Lichnerowicz-Obata Estimate, Almost Parallel $p$-form and Almost Product Manifolds

We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give a pinching result about the almost equality case when $2\leq p


Introduction
In this paper we give an estimate for the first eigenvalue of the Laplacian of closed Riemannian manifolds with positive Ricci curvature and an almost parallel form, and give a pinching result about the almost equality case.
One of the most famous theorem about the estimate of the first eigenvalue of the Laplacian is the Lichnerowicz-Obata theorem. Lichnerowicz showed the optimal comparison result for the first eigenvalue when the Riemannian manifold has positive Ricci curvature, and Obata showed that the equality of the Lichnerowicz estimate implies that the Riemannian manifold is isometric to the standard sphere. In the following, λ k (g) denotes the k-th eigenvalue of the Laplacian ∆ := − tr g Hess acting on functions.
Theorem 1.1 (Lichnerowicz-Obata theorem). Take an integer n ≥ 2. Let (M, g) be an n-dimensional closed Riemannian manifold. If Ric ≥ (n−1)g, then λ 1 (g) ≥ n. The equality holds if and only if (M, g) is isometric to the standard sphere of radius 1.
Note that Petersen considered the pinching condition on λ n+1 (g), and Aubry and Honda improved it independently.
We mention some improvements of the Lichnerowicz estimate when the Riemannian manifold has a special structure. If (M, g) is a real n-dimensional Kähler manifold with Ric ≥ (n−1)g, then the Lichnerowicz estimate is improved as follows: (1) λ 1 (g) ≥ 2(n − 1).
See [3] for the proof. For these cases, the Riemannian manifold (M, g) has a nontrivial parallel 2 and 4-form, respectively. When (M, g) is an n-dimensional product Riemannian manifold (N 1 × N 2 , g 1 + g 2 ) with Ric ≥ (n − 1)g, then we have and M has a non-trivial parallel form if either N 1 or N 2 is orientable. Grosjean [14] gave a unified proof of the improvements of the Lichnerowicz estimate when the Riemannian manifold has a non-trivial parallel form. Theorem 1.3 ([14]). Let (M, g) be an n-dimensional closed Riemannian manifold. Assume that Ric ≥ (n − p − 1)g and that there exists a nontrivial parallel p-form on M (2 ≤ p ≤ n/2). Then, we have Moreover, if p < n/2 and if in addition M is simply connected, then the equality in (3) implies that (M, g) is isometric to a product S n−p × (X, g ′ ), where (X, g ′ ) is some p-dimensional closed Riemannian manifold.
Remark 1.1. We give several remarks on this theorem.
• Grosjean also showed this type theorem when M has a convex smooth boundary.
• Though Grosjean originally assumed the manifold is orientable, the assumption can be easily removed by taking the orientable double covering. • If M is simply connected, p = n/2 and n ≥ 6, then it is not difficult to show that the equality in (3) also implies that M is isometric to a product S n/2 × X (see Corollary 3.4). • If (M, g) is either a Kähler manifold or a quaternionic Kähler manifold, then the estimate (1) or (2) is better. • If there exists a non-trivial parallel p-form ω (1 ≤ p ≤ n − 1) on an ndimensional Riemannian manifold (M, g), then ω(x) ∈ p T * x M (x ∈ M ) is invariant under the Holonomy action, and so the Holonomy group coincides with neither SO(n) nor O(n).
Remark 1.2. In fact, we prove that there exist constants C(n, p) > 0 and α(n) > 0 such that d GH (M, S n−p × X) ≤ C(n, p)δ α(n) under the assumption of Main Theorem 2. One can easily find the explicit value of α(n) (see Notation 5.38 and Theorem 5.51). However, it might be far from the optimal value. By the Gromov's pre-compactness theorem, we can take X to be a geodesic space. However, we lose the information about the convergence rate in that case.
We would like to point out that our work was motivated by Honda's spectral convergence theorem [19], which asserts the continuity of the eigenvalues of the connection Laplacian ∆ C,p acting on p-forms with respect to the non-collapsing Gromov-Hausdorff convergence assuming the two-sided bound on the Ricci curvature. By virtue of his theorem, we can generalize our main theorems to Ricci limit spaces under such assumptions. See Appendix A for detail. Note that we show our main theorems without the non-collapsing assumption, i.e., without assuming the lower bound on the volume of the Riemannian manifold.
Our work was also motivated by the Cheeger-Colding almost splitting theorem (see [8,Theorem 9.25]), whose conclusion is the Gromov-Hausdorff approximation to a product R × X. As the almost splitting theorem, we need to show the almost Pythagorean theorem under the assumption of Main Theorem 2. One step of the proof (Lemma 5.41) is similar to the final step of the almost splitting theorem [8,Lemma 9.16].
The structure of this paper is as follows.
In section 2, we recall some basic definitions and facts, and give calculations of differential forms.
