MANIFOLDS WITH SMALL DIRAC EIGENVALUES ARE NILMANIFOLDS

BERND AMMANN AND CHAD SPROUSE

Abstract. Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2, 3 and r = 2^{[n/2]−1} + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.

Keywords: Dirac operator, nilmanifold, small eigenvalues, almost positive scalar curvature MSC: 53C27 (Primary), 58J50, 53C20, 53C21 (Secondary)

Contents

1. Introduction 1 2. General estimates on vector bundles 6 3. The first Dirac eigenvalue 7 4. The fixing dimension of a faithful representation 8 5. Eigenvalue pinching on vector bundles with special holonomy 9 6. The fixing dimension for the spinor representation 11 7. Proof of Theorem 1.1 12 A. Some analytical tools 14 B. Connection Laplacians under perturbations of the metric 16 References 17

1. Introduction

The theorem of Bochner implies that if a connected compact Riemannian manifold has non-negative curvature operator, then b^{p}(M) ≤ \binom{n}{p}. Furthermore if b^{p}(M) = \binom{n}{p} for some p between 0 and n then M is isometric to a flat torus. In [35] it was shown that Riemannian manifolds with almost-nonnegative curvature operator, and \binom{n}{p}

Research at MSRI is supported in part by NSF grant DMS-9810361.

eigen-p-forms with small eigenvalue must be diffeomorphic to a nilmanifold. Given the Hodge-de Rham theorem, this could be viewed as a quantitative generalization of Bochner’s theorem.

Here we discuss a similar result for the Dirac operator on Riemannian spin manifolds. Let λ_{i}(D̸^{2}) denote the i-th eigenvalue of the square of the Dirac operator, and let λ_{i}(∇^{∗}∇) denote the i-th eigenvalue of the connection Laplacian on spinors. Here and throughout the article we assume that all eigenvalues are counted with multiplicity. All manifolds are connected. Let r(n) = 2^{[ \frac{n}{2} ]−1} + 1 for n ≥ 4 and r(n) = 1 for n ≤ 3. Our main result is:

THEOREM 1.1. Let (M^{n}, g, χ) be a compact Riemannian spin manifold with | sec | < K, diam < D. Then there is ε = ε(n, K, D) > 0, such that if λ_{r}(∇^{∗}∇) < ε, then M is diffeomorphic to a nilmanifold. Furthermore, χ is the trivial spin structure on M.

Using the Schrödinger-Lichnerowicz formula D̸^{2} = ∇^{∗}∇ + scal /4 this implies:

COROLLARY 1.2. Let (M^{n}, g, χ) be a compact Riemannian spin manifold with | sec | < K, diam < D. Then there is ε = ε(n, K, D) > 0, such that if scal > −ε and λ_{r}(D̸^{2}) < ε then M is diffeomorphic to a nilmanifold. Furthermore, χ is the trivial spin structure on M.

By reformulating we obtain.

COROLLARY 1.3. If (M, χ) is not spin-diffeomorphic to a nilmanifold with a trivial spin structure, then among all metrics with bounded diameter and curvature, there is a uniform lower bound on the r-th ∇^{∗}∇-eigenvalue:

λ_{r}(∇^{∗}∇) ≥ ε = ε(n, max | sec |, diam) > 0.

In particular, any metric g on M with scal > −4ε(n, max | sec |, diam) has a nontrivial uniform lower bound on the r-th Dirac eigenvalue

(1.4)  λ_{r}(D̸^{2}) ≥ ε(n, max | sec |, diam) + \frac{min scal}{4}.

Recall that the Atiyah-Singer index theorem implies dim kerD̸ ≥ |Â(M)|. Therefore we obtain as a corollary: If M is an n-dimensional compact spin manifold with |Â(M)| ≥ r(n), then M does not carry a metric with scal > −4ε(n, max | sec |, diam) for the above ε > 0. This corollary already appears in Gallot’s article [21] with r(n) replaced by 2^{[n/2]}.

We will give some examples that explain the special role of nilmanifolds and why we cannot replace r by a smaller number.

Examples.

(1) Any nilmanifold M^{n} carries a sequence of “left-invariant” metrics g_{i} with max | sec_{i} | → 0, diam_{i} → 0 and vol_{i} → 0. If the spin structure on M is trivial, then for this sequence of metrics λ_{s}(D̸_{i}^{2}) → 0 where s = rank(Σ_{i}M) = 2^{[n/2]} > r.

(2) Let N be a K3-surface with a Calabi-Yau metric g^{N} and its unique spin structure. In particular, g^{N} is ricci-flat and kerD̸ = ker ∇^{∗}∇ has complex dimension 2. For an integer n ≥ 4, let M be the Riemannian product N × T^{n−4}, where the n − 4 dimensional torus carries an arbitrary flat metric. We equip M with the product spin structure of the unique spin structure on N and the trivial spin structure on T^{n−4}. Then the spinor bundle ΣM on M is isomorphic (as a metric bundle with connection) to π_{1}^{∗}(ΣN) ⊗_{ℂ} ℂr̃ where r̃ = 2^{[(n−4)/2]} = 2^{[n/2]−2} and π_{1} : N × T^{n−4} → N is the projection to the first component. If ψ ∈ Γ(ΣN) is a parallel spinor on N and v is a constant section of ℂr̃, then π_{1}^{∗}(ψ) ⊗ v is a parallel spinor on M. As N carries a 2-dimensional space of parallel spinors, the dimension of the space of parallel spinors on M is at least 2r̃ = r − 1. And hence

λ_{1}(D̸^{2}) = … = λ_{r−1}(D̸^{2}) = 0.

This example shows that we cannot replace r by r − 1 in the above theorem.

The next example will show that the sectional curvature bounds in Theorem 1.1 are necessary. We need a lemma.

LEMMA 1.5. Let h be the standard metric on S^{3}. The ball of radius R around 0 in Euclidean space ℝ^{4} will be denoted as B_{R}^{ℝ^{4}}(0). For any ε, R, ρ > 0 there is a > 0 and a metric g = dt^{2} + φ^{2}(t)h on S^{3} × (−(a + ρ), (a + ρ)) such that

(a) g|_{(−(a+ρ),−a]×S^{3}} and g|_{[a,(a+ρ))×S^{3}} are isometric to B_{R+ρ}^{ℝ^{4}}(0) \ B_{R}^{ℝ^{4}}(0),

(b) scal_{g} ≥ −ε, and

(c) diam(M, g) ≤ 6(R + ρ)

For the proof one translates the desired properties into an ordinary differential inequality for φ. Details are available in [4].

