals over B and ∂B. One has a_{0}^{D}(B) = a_{0}^{N}(B) = vol(B) and a_{0.5}^{D}(B) = −a_{0.5}^{N}(B) = −\frac{\sqrt{π}}{2} vol(∂B) (see [3]). In the proof of Theorem 5.1 we will use the explicit formulas for a_{2}^{D}(B) and a_{2}^{N}(B) from [3]. Let ν denote the outward pointing unit vector field on the boundary ∂B of B, and let σ = ∇ν be the associated shape operator. Let scal, Ric, R always refer to the usual objects on M (not to the ones associated with the induced metric on ∂B). Then

a_{2}^{D}(B) = \frac{1}{360}[∫_{B}(−12Δ(scal) + 5scal^{2} − 2|Ric|^{2} + 2|R|^{2})dvol_{B}
+ ∫_{∂B}(18ν(scal) + 20scal · Tr(σ) − 4Tr(R_{ν})Tr(σ) + 12Tr(R_{ν} ∘ σ)

− 4Tr((Ric − R_{ν}) ∘ σ) + \frac{40}{21}(Tr(σ))^{3} − \frac{88}{7}Tr(σ)Tr(σ^{2}) + \frac{320}{21}Tr(σ^{3}))dvol_{∂B}],

a_{2}^{N}(B) = \frac{1}{360}[∫_{B}(−12Δ(scal) + 5scal^{2} − 2|Ric|^{2} + 2|R|^{2})dvol_{B}
+ ∫_{∂B}(−42ν(scal) + 20scal · Tr(σ) − 4Tr(R_{ν})Tr(σ) + 12Tr(R_{ν} ∘ σ)
− 4Tr((Ric − R_{ν}) ∘ σ) + \frac{40}{3}(Tr(σ))^{3} + 8Tr(σ)Tr(σ^{2}) + \frac{32}{3}Tr(σ^{3}))dvol_{∂B}].

If M is harmonic then, by the results of Section 2, the previous formulas simplify to

a_{2}^{D}(B) = \frac{1}{360}[vol(B) · (5(nC)^{2} − 2nC^{2} + \frac{4}{3}n((n + 2)H − C^{2}))
+ ∫_{∂B}(20nCTr(σ) − 8CTr(σ) + 16Tr(R_{ν} ∘ σ)

+ \frac{40}{21}(Tr(σ))^{3} − \frac{88}{7}Tr(σ)Tr(σ^{2}) + \frac{320}{21}Tr(σ^{3}))dvol_{∂B}],

(19)

a_{2}^{N}(B) = \frac{1}{360}[vol(B) · (5(nC)^{2} − 2nC^{2} + \frac{4}{3}n((n + 2)H − C^{2}))
+ ∫_{∂B}(20nCTr(σ) − 8CTr(σ) + 16Tr(R_{ν} ∘ σ)
+ \frac{40}{3}(Tr(σ))^{3} + 8Tr(σ)Tr(σ^{2}) + \frac{32}{3}Tr(σ^{3}))dvol_{∂B}].

(20)

In the proof of Theorem 5.1 we will follow a similar strategy as in the proof of Theorem 1.1. To this end, we need some preliminary results in the special case that B = B_{r}(p) with r ∈ (0, i(p)) and M is harmonic. We remark – without going into details this time – that one can compute in this case, using equation (4), Proposition 2.1, and the Taylor series expansion

P_{γ_{u}}^{r,0} ∘ (R_{ν})_{γ_{u}(r)} ∘ P_{γ_{u}}^{0,r} = ∑_{k=0}^{∞} \frac{r^{k}}{k!} R_{u}^{(k)},

that the r^{3}-coefficient in the power series expansion of Tr(R_{ν} ∘σ) equals −\frac{1}{1440}L+ \frac{1}{96}Tr(R′_{u}R′_{u}), and that the r^{3}-coefficient in the power series expansion of Tr(σ^{3}) equals \frac{1}{30240}L − \frac{1}{96}Tr(R′_{u}R′_{u}). (That the contributions of Tr(R′_{u}R′_{u}) in these terms are negatives of each other can also be checked as follows: Using (3) twice, we have Tr(R_{ν} ∘ σ) + Tr(σ^{3}) = −Tr(σ′σ) = −\frac{1}{2}Tr(σ^{2})′ = \frac{1}{2}Tr(R_{ν} + σ′)′ = \frac{1}{2}Tr(σ)′′ whose r^{3}-coefficient indeed depends only on L by (5).) Using (16), we conclude that the r^{3}-coefficient in the power series expansion of r ↦ \frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} Tr(R_{ν} ∘ σ)dvol_{S_{r}(p)} is

(21)  −\frac{1}{1440}L + \frac{1}{32n(n + 2)(n + 4)}|∇R|_{p}^{2} ,

Similarly, the r^{3}-coefficient in the power series expansion of r ↦ \frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)} Tr(σ^{3})dvol_{S_{r}(p)} is

(22)  \frac{1}{30240}L − \frac{1}{32n(n + 2)(n + 4)}|∇R|_{p}^{2} .