In section 3, we assume that the Riemannian manifold has a non-trivial parallel p-form. We give an easy proof of the formula used by Grosjean to prove Theorem 1.3. In section 4, we estimate the error terms of the Grosjean's formula when the Riemannian manifold has a non-trivial almost parallel p-form. As a consequence, we prove Main Theorem 1 and Main Theorem 3. In section 5, we prove Main Theorem 2 and Main Theorem 4. In subsection 5.1, we list some useful technique for pinching problems. In subsection 5.2, we show some pinching conditions on the eigenfunctions along geodesics under the assumption λ k (g) ≤ n − p + δ and λ 1 (∆ C,p ) ≤ δ. In subsection 5.3, we show that similar results hold under the assumption λ k (g) ≤ n − p + δ and λ 1 (∆ C,n−p ) ≤ δ. In subsection 5.4, we show that the eigenfunctions are almost cosine functions in some sense under our pinching condition. In subsection 5.5, we construct an approximation map and show Main Theorem 2 except for the orientability. In subsection 5.6, we give some lemmas to prove the remaining part of main theorems. In subsection 5.7, we show the orientability of the manifold under the assumption of Main Theorem 2, and complete the proof of it. In subsection 5.8, we show that the assumption of Main Theorem 4 implies that λ n−p+1 (g) is close to n − p, and complete the proof of Main Theorem 4. In Appendix A, we discuss Ricci limit spaces. We show a gap theorem of the first eigenvalue of the Laplacian acting on n-forms for n-dimensional unorientable closed Riemannian manifolds. As a consequence, we show the stability of unorientability under the non-collapsing Gromov-Hausdorff convergence assuming the two-sided bound on the Ricci curvature and the upper bound on the diameter. This enable us to generalize our main theorems to Ricci limit spaces under such assumptions.
In Appendix B, we give the almost version of the estimate (1) assuming that there exists a 2-form ω which satisfies that ∇ω 2 and J 2 ω + Id 2 are small, where J ω ∈ Γ(T * M ⊗ T M ) is defined so that ω = g(J ω ·, ·).
Acknowledgments. I am grateful to my supervisor, Professor Shinichiroh Matsuo for his advice. I also thank Professor Shouhei Honda for helpful discussions about the orientability of Ricci limit spaces.   The Hausdorff distance defines a metric on the collection of compact subsets of X.
Definition 2.2 (Gromov-Hausdorff distance). Let (X, d X ), (Y, d Y ) be metric spaces. Define The Gromov-Hausdorff distance defines a metric on the set of isometry classes of compact metric spaces (see [22,Proposition 11.1.3]). Definition 2.3 (ǫ-Hausdorff approximation map). Let (X, d X ), (Y, d Y ) be metric spaces. We say that a map f : X → Y is an ǫ-Hausdorff approximation map for ǫ > 0 if the following two conditions hold.
(i) For all a, b ∈ X, we have |d , for all y ∈ Y , there exists x ∈ X with d Y (f (x), y) < ǫ.
If there exists an ǫ-Hausdorff approximation map f : X → Y , then we can show that d GH (X, Y ) ≤ 3ǫ/2 by considering the following metric d on X Y : If d GH (X, Y ) < ǫ, then there exists a 2ǫ-Hausdorff approximation map from X to Y . Let C(u 1 , . . . , u l ) > 0 denotes a positive function depending only on the numbers u 1 , . . . , u l . For a set X, Card X denotes a cardinal number of X.
Let (M, g) be a closed Riemannian manifold. For any p ≥ 1, we use the normalized L p -norm: and f ∞ := sup ess x∈M |f (x)| for a measurable function f on M . We also use this notation for tensors. We have f p ≤ f q for any p ≤ q ≤ ∞. Let ∇ denotes the Levi-Civita connection. Throughout in this paper, 0 = λ 0 (g) < λ 1 (g) ≤ λ 2 (g) ≤ · · · → ∞ denotes the eigenvalues of the Laplacian ∆ = − i,j g ij ∇ i ∇ j acting on functions. We sometimes identify T M and T * M using the metric g. Given points x, y ∈ M , let γ x,y denotes one of minimal geodesics with unit speed such that γ x,y (0) = x and γ x,y (d(x, y)) = y. For given x ∈ M and u ∈ T x M with |u| = 1, let γ u : R → M denotes the geodesic with unit speed such that γ u (0) = x andγ u (0) = u.
For any x ∈ M and u ∈ T x M with |u| = 1, put t(u) := sup{t ∈ R >0 : d(x, γ u (t)) = t}, and define the interior set I x ⊂ M at x (see also [23, p.104]) by Then, I x is open and Vol(M \ I x ) = 0 [23,III Lemma 4.4]. For any y ∈ I x \ {x}, the minimal geodesic γ x,y is uniquely determined. The function d(x, ·) : M → R is differentiable in I x \ {x} and ∇d(x, ·)(y) =γ x,y (d(x, y)) holds for any y ∈ I x \ {x} [23,III Proposition 4.8].
Let V be an n-dimensional real vector space with an inner product , . We define inner products on k V and V ⊗ k V as follows: Then, ι defines a bi-linear map: By identifying V and V * using , , we also use the notation ι for the bi-linear map: For any Riemannian manifold (M, g), we define operators ∇ * : and ω ∈ Γ( k T * M ), where n = dim M and {e 1 , . . . , e n } is an orthonormal basis of T M . If M is closed, then we have Finally, we list some important notation. Let (M, g) be a closed Riemannian manifold.
• d denotes the Riemannian distance function.
• Ric denotes the Ricci curvature tensor.
• diam denotes the diameter.
• Vol or µ g denotes the Riemannian volume measure. • · p denotes the normalized L p -norm for each p ≥ 1, which is defined by for any measurable function f on M . • f ∞ denotes the essential sup of |f | for any measurable function f on M .
denotes the Hodge Laplacian defined by ∆ := dd * + d * d. We frequently use the Laplacian acting on functions. Note that ∆ = − tr g ∇ 2 holds for functions under our sign convention.
• γ x,y : [0, d(x, y)] → M denotes one of minimal geodesics with unit speed such that γ x,y (0) = x and γ x,y (d(x, y)) = y for any x, y ∈ M . • γ u : R → M denotes the geodesic with unit speed such that γ u (0) = x anḋ γ(0) = u for any x ∈ M and u ∈ T x M with |u| = 1.
• I x denotes the interior set at x ∈ M . We have Vol(M \ I x ) = 0. We have that γ x,y is uniquely determined and ∇d(x, ·) =γ x,y (d(x, y)) holds for any denotes the connection Laplacian acting on k-forms.
Note that the lowest eigenvalue of the Laplacian ∆ acting on function is always equal to 0, and so we start counting the eigenvalues of it from i = 0. This is not the case with the connection Laplacian ∆ C,k acting on k-forms, and so we start counting the eigenvalues of it from i = 1. For any i ∈ Z >0 , we have