Using the lemma we can construct an example showing that the curvature bound is necessary.

Example.

(3) Consider the flat torus T^{4}. Let Z := Z(R_{j}, ε_{j}) := (−c, c) × S^{n−1} carry a metric as in the above lemma for ε_{j} and R_{j} sufficiently small, that we will choose later, and for ρ_{j} = R_{j}. Let (M_{j}, g_{j}) be given by removing 2j small disks from T^{4} and attaching j handles isometric to Z. The trivial spin structure on T^{4} can be extended to a spin structure on M_{j}.^{1} For a suitable choice of R_{j} and ε_{j} we obtain a family of Riemannian manifolds (M_{j}, g_{j}) with uniformly bounded diameter and lim inf_{j→∞} min scal_{j} = 0. They are pairwise non-diffeomorphic, and the sectional curvature is not uniformly bounded. Following the lines of [10] we use a cut-off function vanishing in the handles to construct a (rank ΣM_{j} = 4)-dimensional space of test spinors. From this we see that D̸^{2} has at least 4 eigenvalues arbitrarily close to 0. This example shows that the sectional bound in Theorem 1.1 is necessary if the dimension of M is 4. We obtain similar examples for higher dimensions n by taking the product with a flat (n − 4)-dimensional torus equipped with the trivial spin structure. See [4] for more details.

1This extension is not unique, but our construction works for any choice of spin structure.

Finally, we will give several examples in order to show that the bound (1.4) generalizes previously known bounds in dimension 2 and 3.

Examples.

(4) If M is diffeomorphic to the 2-dimensional sphere, then such a lower bound is already known. It is a result of Bär [7] that

λ_{1}(D̸^{2}) ≥ \frac{4π}{area(M)}.

If K ≥ −δ^{2}, δ > 0 then area(M) ≤ \frac{2π}{δ^{2}} [cosh (δ diam M) − 1]. Hence,

λ_{1}(D̸^{2}) ≥ \frac{2δ^{2}}{cosh (δ diam M) − 1} = \frac{4}{diam M^{2}} − O(δ^{2} diam M^{2}).

(5) Let (M, g) be diffeomorphic to the 2-dimensional torus T^{2} equipped with a nontrivial spin structure and with | sec | < K, diam < D. It is not difficult to derive an explicit lower bound on λ_{1}(D̸^{2}) from previously known estimates. To derive this bound, we use the uniformization theorem to find u ∈ C^{∞}(T^{2}) and a flat metric g_{0} with g = e^{2u}g_{0}. Using the estimates in [2] together with some elementary calculations [4] one obtains

max u − min u ≤ τ(K|D),

where τ is an explicitly known, but long expression [2, Theorem 6.1]. Then one easily derives from [2, Corollary 2.3] that

λ_{1}(D̸^{2}) ≥ \frac{π^{2}}{4D^{2}} e^{−4τ(K|D)}.

(6) Surfaces of genus greater than 1 cannot have almost non-negative curvature in the above sense. Hence, (4) and (5) yield an explicit, but long formula for ε in dimension n = 2.

In higher dimension one expects ε to be an even more complicated expression. Thus, we want to restrict our attention to results that show only existence of a positive ε.

(7) Let (M, g, χ) be a compact spin 3-manifold with scal ≥ 0 and kerD̸ ≠ {0}. Because of the Schrödinger-Lichnerowicz formula, any φ ∈ ker D\{0} is a nontrivial parallel spinor, which implies that (M, g) is Ricci-flat, and hence flat. However, any flat compact 3-manifold admitting a nontrivial parallel spinor is diffeomorphic to a torus (see [36, Theorem 5.1]) and the spin structure is the trivial one.

We compare Corollary 1.2 which gives a uniform lower r-th eigenvalue bound to a theorem of J. Lott which provides a uniform upper bound on all eigenvalues.

THEOREM (Lott [27, Theorem 4]). Let k ∈ ℤ^{+}. Then there is an E_{k} = E(n, K, D, k) such that any compact Riemannian spin manifold (M, g, χ) with | sec | < K, diam ≤ D satisfies either

(a) λ_{k}(D̸^{2}) ≤ E_{k} (b) M is the total space of an affine fiber bundle M → B with possible singularities,

whose generic fiber is an infranilmanifold, and the spin structure along the generic fibers is not trivial.

Another result which will be proven in section 3 gives a different conclusion for manifolds with only one small Dirac eigenvalue, and additionally, a lower volume bound.

THEOREM 1.6. Let (M, g) have | sec | < K, diam < D, vol > v. Let λ_{1}(∇^{∗}∇) denote the first eigenvalue of the connection Laplacian ∇^{∗}∇ on the spinor bundle with respect to a spin structure χ. Then for all δ > 0, there is an ε = ε(n, v, K, D, δ) > 0 such that if λ_{1}(∇^{∗}∇) < ε, then (M, g, χ) has C^{1,α}-distance ≤ δ to a Ricci-flat Einstein metric with a nontrivial parallel spinor.

COROLLARY 1.7. For δ > 0, there is an ε = ε(n, v, K, D, δ) > 0 such that the following holds: Let (M, g) have | sec | < K, diam < D, vol > v and scal > −ε. Let λ_{1}(D̸^{2}) denote the first eigenvalue of D̸^{2} with respect to a spin structure χ. If λ_{1}(D̸^{2}) < ε, then (M, g, χ) has C^{1,α}-distance ≤ δ to a Ricci-flat Einstein metric with a nontrivial parallel spinor.

A compact 4-dimensional manifolds M carrying a parallel spinor is either a flat torus or a K3-surface. Hence, any 4-manifold with one small Dirac eigenvalue is either diffeomorphic to a torus or a K3-surface, or is collapsed.

Example (1) shows that the volume bound in the above theorem and corollary is necessary.