Proof of Theorem 5.1: Let M_{1}, M_{2} be harmonic spaces, p_{1} ∈ M_{1}, p_{2} ∈ M_{2}, and assume there exists ε in the interval (0, min{i(p_{1}), i(p_{2})}) such that for each 0 < r < ε the geodesic spheres B_{r}(p_{1}) and B_{r}(p_{2}) are Dirichlet isospectral (resp. Neumann isospectral). Then dim M_{1} =: n = dim M_{2}, and the heat invariants of the geodesic spheres coincide:

a_{k}^{D}(B_{r}(p_{1})) = a_{k}^{D}(B_{r}(p_{2})), resp. a_{k}^{N}(B_{r}(p_{1})) = a_{k}^{N}(B_{r}(p_{2}))

for each r ∈ (0, ε) and all k ∈ ℕ_{0}. We want to deduce that |∇R|_{p_{1}}^{2} = |∇R|_{p_{2}}^{2}. Actually this will follow using just a_{0} and a_{2}. (We remark without proof here that, viewed as functions of r, the heat coefficients a_{0.5}, a_{1}, and a_{1.5} do actually not contain more information than a_{0} in our situation.)

We first consider the case of Dirichlet conditions. Similarly as in the proof of Theorem 1.1, we reformulate the problem as follows: Let M be an n-dimensional harmonic space and p ∈ M. We want to show that for any ε ∈ (0, i(p)), the two functions

ψ_{k}^{D} : (0, ε) ∋ r ↦ a_{k}^{D}(B_{r}(p)) ∈ ℝ

with k ∈ {0, 2} together determine the value of |∇R|_{p}^{2} . Note that the function

ψ_{0}^{D} : r ↦ a_{0}^{D}(B_{r}(p)) = vol(B_{r}(p))

determines its own derivative which is just

r ↦ vol(S_{r}(p)) = v(r)ω_{n−1}

(see the previous section). By Remark 2.3(i), we conclude that ψ_{0}^{D} again determines the constants C, H, L associated with M. Moreover, the function v : (0, ε) → ℝ determines the radial functions Tr(σ) = v′/v and Tr(σ^{2}) = −(v′/v)′ − C (compare Remark 3.3). By (19) it now follows that ψ_{0}^{D} and ψ_{2}^{D} together determine the function

(0, ε) ∋ r ↦ \frac{1}{vol(S_{r}(p))} ∫_{S_{r}(p)}(16Tr(R_{ν} ∘ σ) + \frac{320}{21}Tr(σ^{3}))dvol_{S_{r}(p)} ∈ ℝ.

Recalling (21) and (22), we see that the r^{3}-coefficient in the power series expansion of the latter function is the sum of

\frac{1}{32n(n + 2)(n + 4)}(16 − \frac{320}{21})|∇R|_{p}^{2} = \frac{1}{42n(n + 2)(n + 4)}|∇R|_{p}^{2}

and a term depending only on L. Since L is determined by ψ_{0}^{D}, we conclude that ψ_{0}^{D} and ψ_{2}^{D} together determine |∇R|_{p}^{2} , as claimed.

In the Neumann case, letting ψ_{k}^{N}(r) := a_{k}^{N}(B_{r}(p)), we again have ψ_{0}^{N}(r) = vol(B_{r}(p)) = ψ_{0}^{D}(r). Proceeding exactly as in the Dirichlet case, using (20) this time, we see that ψ_{0}^{N} and ψ_{2}^{N} together determine the sum of

\frac{1}{32n(n + 2)(n + 4)}(16 − \frac{32}{3})|∇R|_{p}^{2} = \frac{1}{6n(n + 2)(n + 4)}|∇R|_{p}^{2}

and a term depending only on L. Hence they determine |∇R|_{p}^{2} , as claimed. □

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Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain

E-mail address: ariasmarco@unex.es

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany

E-mail address: schueth@math.hu-berlin.de