2.2.
Calculus of Differential Forms. In this subsection, we recall some facts about differential forms, and do some calculations.
We first recall the decomposition: See also [25,Section 2].
Definition 2.5. Let (M, g) be an n-dimensional Riemannian manifold. We define a homomorphism R k : k T * M → k T * M as for any ω ∈ k T * M , where {e 1 , . . . , e n } is an orthonormal basis of T M , {e 1 , . . . , e n } is its dual and R(e i , e j )ω is defined by Note that if k = 1, then we have R 1 ω = Ric(ω, ·) for any ω ∈ Γ(T * M ). The Bochner-Weitzenböck formula is stated as follows: Theorem 2.6 (Bochner-Weitzenböck formula). For any ω ∈ Γ( k T * M ), we have In particular, we have the following theorem when k = 1: Let us do some calculations of differential forms.
Lemma 2.8. Let (M, g) be a Riemannian manifold of dimension n. Take a vector field X ∈ Γ(T M ), a p-form ω ∈ Γ( p T * M ) (p ≥ 1) and a local orthonormal bases {e 1 , . . . , e n } of T M .
Let us show (ii). We have Thus, by (i), we get This gives (ii). Finally, we show (iii). We have This gives (iii).
When ω is parallel, we have the following corollary.
Corollary 2.9. Let (M, g) be a Riemannian manifold of dimension n. Take a vector field X ∈ Γ(T M ) and a parallel p-form ω ∈ Γ( p T * M ) (p ≥ 1).