If one applies the techniques that we will present in this paper to the Friedrich connection on the spinor bundle instead of the standard connection, one obtains analogs of Theorem 1.1, Corollary 1.2, Theorem 1.6 and Corollary 1.7. In particular, we obtain the following theorem. Here, once again, we define r = 1 if n = 2, 3 and r = 2^{[n/2]−1} + 1 if n ≥ 4.

THEOREM 1.8. Let (M^{n}, g, χ) be a compact Riemannian manifold with diam < D, | sec | < K, and scal ≥ n(n − 1)ρ^{2} with a constant ρ > 0. Let D̸ be the Dirac operator on M. Then for any δ > 0, there is ε = ε(n, K, D, ρ, δ) > 0 such that if D̸ has r = r(n) eigenvalues λ_{i} ∈ [0, \frac{nρ}{2} + ε), then M has C^{1,α}-distance ≤ δ to a manifold of constant curvature sec ≡ ρ^{2}.

However, motivated by Bär’s classification of manifolds with real Killing spinors [8], we conjecture that the theorem still holds for a smaller number r.

In the following example, one sees that the bound on the sectional curvature is necessary.

Example.

(8) On any compact manifold M that admits a metric of positive scalar curvature, equipped with an arbitrary spin structure on M, C. Bär and M. Dahl [11] constructed a sequence of metrics g_{i} on M with scalar curvature ≥ n(n − 1), but with λ_{2^{[n/2]}}(D̸_{g_{i}}^{2}) → n^{2}/4.

The structure of our article is as follows. In Section 2 we will reformulate some previously known estimates on vector bundles. In Section 3 we will apply these estimates to prove Theorem 1.6. In the following sections Theorem 1.1 is proved. We will develop most of the tools in such a generality that we can easily replace the Dirac operator (acting on spinors) by other elliptic operators acting on sections of bundles with special holonomy. We begin this in Section 4 by defining the fixing dimension r of a faithful representation, which immediately gives the fixing dimension r of a vector bundle with special holonomy. In Section 5 we show that if there are r almost parallel sections on such a bundle, then the bundle is trivialized by almost parallel sections. Section 6 determines the fixing number of the spinor bundle, and we finally prove Theorem 1.1 in the last section.

Throughout the paper we adopt the convention that τ(x_{1}|x_{2}, …, x_{m}) represents a continuous function in x_{1}, …, x_{m} such that τ → 0 as x_{1} → 0 with x_{2}, …, x_{m} fixed.

Acknowledgement. We want to thank C. Bär, A. Degeratu, M. Dahl, P. Petersen, and W. Tuschmann for several helpful discussions. The first named author wants to thank MSRI Berkeley, CA, USA for their hospitality. Research at MSRI is supported in part by NSF grant DMS-9810361.

2. General estimates on vector bundles

Let V be a complex vector bundle of rank k over M equipped with a connection ∇ and a metric ⟨·, ·⟩. Recall that the second covariant derivative on sections of V is given by ∇_{X,Y}^{2} S = ∇_{X}∇_{Y} S − ∇_{∇_{X}Y} S, and the curvature tensor on sections of V is given by R^{V} (X, Y)S = ∇_{X,Y}^{2} S − ∇_{Y,X}^{2}S.

Furthermore, we consider the connection Laplacian on V, which is given by

∇^{∗}∇S = − ∑_{i=1}^{n} ∇_{e_{i},e_{i}}^{2}S,

where {e_{i}} is an orthonormal set of vectors at any point p ∈ M. We say that S is an eigensection of V with eigenvalue λ if ∇^{∗}∇S = λS. In this general situation, we recall the eigenvalue pinching theorems from ([33], [35]), which characterize eigensections with small eigenvalues.

Notation. We use the volume-normalized L^{p}-norm given by

∥u∥_{p} = (\frac{1}{vol M} ∫_{M} |u|^{p} dvol)^{1/p}

and the volume-normalized L^{2}-scalar product

(u, v) = \frac{1}{vol M} ∫ ⟨u, v⟩ dvol .

THEOREM 2.1. Suppose that V satisfies |R^{V} | < K, |∇R^{V} | < K, and M satisfies | sec | < K, diam < D. Suppose S is an eigensection of V with eigenvalue λ, normalized so that ∥S∥_{2} = 1. Then,

∥S∥_{∞} ≤ 1 + τ(λ|n, K, D)

∥∇S∥_{∞} ≤ τ(λ|n, K, D)
∥∇∇S∥_{2} ≤ τ(λ|n, K, D)

And furthermore,

THEOREM 2.2. Suppose that S_{1}, …, S_{m} are L^{2}-orthonormal eigensections of V, with eigenvalues λ_{1} ≤ ⋯ ≤ λ_{m}. Then with K, D as above,

∥⟨S_{i}, S_{j}⟩ − δ_{ij}∥_{∞} ≤ τ(λ_{m}|n, K, D).

We outline the proofs of these facts in Appendix A.

3. The first Dirac eigenvalue

In this section we characterize manifolds with | sec | < K, diam < D, almost non-negative scalar curvature, and a single small Dirac eigenvalue. Our result is that such manifolds are either collapsed in the sense of Cheeger-Fukaya-Gromov or C^{1,α}-close to an Einstein manifold with a parallel spinor. Note that in the case of the first eigenvalue on differential p-forms λ_{1}^{+}(Δ_{p}), such a manifold would always be collapsed. Or in other words, there is a lower bound on λ_{1}^{+}(Δ_{p}) in terms of a lower volume bound and the above curvature and diameter bounds ([17]). This is proved as follows. Suppose (M_{i}, g_{i}) is a sequence of manifolds as above with λ_{1}^{+}(Δ^{p}) → 0. Then the above conditions imply that there is a subsequence of M_{i} that converges to a limit manifold \overline{M} in the C^{1,α} topology. But this is not possible since \overline{M} would have a higher p-th Betti number than the limiting manifolds M_{i}.

On the other hand, since the number of harmonic spinors is not topologically invariant, this argument will not work in the spinor case. The reader interested in harmonic spinors may consult the classical reference [25] or several articles containing recent results [9],[10], [28], [3] about the dependence of dim kerD̸ on the metric.

We will now prove the following theorem from the introduction.