Parallel p-form
In this section, we consider Riemannian manifolds with a non-trivial parallel differential form. The reader who is interested only in the proof of the main theorems can skip this section.
3.1. Bochner-Reilly-Grosjean Formula. The aim of this subsection is to give an easy proof of what Grosjean called a new Bochner-Reilly formula [14, Proposition 3.1] when the Riemannian manifold has a non-trivial parallel p-form ω. In section 4, we estimate the error terms when the manifold has no boundary and ω is not parallel.
Proposition 3.1 (Bochner-Reilly-Grosjean formula [14]). Let (M, g) be a compact n-dimensional Riemannian manifold possibly with a smooth boundary (∂M, g ′ ), and let ν be the outward unit normal vector field. For any f ∈ C ∞ (M ) and any parallel The following proposition is the main result of this subsection. Assume that there exists a non-trivial parallel p-form ω on M (1 ≤ p ≤ n/2). Then, we have Moreover, if either p = n 2 or n ≥ 6 and if in addition M is simply connected, then the equality in (14) implies that (M, g) is isometric to a product (S n−p , g r ) × (X, g ′ ), where g r is some rotationally symmetric metric on S n−p and (X, g ′ ) is a p-dimensional closed Riemannian manifold.
Proof. We first show (14). By taking the two-sheeted orientable Riemannian covering of (M, g) if necessary, we can assume that (M, g) is oriented. Then, we have (9). This implies the estimate (14). We next consider the equality case. Suppose that M is simply connected. Let be the irreducible decomposition of the holonomy representation, and let (M, g) = (M 1 , g 1 ) × · · · × (M k , g k ) be the corresponding de Rham decomposition. There exist non-negative integers p 1 , . . . , p k ∈ Z ≥0 such that p 1 + · · · + p k = p and the p1 E 1 ⊗ · · · ⊗ p k E kcomponent of ω is non-zero and parallel, where E i is the sub-bundle of T * M that corresponds to E i . Thus, we can assume ω ∈ Γ( p1 E 1 ⊗ · · · ⊗ p k E k ). Take i with p i = 0. Let us show that there exists a non-trivial parallel p i -form on M i . Take some x ∈ M and decompose ω x as Thus, we get for all f ∈ C ∞ (M i ) by Proposition 3.1, and so By considering * ω, we also have By (16), (17) and [1, Proposition 2.4], we get where we put Suppose that (19) Ω 1 (g) = n − p − 1 n − p and either p = n 2 or n ≥ 6. Without loss of generality, we can assume that dim M 1 = max i {dim M i }. If dim M 1 < n − p, then we get (18). This contradicts to (19), and so we have dim M 1 ≥ n − p.
We consider the following three cases: • dim M 1 = n. We first suppose that n−p < dim M 1 < n. Then, p 2 +· · ·+p k ≤ n−dim M 1 < p, and so p 1 = 0. Moreover, we have p 1 ≤ p < dim M 1 . Thus, we have Since dim M i < n − p for all i = {2, . . . , k}, we get (18). This contradicts to (19). We next suppose that dim M 1 = n. Then, we have M = M 1 . Since we have Ω 1 (g) ≤ (p − 1)/p and p ≤ n − p, we get (20) p = n − p = n/2 ≥ 3 by (19). Since there exists a non-trivial parallel p-form, the holonomy group of (M, g) is not equal to SO(n). If Ric ≤ 0, then Ω 1 (g) = 0, and so we have one of the following by the Berger classification theorem: • (M, g) is a Kähler manifold, • (M, g) is a quaternionic Kähler manifold, • (M, g) is a symmetric space.
As a corollary, we get the following: Corollary 3.4. Let (M, g) be an n-dimensional closed Riemannian manifold. Assume that Ric ≥ (n − p − 1)g and there exists a non-trivial parallel p-form on M (2 ≤ p ≤ n/2). Then, we have Moreover, if either p = n/2 or n ≥ 6 and if in addition M is simply connected, the equality in (21) implies that (M, g) is isometric to a product S n−p × X, where X is a p-dimensional closed Riemannian manifold.
If we assume more strong condition on eigenvalues, then the assumption that the manifold is simply connected can be removed. Corollary 3.5. Let (M, g) be an n-dimensional closed Riemannian manifold. Assume that Ric ≥ (n − p − 1)g and there exists a non-trivial parallel p-form on M (2 ≤ p < n/2). If (23) λ n−p+1 (g) = n − p, Proof. Let f k be the k-th eigenfunction of the Laplacian on S n−p . Note that the functions f 1 , . . . , f n−p+1 are height functions. By Corollary 3.4, the universal cover ( M ,g) of (M, g) is isometric to a product S n−p × (X, g ′ ), where (X, g ′ ) is a p-dimensional closed Riemannian manifold. We regard the function f i as a function on M . Since λ n−p+1 (g) = n − p, each f i ∈ C ∞ ( M ) (i = 1, . . . , n − p + 1) is a lift of some function on M . Thus, the covering transformation preserves f 1 , . . . , f n−p+1 . Therefore, the covering transformation does not act on S n−p , and so we get the corollary.
The almost version of this corollary is Main Theorem 2.

Examples.
In this subsection, we show that the assumption of Corollary 3.5 is optimal in some sense by giving examples.
• (M, g) is not isometric to any product Riemannian manifolds.
is not isometric to any product Riemannian manifolds.

Almost Parallel p-form
In this section, we show Main Theorem 1 and Main Theorem 3. Recall that λ 1 (∆ C,p ) denotes the first eigenvalue of the connection Laplacian acting on p-forms, and It is enough to show Main Theorem 1 when λ 1 (∆ C,p ) ≤ 1. Note that we always have if Ric g ≥ (n − 1)g.

Error Estimates.
In this subsection, we give error estimates about Proposition 3.1. Lemma 4.5 (vii) corresponds to Proposition 3.1.
We list the assumptions of this subsection. We mention that most technique in this paper can be used under the assumption Ric g ≥ −Kg and diam(M ) ≤ D.
Assumption 4.1. In this subsection, we assume the following: • (M, g) is an n-dimensional closed Riemannian manifold with Ric g ≥ −Kg and diam(M ) ≤ D for some positive real numbers K > 0 and D > 0.
Note that we have by the Bochner formula. We first show the following: There exists a positive constant C(n, K, D) > 0 such that |ω|−1 2 ≤ Cλ 1/2 holds.

Proof.
Put Since we have |ω| ∈ W 1,2 (M ), we get by the Kato inequality. Thus, by the Li-Yau estimate [24, p.116], we have Let us give error estimates about Proposition 3.1.
There exists a positive constant C = C(n, k, K, D, L 1 , L 2 ) > 0 such that the following properties hold: Although an orthonormal basis {e 1 , . . . , e n } of T M is defined only locally, To prove (ii) and (iii), we estimate following terms: Thus, we get Thus, we get By Theorem 2.6, (25) and (26), we have we estimate the following terms: 1 Thus, by Lemma 2.8 (iii), we get

Eigenvalue Estimate.
In this subsection, we complete the proofs of Main Theorem 1 and Main Theorem 3. We need the following L ∞ estimates.
Let us show (ii). Since we have we get ω ∞ ≤ C by [22, Proposition 9.2.7] (see also Proposition 7.1.13 and Proposition 7.1.17 in [22]). Note that our sign convention of the Laplacian is different from [22].
We use the following proposition not only for the proofs of Main Theorem 1 and Main Theorem 3 but also for other main theorems.
Proposition 4.5. For given integers n ≥ 4 and 2 ≤ p ≤ n/2, there exists a constant C(n, p) > 0 such that the following property holds. Let (M, g) be an n-dimensional closed oriented Riemannian manifold with Ric g ≥ (n − p − 1)g. Suppose that an integer i ∈ Z >0 satisfies λ i (g) ≤ n − p + 1, and there exists an eigenform ω of the connection Laplacian ∆ C,p acting on p-forms with ω 2 = 1 corresponding to the eigenvalue λ with 0 ≤ λ ≤ 1. Then, we have where f i denotes the i-th eigenfunction of the Laplacian acting on functions.
Proof. By Lemma 4.3 (viii), we have Thus, we get the proposition by Lemma 4.2.

Pinching
In this section, we show the remaining main theorems. Main Theorem 2 is proved in subsection 5.5 except for the orientability, and the orientability is proved in subsection 5.7. Main Theorem 4 is proved in subsection 5.8.
We list assumptions of this section.
Assumption 5.1. Throughout in this section except for subsection 5.1, we assume the following: • n ≥ 5 and 2 ≤ p < n/2.
• (M, g) is an n-dimensional closed Riemannian manifold with Ric g ≥ (n − p − 1)g. • C = C(n, p) > 0 denotes a positive constant depending only on n and p.
Note that, for given real numbers a, b with 0 < b < a and a positive constant C > 0, we can assume that Cδ a ≤ δ b . For most subsections, we list additional assumptions at the beginning of them.

Useful Technique.
In this subsection, we list some useful technique for the pinching problems.
The following lemma is a variation of the Cheng-Yau estimate. See [2, Lemma 2.10] for the proof (see also [8,Theorem 7.1]).
The following theorem is an easy consequence of the Bishop-Gromov inequality.
The following theorem is due to Cheeger-Colding [9] (see also [22,Theorem 7.1.10]). By this theorem, we get integral pinching conditions along the geodesics under the integral pinching condition for a function on M .
Lemma 5.5. Take positive real numbers l, ǫ > 0 and a non-negative real number Then, we have The following lemma is standard.
For any t ∈ [−π, π], we have cos t ≤ 1 − 1 9 t 2 , and so |t| ≤ 3(1 − cos t) 1/2 . For any Finally, we recall some facts about the geodesic flow. Let (M, g) be a closed Riemannian manifold and let U M denotes the sphere bundle defined by There exists a natural Riemannian metric G on U M , which is the restriction of the Sasaki metric on T M (see [23, p.55]). The Riemannian volume measure µ G satisfies for any u ∈ U M . Though φ t does not preserve the metric G in general, it preserves the measure µ G . This is an easy consequence of [23,Lemma 4.4], which asserts that the geodesic flow on T M preserve the natural symplectic structure on T M . We use the following lemma.
Lemma 5.7. Let (M, g) be a closed Riemannian manifold. For any f ∈ C ∞ (M ) and l > 0, we have Proof. Let π : U M → M be the projection. Since the geodesic flow φ t preserves the measure µ G , we have This gives the lemma.
This kind of lemma was used by Colding [12] to prove that the almost equality of the Bishop comparison theorem implies the Gromov-Hausdorff closeness to the standard sphere.