THEOREM 1.6. Let (M, g) have | sec | < K, diam < D, vol > v > 0. Let λ_{1}(∇^{∗}∇) denote the first eigenvalue of the connection Laplacian ∇^{∗}∇ with respect to any spin structure. Then for any δ > 0, there is ε = ε(n, v, K, D, δ) > 0 such that if λ_{1}(∇^{∗}∇) < ε, then (M, g) has C^{1,α}-distance ≤ δ to a Ricci-flat Einstein metric with a nontrivial parallel spinor.

Proof. In the proof of the theorem we will use the following proposition about smoothing Riemannian metrics.

PROPOSITION 3.1. For any δ = δ(K, n) > 0 there is K_{1} = K_{1}(n, K, δ) such that on any n-dimensional complete Riemannian manifold (M, g) with | sec | ≤ K there is a Riemannian metric g̃ on M such that

e^{−δ}g̃(X, X) ≤ g(X, X) ≤ e^{δ}g̃(X, X) ∀X ∈ T M,
|∇^{g} − ∇g̃|_{g} ≤ δ.
| secg̃ |g̃ ≤ K + δ, |∇g̃Rg̃|g̃ ≤ K_{1}.

As far as we know, the proposition is due to Abresch [1], it also appears in [15, Theorem 1.12]. A modern proof using the Ricci flow is explained for example in the textbook [16].

Together with Proposition B.1 we see that one can assume in the proof of the theorem that |∇R| < K_{1}(K, n, D). Let σ denote an eigenspinor to the eigenvalue λ_{1} with ∥σ∥_{2} = 1. Then ∥|σ| − 1∥_{∞} < τ(λ_{1}|n, K, D) and ||∇∇σ||_{2} < τ(λ_{1}|n, K, D). If e_{1}, …, e_{n} denotes a local orthonormal frame, then from the curvature formula for spinors R(X, e_{i})σ = \frac{1}{4} ∑_{j,k}⟨R(X, e_{i})e_{j}, e_{k}⟩ e_{i} · e_{j} · σ one deduces (see e.g. [18])

Ric(X) · σ = −2 ∑_{i=1}^{n} e^{i} · R(X, e_{i})σ,

and the fact that R(·, ·)σ is clearly bounded by ∇∇σ. This implies ∥ Ric ∥_{2} < τ(λ_{1}|n, K, D). The lower bound on the volume together with the upper bounds on | sec | and diam imply a lower bound on the injectivity radius of (M, g). Then from [32, Theorem 6.1] (see also [5], [22]), we have that for λ_{1} < ε(n, v, K, D), M is C^{1,α}-close to a C^{∞} Einstein manifold \overline{M} with Ric ≡ 0. As \overline{M} is diffeomorphic to M if M and \overline{M} are C^{1,α}-close, we may assume in the following that M and \overline{M} are equipped with the same spin structure.

It remains to show, that if ε has been chosen small enough, then \overline{M} must carry a parallel spinor. Assume the opposite, then we have a sequence of manifolds M_{i} converging to \overline{M} in the C^{1,α} topology, with λ_{1}(∇^{∗}∇, M_{i}) → 0. Proposition B.1 implies λ_{1}(∇^{∗}∇, \overline{M}) = 0, or in other words \overline{M} admits a parallel spinor. □

Remark. The above proof can be slightly simplified by using spinors on manifolds with a C^{1,α}-metric. We avoided this for technical reasons.

4. The fixing dimension of a faithful representation

Here we discuss the fixing dimension for representations which we will need in Section 5. Let G be a compact Lie group, and let ρ : G → End(V) be a faithful (i.e. injective) complex representation. For any subspace W of V let

Stab^{G}(W) := {g ∈ G | ρ(g)w = w ∀w ∈ W }

be the stabilizer. Note that faithfulness of ρ means that Stab^{G}(V) = {1}.

Definition. The fixing dimension F(ρ) of ρ is defined to be the smallest number r ∈ {0, …, dim V } with the property that any r-dimensional subspace W ⊂ V has a finite stabilizer Stab^{G}(W).

For the standard representation of U(n) on ℂ^{n}, we have

F(U(n) ↪ GL(ℂ^{n})) = n,

whereas

F(SU(n) ↪ GL(ℂ^{n})) = n − 1.

PROPOSITION 4.1. Let ρ : G → End(V) be a faithful unitary representation with fixing dimension F(ρ). Then there is N ∈ ℕ such that for all F(ρ)-dimensional subrepresentations W ⊂ V, we have #Stab^{G}(W) ≤ N. We denote by N(ρ) the smallest such N.

Proof. Let k = F(ρ), and let K be the set of all orthonormal k-frames F = (v_{1}, …, v_{k}) ∈ V^{k}. The compact group G acts on the compact manifold K differentiably. The isotropy group of F = (v_{1}, …, v_{k})

G_{F} := {g ∈ G | gF = F }

coincides with Stab^{G}(span(v_{1}, …, v_{k})). Due to the definition of the fixing dimension, these isotropy groups are finite. In order to show that sup{#G_{F} | F ∈ K} < ∞ one can either suppose that there exists a sequence of F (i) ∈ K with #G_{F (i)} → ∞ and easily prove that this yields a contradiction, or one can alternatively use the following theorem due to Mostow.

THEOREM 4.2 ([30, Sec. 4],[31]). Let G be a compact Lie group acting on a compact manifold K, then there is a finite set {H_{1}, …, H_{j}} of subgroups of G such that any isotropy group of the action is conjugated to a H_{i}, 1 ≤ i ≤ j.

□

In section 6 we will determine the fixing dimension of the spinor representation.

5. Eigenvalue pinching on vector bundles with special holonomy

Let V be a vector bundle of rank k > 0 over M equipped with a connection ∇ and a metric ⟨·, ·⟩. We fix p ∈ M. We assume that the holonomy group of the bundle is contained in a closed Lie group H ⊂ U(V_{p}). For any q ∈ M let H_{q} be the parallel transport of H to q.