5.2.
Estimates for the Segments. The goal of this subsection is to give error estimates along the geodesics.
Assumption 5.8. In this subsection, we assume the following in addition to Assumption 5.1.
is an eigenfunction of the Laplacian acting on functions with f i • ω ∈ Γ( p T * M ) is an eigenform of the connection Laplacian ∆ C,p with ω 2 = 1 corresponding to the eigenvalue λ with 0 ≤ λ ≤ δ. Lemma 4.4). By Main Theorem 1, we have for all i. If the manifold is orientable, one can regard f i as the i-th eigenfunction and λ i as λ i (g). However, for the unorientable case, our assumption is convenient when we consider f i •π, where π( M ,g) → (M, g) denotes the two-sheeted orientable Riemannian covering.
We first list some basic consequences of our pinching condition.
Proof. It is enough to consider the case when M is orientable.
Finally, we prove (iii). Take arbitrary f ∈ Span R {f 1 , . . . , f k }. We have Thus, we have and so we get
We use the following notation.
For each y 1 ∈ M , we define Now, we use the segment inequality and Lemma 5.7. We show that we have the integral pinching condition along most geodesics.
Under the pinching condition along the geodesic, we get the following: Suppose that a geodesic γ : [0, l] → M satisfies one of the following: • There exist x ∈ M and y ∈ D f (x) such that l = d(x, y) and γ = γ x,y , • There exist x ∈ M and u ∈ E f (x) such that l = π and γ = γ u .
Then, we have for all s ∈ [0, l], and at least one of the following: for all s ∈ [0, l], and In particular, for both cases, there exists a parallel orthonormal basis
Let {E 1 , . . . , E n } be the parallel orthonormal basis of T M along γ such that E i (t) = e i , and {E 1 , . . . , E n } a its dual. Because for all s ∈ [0, l]. Thus, we get for all i = 1, · · · , n and j = 1, . . . , n − p, and for all i, j = n − p + 1, · · · , n. Therefore, we get Thus, for all i = 1, · · · , n and j = 1, . . . , n − p, we get for all i = 1, · · · , n and j = 1, . . . , n − p. Because This implies (ii). Let us show the final assertion. It is trivial that |γ E | is constant along γ. Since we have Thus, we get the lemma by Lemma 5.5.

5.3.
Almost Parallel (n− p)-form I. In this subsection, we consider the pinching condition on λ 1 (∆ C,n−p ) for 2 ≤ p < n/2. If M is oriented, then this is coincide with the pinching condition on λ 1 (∆ C,p ). Thus, we only consider the case when M is not orientable.
Assumption 5.13. In this subsection, we assume the following in addition to Assumption 5.1.
• M is not orientable. Under these assumptions, we use the following notation.
We immediately have the following lemmas by Lemma 5.11 and Lemma 5.12.

Eigenfunction and Distance.
In this subsection, we show that the function is an almost cosine function in some sense under our pinching condition.
Assumption 5.17. In this subsection, we assume the following in addition to Assumption 5.1.
is an eigenfunction of the Laplacian acting on functions with f i The following proposition is the goal of this subsection. See Notation 5.10 and Notation 5.14 for the definitions of D f , Q f , E f and R f .
There exists a point p f ∈ Q f such that the following properties hold: Then, we have for all x ∈ M , and Similarly to p f , we take a point Claim 5.20. Take x ∈ M and y ∈ D f (x). Let {E 1 , . . . , E n } be a parallel orthonormal basis along γ x,y in Lemma 5.12 or Lemma 5.16. If (i) holds in the lemmas, we can assume that E 1 =γ x,y . Then, we have Proof of Claim 5.20. If (i) holds in the lemmas,γ x,y =γ E x,y , and so (45) and (46) are trivial. If (ii) in the lemma holds, we have | ∇f (x), E i | ≤ Cδ 1/25 for all i = n − p + 1, . . . , n. This gives (45) and (46). We get the remaining part of the claim by (45) and Lemma 5.12 or Lemma 5.16.
By Claim 5.19 and Claim 5.21, we get

Gromov-Hausdorff Approximation.
In this subsection, we construct a Hausdorff approximation map, and show that the Riemannian manifold is close to the product metric space S n−p × X in the Gromov-Hausdorff topology under our pinching condition.
Assumption 5.23. In this subsection, we assume the following in addition to Assumption 5.1.
We need the following claim [21, Theorem 7.1]. Note that our sign convention of the Laplacian is different from [22].
Claim 5.26. For a smooth functions u ∈ C ∞ (M ) and a non-negative continuous function F with ∆u ≤ F , we have To apply Claim 5.26 to −| Ψ| 2 , we compute ∆| Ψ| 2 .
We estimate each component. By the assumption, we have . Combining this and Ψ 2 = 1, we get Finally, we estimate f 2 i + |∇f i | 2 − 1 n . Since we have we get Notation 5.27. In the remaining part of this subsection, we use the following notation.
• Let d S denotes the intrinsic distance function on S n−p (1). Note that we have cos d S (x, y) = x · y and for all x, y ∈ S n−p ⊂ R n−p+1 .
• For each f ∈ Span R {f 1 , . . . , f n−p+1 }, we use the notation p f and A f of Proposition 5.18. Recall that we defined • For each x ∈ M , put The goal of this subsection is to show that is an approximation map. This and d(y, A f ) ≤ π + Cδ 1/100n give the lemma.
By the definition of A y , we immediately get the following corollaries:  We need to show the almost Pythagorean theorem for our purpose. To do this, we regard |γ E |s in Lemma 5.12 or Lemma 5.16 as a moving distance in S n−p . We first approximate their cosine.
Notation 5.34. We use the following notation: • For any y 1 , y 2 ∈ M and f ∈ Span R {f 1 , . . . , f n−p+1 } with • For any y 1 , y 2 ∈ M , define • For any y 1 , f plays a crucial role for our purpose. Let us estimate G y1 f .
Let us show the integral pinching condition.
Notation 5.38. We use the following notation.
We use Lemma 5.37 to give the almost Pythagorean theorem for the special case (see Lemma 5.47). For the general case, we need to estimate Gỹ 1 f 1 . To do this, we show that |γ Ẽ y1,y2 |d(ỹ 1 , y 2 ) ≤ π + L under the assumption of Lemma 5.33 in Lemma 5.49. Then, we can estimate Gỹ 1 f 1 similarly to Lemma 5.37. After proving that, we use Lemma 5.41 again to give the almost Pythagorean theorem for the general case. The following lemma, which guarantees that an almost shortest pass from a point in M to A f almost corresponds to a geodesic in S n−p through Ψ under some assumptions, is the first step to achieve these objectives.
The following lemma asserts that the differential of an almost shortest pass from a point in M to A f is in the direction of ∇f under some assumptions.
Let {E 1 , . . . , E n } be a parallel orthonormal basis of T * M along γ y,x in Lemma 5.12 or Lemma 5.16 for f . Then, we have the following for all s ∈ [0, d(x, y)]:
The following lemma is crucial to show the almost Pythagorean theorem. There exist w ∈ S n−p and x 1 , Then, Note that we have r(s) ≤ π by diam(S n−p ) = π. By (95), we get