PROPOSITION 5.1. We assume that M and V satisfy the conditions of Theorem 2.1. Assume that the rank-k bundle V has holonomy contained in H. Let r = F(H ⊂ U(T_{p}M)) be the fixing dimension of the holonomy. Let S_{1}, …, S_{r} be L^{2}-orthonormal sections of V such that

∇^{∗}∇S_{i} = λ_{i}S_{i},

1 ≤ i ≤ r, 0 ≤ λ_{i} ≤ ε. Then for small ε > 0, there is a finite cover π : M̃ → M and smooth sections e_{1}, …, e_{k} of π^{∗}(V) with the following properties:

(1) E := (e_{1}, …, e_{k}) is a frame, i.e. E(q) is a basis of π^{∗}(V)_{q} for all q ∈ M̃. (2)

|S_{i}(π(q)) − e_{i}(q)| ≤ τ(ε|n, K, D) ∀q ∈ M̃, i = 1, …, r.

(3)

(∇E)_{q} = (∇e_{1}, …, ∇e_{k})_{q} ∈ T_{q}^{∗}M̃ ⊗ Lie(H_{π(q)})

(4)

|∇e_{i}(q)| ≤ τ(ε|n, K, D) ∀i = 1, …, k ∀q ∈ M̃.

Furthermore, if N(ι) = 1 where ι : H ⊂ U(T_{p}M)) = 1 is the inclusion, then we can choose M̃ = M.

Proof. According to Theorems 2.1 and 2.2 we have the estimates

(5.2)  ∥S_{i}∥_{∞} ≤ 1 + τ(ε|n, K, D)
(5.3)  ∥∇S_{i}∥_{∞} ≤ τ(ε|n, K, D)

and

(5.4)  ∥⟨S_{i}, S_{j}⟩ − δ_{ij}∥_{∞} < τ(ε|n, K, D).

We apply pointwise the Hilbert-Schmidt orthogonalization procedure to S_{1}, …, S_{r} and obtain new sections S̃_{1}, …, S̃_{r}. All functions in this procedure and their first derivatives are controlled in terms of ε, n, K, and D. As a consequence these new sections also satisfy (5.2) and (5.3) and are pointwise orthonormal. We fix a point p ∈ M. The parallel transports of (S̃_{1}(p), …, S̃_{r}(p)) define a principal bundle over M whose structure group is the holonomy group. By enlarging the structure group to H we obtain an H-principal bundle which we will denote by P_{H}(M). The bundle P_{H}(M) is a parallel subbundle of the frame bundle P_{U(k)}(V).

We denote G := Stab^{U(k)}(ℝ^{r}) ≅ U(k−r). Let P_{G}(M) be the bundle of orthonormal bases of V such that the first r basis vectors coincide with S̃_{1}, …, S̃_{r} at each base point. Note that P_{G}(M) is an U(k − r) principal bundle. The bundles P_{G}(M) and P_{H}(M) have a common point (S̃_{1}(p), …, S̃_{r}(p)). Because r is the fixing dimension of H ⊂ U(T_{p}M) we get #G ∩ H < ∞.

Choose a bi-invariant metric on U(k). This induces a metric on each fiber of P_{U(k)}(V). For x ∈ M let δ(x) be the distance of the fiber of P_{G}(M) over x to the fiber of P_{H}(M) over x with respect to this metric. Then, δ : M → [0, ∞) is a function with δ(p) = 0. Using (5.3) one sees that δ is a Lipschitz function with Lipschitz constant of the form τ(ε|n, K, D). We assume that ε is so small that 2δ is smaller than the injectivity radius of U(k) and smaller than inf{d(A, e) | A ∈ G ∩ H, A ≠ e}. Let M̃ be the set of all points p ∈ P_{H}(M) having minimal distance to P_{G}(M), more exactly the points such that d_{π(p)}(p, P_{G}(M)) = δ(π(p)), where π(p) ∈ M is the basepoint of M and where d_{π(p)} denotes the distance in the fiber of P_{U(k)} over π(p). Because of symmetry we have #G ∩ H many points in M̃ over each point in M. As δ is chosen as above, M̃ is a smooth manifold and π : M̃ → M is a covering of M with #G ∩ H many leaves. Any q ∈ M̃ ⊂ P_{U(k)}V can be written as q = (e_{1}(q), …, e_{k}(q)) with e_{j} ∈ V_{π(q)}, and e_{j} are clearly smooth sections of π^{∗}(V) satisfying (1).

Because of our construction the distance between e_{i} and S̃_{i} is bounded by δ, and hence we obtain (2). As E := (e_{1}, …, e_{k}) is a section of π^{∗}(P_{H}(M)), we see that (3) holds.

For (4) we have to prove that for q in M and an arclength-parametrized curve c with c(0) = q,

|∇_{ċ(0)}e_{i}(q)| ≤ τ(ε|n, K, D). (5.5)

Let Ê be the parallel transport of E_{q} along c. Obviously, Ê ∈ P_{H}(M). Suppose that A ∈ U(n) is the unique matrix such that E_{q} · A is the closest point to E_{q} in (P_{G}(M))_{q}. Then Ê · A is also a parallel frame of V along c, and by the construction of P_{G}(M) and also (5.3) one sees that the distance from Ê(t) · A to (P_{G}(M))_{c(t)} is bounded by |t|τ(ε|n, K, D). On the other hand, E(c(t)) is by definition a point such that it is — among all points in (P_{H}(M))_{c(t)} — of minimal distance to (P_{G}(M))_{c(t)}. By applying the implicit function theorem one concludes that the distance between E(c(t)) and Ê(t) is bounded by |t|τ(ε|n, K, D) for small t. This implies (5.5). □

E E · A Ê Ê · A  (P_{H}(M))_{c(t)} (P_{G}(M))_{c(t)}

Figure 1. The shortest line between (P_{G}(M))_{c(t)} and (P_{H}(M))_{c(t)}.

6. The fixing dimension for the spinor representation

Let V be an n-dimensional real vector space. We view Spin(V) as a subgroup of the group of invertible elements of the Clifford algebra of V (see e.g. [26], [19] or [24]). Let e_{1}, …, e_{n} be an orthonormal basis of V. The complex spinor representation Σ of Spin(V) has dimension 2^{[n/2]}.

PROPOSITION 6.1. Let g ∈ Spin(V), g ≠ 1. Then the multiplicity of the eigenvalue 1 of the endomorphism g ∈ End(Σ) is at most 2^{[n/2]−1}.

Proof. We set A_{j} := e_{2j−1} · e_{2j} ∈ Cl(V) for j = 1, …, m, m := [n/2].

Any g ∈ Spin(V) is contained in a maximal torus, i.e. there is an h ∈ Spin(V) and t_{j} ∈ ℝ such that

g = h · exp(t_{1}A_{1}) · exp(t_{2}A_{2}) · · · · · exp(t_{m}A_{m}) · h^{−1}.

Let h′ be the image of h under the map Spin(V) → SO(V). Then h·exp(t_{j}A_{j})·h^{−1} = exp(t_{j}h · A_{j} · h^{−1}). We set

Â_{j} := h · A_{j} · h^{−1} = h′(e_{2j−1}) · h′(e_{2j})

and

g_{j} := exp(t_{j}Â_{j}) = cos t_{j} + sin t_{j}Â_{j}.

The Â_{j} are pairwise commuting anti-self-adjoint endomorphisms. Hence they are simultaneously diagonalizable, with eigenvalues i and −i. Furthermore h′(e_{2j}) anticommutes with Â_{j} and commutes with Â_{k}, j ≠ k. Hence, all simultaneous eigenspaces have the same dimension, which is 1.

We conclude that all g_{j} are simultaneously diagonalizable with eigenvalues exp(it_{j}) and exp(−it_{j}), having 1-dimensional simultaneous eigenspaces. Thus g has the eigenvalues

e^{i(±t_{1}±t_{2}±···±t_{m})}

where the signs vary independently, each sign combination providing an eigenspace of multiplicity 1. As a consequence, for any g ≠ 1, the multiplicity of the eigenvalue 1 is at most 2^{m−1}. □

PROPOSITION 6.2. The fixing dimension of the complex spinor representation of Spin(n) is

r := \begin{cases}1 & if n = 2, 3 \\ 2^{[^{\frac{n}{2}}]−1} + 1 & if n ≥ 4 \end{cases}

Furthermore, for any r-dimensional subspace W of Σ we have #Stab^{G}(W) = 1, i.e. the N in Proposition 4.1 equals 1.

Proof. If n = 2, the spinor representation is

S^{1} → SU(2), z ↦ (\begin{matrix}z & 0 \\ 0 & z^{−1} \end{matrix}) .

There are no invariant subspaces, hence r = 1 and N = 1.

If n = 3, then the spinor representation is the identity Spin(3) = SU(2) → SU(2). Let W be a 1-dimensional subspace of ℂ^{2}. Any h ∈ Stab^{SU(2)}(W) can be diagonalized with eigenvalues λ and λ^{−1}. However, as W is fixed by h, we obtain λ = 1, and hence h = 1. We have thus shown that r ≤ 1. Obviously r ≥ 1. We also see N = 1.

If n ≥ 4, then example (2) in the introduction shows that r > 2^{[^{\frac{n}{2}}]−1}. The previous proposition implies r ≤ 2^{[^{\frac{n}{2}}]−1} + 1 and N = 1. □

7. Proof of Theorem 1.1

We briefly recall some definitions from spin geometry. Details can be found for example in [26].

Definition. Let (M, g) be a Riemannian manifold. Let P_{SO}(M) be the frame bundle over M. A spin structure is a Spin(n)-principal bundle P_{Spin}(M) together with a Θ : Spin(n) → SO(n)-equivariant fiber map χ : P_{Spin}(M) → P_{SO}(M) over the identity M → M.

Example. Let G be an n-dimensional Lie group, and Γ a lattice in G. The frame bundle of G is trivialized by left invariant frames, i.e. SO(G) = G × SO(n). Hence, there is a spin structure on G given by Spin(G) = G × Spin(n) where χ is the identity in the first component and the standard map Spin(n) → SO(n) in the second component. The frame bundle of Γ\G is (Γ\G) × SO(n). One possible spin structure on Γ\G is (Γ\G) × Spin(n) together with the equivariant map id ×Θ. This spin structure is called the trivial spin structure.

Definition. Let ρ : Spin(n) → U(Σ) be the complex spinor representation. The spinor bundle is defined as the associated vector bundle

ΣM := P_{Spin}(M) ×_{ρ} Σ.

As ρ is a 2^{[n/2]}-dimensional complex representation, the complex vector bundle Σ has rank k := 2^{[n/2]}. The holonomy is contained in Spin(n), the inclusion Spin(n) ↪ U(k) given by the spinor representation.

As before we denote the fixing dimension of the spinor representation by

r = r(n) = \begin{cases}1 & if n = 2, 3 \\ 2^{[^{\frac{n}{2}}]−1} + 1 & if n ≥ 4. \end{cases}

THEOREM 1.1. Let (M^{n}, g, χ) be a compact Riemannian spin manifold with | sec | < K, diam < D. Then there is ε = ε(n, K, D) > 0, such that if λ_{r}(∇^{∗}∇) < ε, then M is diffeomorphic to a nilmanifold. Furthermore, χ is the trivial spin structure on M.

Proof. We will use Abresch’s smoothing theorem (Proposition 3.1). A brief survey of such results is contained in Section 5 of [20].) This theorem states that for any δ > 0 there is a constant K_{1}(K, n, δ) such that any metric g on a compact manifold M^{n} with | sec_{g} | < K can be approximated by another metric g̃ on M with

(1) e^{−δ}g ≤ g̃ ≤ e^{δ}g,

(2) |∇^{g} − ∇g̃| ≤ δ,

(3) | secg̃ | ≤ K + δ, and

(4) |∇Rg̃| ≤ K_{1}. As the eigenvalues of the connection Laplacian ∇^{∗}∇ on the spinor bundle are uniformly continuous under C^{1}-perturbations of the metric (Proposition B.1), this shows that it is sufficient to prove the theorem under the additional assumption that |∇R| is bounded. We will formulate the remaining step as a lemma. □

LEMMA 7.1. Let (M^{n}, g, χ) be a compact Riemannian spin manifold with | sec | < K, |∇R| < K, diam < D. Then there is ε = ε(n, K, D) > 0, such that if λ_{r}(∇^{∗}∇) < ε, then M is diffeomorphic to a nilmanifold. Furthermore, χ is the trivial spin structure on M.