Thus, we have
By (94) and the definition of l, we get the lemma.
The following lemma guarantees that if the images of two points in M under Φ f are close to each other in S n−p × A f , then their distance in M are close to each other under some assumptions.
Proof. We first show the following claim.
Let us show the almost Pythagorean theorem for the special case. Recall that we defined η 1 := η τ 0 .
Take arbitrary i ∈ {0, 1, 2} and suppose that we have chosen a i , b i ∈ M such that (i), (ii) and (iii) hold if i ≥ 1. Let us define a i+1 , b i+1 ∈ M that satisfy our properties. Take a Π-triple (b i ,ã i ,ỹ i ) for (b i , a i , y, f ). Define by Lemma 5.28 and Lemma 5.44. Take a Π-triple (a i+1 , b i , by Lemma 5.28 and Lemma 5.44. By (112) and (113), we get (iv).
By the assumptions and Lemma 5.39, we get (ii) for a i+1 and b i+1 . By the assumptions, we have Thus, we get (iii) for a i+1 and b i+1 . By definition, we have Thus, we get (v).
Let us show that the map Φ f : Proposition 5.18. In the following, we show that a v := a Fv (a) ∈ A Fv has the desired property. By Lemma 5.29, we get Thus, by Lemma 5.29, we get and so we get by Lemma 5.47 putting x = z = a v , y = a and w = a f (a v ).
By the above theorem, we get Main Theorem 2 except for the orientability, which is proved in subsection 5.7. 5.6. Further Inequalities. In this subsection, we show two lemmas to prove the remaining part of main theorems.
Assumption 5.52. In this subsection, we assume the following in addition to Assumption 5.1. • ω ∈ Γ( p T * M ) is an eigenform of the connection Laplacian ∆ C,p with ω 2 = 1 corresponding to the eigenvalue λ with 0 ≤ λ ≤ δ.
Then, we have the following properties.
for all i = j. Therefore, we get for all i = j. Thus, we get (i).
For all x ∈ G and i, j with i = j, we have Thus, we get (ii).