Proof of the lemma. We apply Proposition 5.1 for V = ΣM, H = Spin(n). We obtain a frame E of ΣM with |∇E| = τ(λ|n, K, D). The spin structure χ : P_{Spin}(M) → P_{SO}(M) maps E to χ(E) with

|∇χ(E)| ≤ τ(ε|n, K, D),

i.e. an almost parallel frame of T M. Now, using [23] we see that M is C^{0}-close and diffeomorphic to a nilmanifold Γ\N with Γ a cocompact lattice in the nilpotent Lie group N. Let F′ be a frame on M which is sufficiently close to χ(E) and whose pullback to N is left-invariant. It can be lifted, i.e. there is a frame E′ with χ(E′) = F′, hence the spin structure is trivial. □

A. Some analytical tools

Here we outline the results from [33],[34] which we need. The main analytic tool is the following lemma which follows from Moser iteration. Note that Lemma 3.1 in [33] is incorrect. A correct version is as follows [34]. Similar bounds were also obtained in [6] and [12], where a version of Lemma A.4 was derived which does not depend on | div R^{V} |.

LEMMA A.1. Let (M, g) satisfy Ric ≥ −k^{2}, diam < D. Then for a function u on M satisfying Δu ≤ αu + β, α, β ≥ 0 we have ∥u∥_{∞} ≤ τ(∥u∥_{2}|α, β, n, k, D). If β = 0 then in fact ∥u∥_{∞} ≤ (1 + τ(α|n, k, D))∥u∥_{2}.

Here the diameter and Ricci curvature bounds give a bound on the Sobolev constant used in Moser iteration by a result of Gallot, and the lower Ricci curvature bound is implied by the bounds on sectional curvature which we have assumed. Then, a standard argument yields the following.

LEMMA A.2. Let V be a vector bundle over M. Suppose M has Ric ≥ −k^{2}, diam < D. Then for any section of V satisfying ⟨∇^{∗}∇S, S⟩ ≤ λ|S|^{2}, ∥S∥_{2} = 1, we have ∥S∥_{∞} ≤ 1 + τ(λ|n, k, D).

Proof.

Δ|S|^{2} = 2⟨∇^{∗}∇S, S⟩ − 2|∇S|^{2}
≤ 2⟨∇^{∗}∇S, S⟩ − 2|∇|S||^{2}
≤ 2λ|S|^{2} − 2|∇|S||^{2}

Now we can use that we also have

Δ|S|^{2} = 2|S| Δ|S| − 2|∇|S||^{2}

and solving for Δ|S| we get Δ|S| ≤ λ|S|, and hence A.1 gives the desired result. □

To bound |∇S| we apply the following Bochner formula

LEMMA A.3. Let R^{V} : T M ⊗ T M ⊗ V → V denote the curvature of the vector bundle V. Let div^{1} R^{V} be minus the metric contraction of ∇R^{V} in the first two slots. Then for any section S of V

∇(∇^{∗}∇)S = (∇^{∗}∇)∇S − (div^{1} R^{V})S + ∇_{Ric(.)}S + 2c_{12}(id ⊗R^{V})(∇S)

Proof. First note that, as the metric is parallel, metric contraction is parallel. The metric contraction of the i-th slot with the j-th slot is denoted by c_{ij}. Let τ : T M ⊗T M → T M ⊗T M, X ⊗Y ↦ Y ⊗X. Note that R^{V} S = ∇∇S −(τ ⊗id)∇∇S. We calculate

∇(∇^{∗}∇)S = −∇c_{12}(∇∇S)
= −c_{23}(∇∇∇S)
= −c_{23}(τ ⊗ id ⊗ id)(∇∇∇S + R^{T^{∗}M⊗V} ∇S)
= −c_{23}R^{T^{∗}M⊗V} ∇S − c_{13}∇∇∇S

The first summand gives

−c_{23}R^{T^{∗}M⊗V} ∇S = c_{23}∇_{R(·,·)·}S − c_{13}(id ⊗R^{V})(∇S) = ∇_{Ric(·)}S + c_{12}(id ⊗R^{V})(∇S).

For the second term,

−c_{13}∇∇∇S = −c_{13}∇ ((τ ⊗ id)∇∇S + R^{V} S)
= −c_{13}(id ⊗τ ⊗ id)(∇∇∇S) + c_{12}∇R^{V} S
= (∇^{∗}∇)∇S − div^{1} R^{V} S + c_{12}(id ⊗R^{V})(∇S),

where we have used the definition div^{1} R^{V} S = −c_{12}(∇R^{V})S. □

As a consequence of the Lemmas A.1 and A.3 we have

LEMMA A.4. Suppose that M, V have Ric ≥ −k^{2}, |c_{12}(id ⊗R^{V})|, | div^{1} R^{V} | < K, and diam < D. Let S be an eigensection of V with ∇^{∗}∇S = λS, and ∥S∥_{2} = 1. Then ∥∇S∥_{∞} ≤ τ(λ|n, k, K, D).

Note that | Ric | and |c_{12}(id ⊗R^{V})| are bounded by | sec | and |R^{V} |, and | div^{1} R^{V} | is bounded by |∇R^{V} |.

Proof. First of all

Δ|∇S|^{2} = 2⟨∇^{∗}∇(∇S), ∇S⟩ − 2|∇∇S|^{2}
≤ 2⟨∇^{∗}∇(∇S), ∇S⟩ − 2|∇|∇S||^{2}

Again we can use

(A.5)  Δ|∇S|^{2} = 2|∇S| Δ|∇S| − 2|∇|∇S||^{2}

Which gives

|∇S|Δ|∇S| ≤ ⟨∇^{∗}∇(∇S), ∇S⟩
= ⟨∇(∇^{∗}∇)S − ∇_{Ric(.)}S + div^{1}(R^{V} S) − 2c_{12}(id ⊗R^{V})(∇S), ∇S⟩
≤ (λ + k^{2} + 2|c_{12}(id ⊗R^{V})|)|∇S|^{2} + | div^{1} R^{V} ||S||∇S|.

Hence

Δ|∇S| ≤ (λ + k^{2} + 2|c_{12}(id ⊗R^{V})|)|∇S| + | div^{1} R^{V} ||S|
≤ (λ + k^{2} + 2|c_{12}(id ⊗R^{V})|)|∇S| + | div^{1} R^{V} |(1 + τ(λ|n, k, D)).