5.7.
Orientability. The goal of this subsection is to show the orientability of Main Theorem 2. Note that our assumptions are Assumption 5.1. Proof. To prove the theorem, we use the following claim: In the following, we show that ∇V 2 2 / V 2 2 < (n − p + 1)/(n − 1). Define a vector bundle E := T * M ⊕ Re, where Re denotes the trivial bundle of rank 1 with a global non-vanishing section e. We consider an inner product ·, · on E defined by α + f e, β + he := α, β + f h for all α, β ∈ Γ(T * M ) and f, h ∈ C ∞ (M ). Put for each i, and Then, we have ι(e)(α ∧ ω) = V , and so (153) |α ∧ ω| = |V |.
5.8. Almost Parallel (n − p)-form II. In this subsection, we show that the assumption "λ n−p (g) is close to n − p" implies the condition "λ n−p+1 (g) is close to n − p"under the assumption λ 1 (∆ C,n−p ) ≤ δ.
Note that our assumptions are Assumption 5.1.
Proof. If M is not orientable, we take the two-sheeted oriented Riemannian covering π : ( M ,g) → (M, g), and put Then, we have . . ∧ df n−p , π * ξ . Thus, it is enough to consider the case when M is orientable. In the following, we assume that M is orientable, and we fix an orientation of M . Put Let V g ∈ Γ( n T * M ) be the volume form of (M, g). Then, we have Define a vector bundle E := T * M ⊕ Re, where Re denotes the trivial bundle of rank 1 with a global non-vanishing section e. We consider an inner product ·, · on E defined by α + f e, β + he := α, β + f h for all α, β ∈ Γ(T * M ) and f, h ∈ C ∞ (M ). Put for each i, and β := S 1 ∧ · · · ∧ S n−p ∈ Γ( n−p E).
Theorem 5.59. Take n ≥ 5 and 2 ≤ p < n/2. Let {(M i , g i )} i∈N be a sequence of n-dimensional closed Riemannian manifolds with Ric gi ≥ (n − p − 1)g i that satisfies one of the following: By the correspondence through the Hodge star operator, we have that λ 1 (∆ C,n , g) is the eigenvalue of the Laplacian acting on functions on M . If λ 1 (∆ C,n , g) = 0, then ω defines an orientation of M , and so we have λ 1 (∆ C,n , g) = 0 = λ 0 (g).
We immediately get the following corollary.
Corollary A.2. Let (M, g) be an n-dimensional closed Riemannian manifold. If one of the following properties holds, then M is orientable.
Remark A.1. For any closed Riemannian manifold (M, g) and a positive real number a > 0, we have Ric g = Ric ag and λ k (ag) = 1 a λ k (g) for all k. Thus, we get Claim 5.56 by Corollary A.2.
As an application of Lemma A.1, we show the stability of unorientability under the non-collapsing Gromov-Hausdorff convergence assuming the two-sided bound on the Ricci curvature.
Theorem A.3. Take real numbers K 1 , K 2 ∈ R and positive real numbers D > 0 and v > 0. Let {(M i , g i )} be a sequence of n-dimensional unorientable closed Riemannian manifolds with K 1 g i ≤ Ric gi ≤ K 2 g i , diam(M ) ≤ D and Vol(M ) ≥ v. Suppose that {(M i , g i )} converges to a limit space X in the Gromov-Hausdorff sense. Then, X is not orientable in the sense of Honda [18] (see also the definition below).
Note that Honda [18,Theorem 1.3] showed the stability of orientability without assuming the upper bound on the Ricci curvature.
Before proving Theorem A.3, we fix our notation and recall definitions about limit spaces. By the Myers theorem, we have M 1 (n, K 1 , K 2 , v) ⊂M 2 (n, K 1 , K 2 , (n − 1)/K 1 π, v), M 1 (n, K 1 , K 2 , v) ⊂M 2 (n, K 1 , K 2 , (n − 1)/K 1 π, v) if K 1 > 0. If X i ∈ M 2 (i ∈ N) converges to X ∈ M 2 in the Gromov-Hausdorff topology, then there exist a sequence of positive real numbers {ǫ i } i∈N with lim i→∞ ǫ i = 0, and a sequence of ǫ i -Hausdorff approximation maps φ i : X i → X. Fix such a sequence. We say a sequence x i ∈ X i converges to x ∈ X if lim i→∞ φ i (x i ) = x (denote it by x i GH → x). By the volume convergence theorem [10, Theorem 5.9], (X i , H n ) converges to (X, H n ) in the measured Gromov-Hausdorff sense, i.e., for all r > 0 and all sequence x i ∈ X i that converges to x ∈ X, we have lim i→∞ H n (B r (x i )) = H n (B r (x)), where H n denotes the n-dimensional Hausdorff measure.
For all X ∈ M 2 , we can consider the cotangent bundle π : T * X → X with a canonical inner product by [7] and [11] (see also [17,Section 2] for a short review). We have H n (X \π(T * X)) = 0 and T * x X := π −1 (x) is an n-dimensional vector space for all x ∈ π(T * X). For all Lipschitz function f on X, we can define df (x) ∈ T * x X for almost all x ∈ X, and we have df ∈ L ∞ (T * X).
Let us recall definitions of functional spaces on limit spaces. Note that we can define such functional spaces on more general spaces than our assumption. Some of the following functional spaces are first introduced by Gigli [13].
(i) Let LIP(X) be the set of the Lipschitz functions on X. For all f ∈ LIP(X), we define f 2 H 1,2 = f 2 2 + df 2 2 . Let H 1,2 (X) be the completion of LIP(X) with respect to this norm. (ii) Define D 2 (∆, X) := f ∈ H 1,2 (X) : there exists F ∈ L 2 (X) such that X df, dh dH n = X F h dH n for all h ∈ H 1,2 (X) .
We next suppose λ 1 (∆ C,n , X) = 0 and show that X is orientable. Let {(M i , g i )} i∈N be a sequence in M 2 that converges to X in the Gromov-Hausdorff topology. Then, we have lim i→∞ λ 1 (∆ C,n , g i ) = 0 by Theorem A.6. Thus, by Corollary A.2, we get that M i is orientable for sufficiently large i, and so X is orientable by the stability of orientability [18, Theorem 1.3].
Remark B.1. It is enough to prove the theorem when δ is small. Thus, we can assume that n = 2m is an even integer by Lemma B.2. If n = 2, then λ 1 (g) ≥ 2(n − 1) is the original Lichnerowicz estimate. If n = 4, the conclusion of the theorem can also be deduced from Main Theorem 1.