Then finally we can use Lemma A.1, along with the fact that ∥∇S∥_{2} = λ∥S∥_{2}. □

We then note that Lemma A.3 gives us a bound on ∇^{∗}∇(∇S) from which we can conclude from Lemma A.4 that ∫_{M}⟨∇^{∗}∇(∇S), ∇S⟩dV = ∥∇∇S_{i}∥_{2} is small. Hence, Theorem 2.1 is proven.

Finally we include a proof of Theorem 2.2.

THEOREM 2.2. Suppose that S_{1}, …, S_{m} are L^{2}-orthonormal eigensections of V, with eigenvalues λ_{1} ≤ ⋯ ≤ λ_{m}. Then with for k, K, D as above

∥⟨S_{i}, S_{j}⟩ − δ_{ij}∥_{∞} ≤ τ(λ_{m}|n, K, k, D).

Proof. For any unit length vector X we calculate for

|∇_{X}⟨S_{i}, S_{j}⟩| = |⟨∇_{X}S_{i}, S_{j}⟩ + ⟨S_{i}, ∇_{X}S_{j}⟩|
≤ ∥∇S_{i}∥_{∞}∥S_{j}∥_{∞} + ∥S_{i}∥_{∞}∥∇S_{j}∥_{∞}
≤ τ(max(λ_{i}, λ_{j})|n, k, K, D)

where we used Lemmata A.2 and A.4 in the last inequality. Together with ∫_{M}⟨S_{i}, S_{j}⟩ = δ_{ij} the statement easily follows. □

B. Connection Laplacians under perturbations of the metric

We assume here that M is a compact spin manifold with a fixed (topological) spin structure. The topological spin structure defines for any metric g on M a (metric) spin structure P_{Spin}(M, g) → P_{SO}(M, g).

PROPOSITION B.1. Let ∇^{g} be the Levi-Civita-connection on the spinor bundle with respect to the metric g. Let λ_{1}(g) ≤ λ_{2}(g) … be the eigenvalues of the connection Laplacian ∇^{g}^{∗}∇^{g}. For two metrics g and g̃ let δ = δ(g, g̃) be the smallest number such that

e^{−δ}g̃(X, X) ≤ g(X, X) ≤ e^{δ}g̃(X, X) ∀X ∈ T M,
|∇^{g} − ∇g̃|_{g} ≤ δ.

Then

e^{− \frac{1001}{1000} δ}λ_{k}(g̃) − τ(δ) ≤ λ_{k}(g) ≤ e^{\frac{1001}{1000} δ}λ_{k}(g̃) + τ(δ),

where τ(δ) → 0 for δ → 0.

Proof. Let A be the unique positive selfadjoint endomorphism of T M such that

g̃(AX, AY) = g(X, Y).

As a consequence

e^{−δ/2} Id ≤ A ≤ e^{δ/2} Id

in the sense of symmetric operators and

| ˜∇(A^{2})|g̃ = | ˜∇g|g̃ = |( ˜∇ − ∇)g|g̃ ≤ τ_{1}(δ).

Hence also | ˜∇A|g̃ ≤ τ_{2}(δ). Let e_{1}, …, e_{n} be a local orthonormal frame for g. Then Ae_{1}, …, Ae_{n} is an orthonormal frame for g̃. The connection-1-forms ω and ω̃ are defined as

ω(X)_{j}^{k} := g(∇_{X}e_{j}, e_{k}) ω̃(X)_{j}^{k} := g̃( ˜∇_{X}Ae_{j}, Ae_{k}).

We calculate

|ω(X)_{j}^{k} − ω̃(X)_{j}^{k}| ≤ |g((∇_{X} − ˜∇_{X})e_{j}, e_{k})| + |g(A^{−1}( ˜∇_{X}A)e_{j}, e_{k})|
≤ (|∇ − ˜∇|_{g} + e^{δ/2}| ˜∇A|g̃) |X| ≤ τ_{3}(δ) |X|

The map A induces an SO(n) equivariant fiber map P_{SO}(M, g) → P_{SO}(M, g̃) which lifts to the spin structure (if the spin structures coincide as topological spin structures). The corresponding vector bundle map of the associated bundles is an isomorphism of vector bundle A_{Σ} : Σ(M, g) → Σ(M, g̃) which preserves length fiberwise and such that A_{Σ}(V · φ) = A(V) · A_{Σ}(φ) for all V ∈ T_{p}M, φ ∈ Σ_{p}M.

A_{Σ}(∇_{X}φ) − ˜∇_{X}(A_{Σ}φ) = \frac{1}{4} ∑_{j,k=1}^{n}(ω(X)_{j}^{k} − ω̃(X)_{j}^{k})A_{Σ}(e_{j} · e_{k} · φ).

Hence |A_{Σ}(∇_{X}φ) − ˜∇_{X}(A_{Σ}φ)| ≤ \frac{n^{2}τ_{3}(δ)}{4}.

|d(det A)|g̃ ≤ τ(δ).

We set ψ̃ := (det A)A_{Σ}ψ. The map L^{2}(Σ(M, g)) → L^{2}(Σ(M, g̃)), ψ ↦ ψ̃ is an isometry. Then

⟨∇^{∗}∇ψ, ψ⟩^{1/2} ≤ e^{δ/2}⟨ ˜∇^{∗} ˜∇ψ̃, ψ̃⟩^{1/2} + τ(δ).

From this we deduce

(1 − τ(δ)) e^{−δ}λ_{k}(g̃) − τ(δ) ≤ λ_{k}(g) ≤ (1 + τ(δ)) e^{δ}λ_{k}(g̃) + τ(δ).

As a consequence

e^{− \frac{1001}{1000} δ}λ_{k}(g̃) − τ(δ) ≤ λ_{k}(g) ≤ e^{\frac{1001}{1000} δ}λ_{k}(g̃) + τ(δ).
□

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Bernd Ammann, Institut Élie Cartan, Université Henri Poincaré, Nancy I, B.P. 239, 54506 Vandoeuvre-Lès-Nancy, Cedex, France,

www.berndammann.de/publications

Chad Sprouse, Department of Mathematics, CSU Northridge, 18111 Nordhoff Street, Northridge, CA 91330-8313, USA

 E-mail address: chad.sprouse@csun.edu