Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.


Introduction
In this article we show that if an n-dimensional closed Riemannian manifold with positive Ricci curvature admits an almost parallel p-form (2 ≤ p < n/2) in L 2 -sense and if the first (n + 1)-th eigenvalues of the Laplacian acting on functions take almost optimal values, then the Riemannian manifold is close to the product of the standard spheres S n−p × S p with appropriate radii (Main Theorem below). Before giving the precise statement, we provide some backgrounds.
The Lichnerowicz-Obata theorem is one of the classical theorem about the first eigenvalue of the Laplacian. 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 positive eigenvalue of the minus Laplacian −∆ := −tr g Hess acting on functions.
The equality λ 1 (g) = n holds if and only if (M, g) is isometric to the standard sphere of radius 1.
Petersen [27], Aubry [8] and Honda [22] showed the stability result of the Lichnerowicz-Obata theorem. In the following, d GH denotes the Gromov-Hausdorff distance and S n denotes the n-dimensional standard sphere of radius 1. (see Definition 2.13 for the definition of the Gromov-Hausdorff distance).
Note that Petersen considered the pinching condition on λ n+1 (g), and Aubry and Honda improved it independently.
To state the almost version of Theorem 1.3, we introduce the first eigenvalue of the connection Laplacian acting on p-forms λ 1 (∆ C,p ) for a closed Riemannian manifold (M, g): : ω ∈ Γ( p T * M ) with ω = 0 .
Note that there exists a non zero p-form ω with ∇ω 2 L 2 ≤ δ ω 2 L 2 for some δ > 0 if and only if λ 1 (∆ C,p ) ≤ δ holds. For arbitrary integers n, p with 2 ≤ p ≤ n/2 and a real number ǫ > 0, considering a small perturbation of S n−p (1) × S p (r n,p ), we can find an n-dimensional closed Riemannian manifold with Ric ≥ (n−p−1)g such that 0 < λ 1 (∆ C,p ) < ǫ holds. Here we defined r n,p := (p − 1)/(n − p − 1). In other words, we do not have the gap theorem for the first eigenvalue of the connection Laplacian λ 1 (∆ C,p ) if we only assume a lower Ricci curvature bound.
Let us state the almost version of the eigenvalue estimate.
This theorem recovers the estimate in Theorem 1.3 when λ 1 (∆ C,p ) = 0. We next state the approximation result to the product space.
In this article we study the structure of the metric space X in this theorem and show that X with some appropriate Borel measure satisfies the RCD * (n − p − 1, p) condition (see Proposition 3.2), which means a synthetic notion of "Ric ≥ n − p − 1 and dim ≤ p with Riemannian structure" (see Definition 2.6). As a consequence, we can show the estimate λ n−p+2 (g) ≥ p(n−p−1)/(p−1)−ǫ under the assumption of Theorem 1.5 (see Theorem 4.1) and the following theorem.
Main Theorem . For given integers n ≥ 5 and 2 ≤ p < n/2 and a positive real number ǫ > 0, there exists δ = δ(n, p, ǫ) > 0 such that if (M, g) is an n-dimensional closed Riemannian manifold with Ric g ≥ (n − p − 1)g, We show the main theorem including the case when λ 1 (∆ C,n−p ) ≤ δ (see Theorem 4.2). By the topological stability theorem due to Cheeger-Colding [13, Theorem A.1.12], we get that M is diffeomorphic to S n−p × S p if ǫ is sufficiently small in our main theorem.
Acknowledgments. I am grateful to Professor Shouhei Honda for helpful discussions. I also thank Professor Dario Trevisan for answering my questions about the regular Lagrangian flow. This work was supported by RIKEN Special Postdoctoral Researcher Program.

Preliminaries
2.1. Basic Notation. We first recall some basic definitions and fix our convention.
Let (M, g) be a closed Riemannian manifold. For any p ≥ 1, we use the normalized L p -norm: and f L ∞ := ess sup x∈M |f (x)| for a measurable function f on M . We also use these notation for tensors. We have f L p ≤ f L 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 minus Laplacian −∆ = −trHess acting on functions. For p = 0, 1, . . . , n, let For metric space (X, d) and k ∈ R ≥0 , let H k denotes the k-dimensional Hausdorff measure. If 0 < H k (X) < ∞, let H k denotes the normalized k-dimensional Hausdorff measure: In this article, for metric spaces (X i , d i ) (i = 1, 2), let d 1 ×d 2 denotes the distance on X 1 × X 2 satisfying 2.2. Metric Measure Spaces. In this article we only consider a compact metric measure space with full support and unit total mass for simplicity of the description because it is enough for our purpose.
Definition 2.1. In this article we say that (X, d, m) is a compact metric measure space if (X, d) is a compact metric space and m is a Borel measure with suppm = X and m(X) = 1.
We introduce some functional analytic tools on a metric measure space. Our main references are [2], [16] and [17] Definition 2.2. Let (X, d, m) be a compact metric measure space.
• (Local Lipschitz Constant) Let LIP(X) denotes the set of the Lipschitz functions on X. For each f ∈ LIP(X) and x ∈ X, we define a local Lipschitz constant Lip(f )(x) by We have that W 1,2 (X) is a Banach space with the norm f W 1,2 = ( f 2 For any f ∈ W 1,2 (X), the minimal relaxed gradient |Df | ∈ L 2 (X) exists and unique. See [2, Definition 4.2, Lemma 4.3]. • (Sobolev-to-Lipschitz Property) We say that (X, d, m) satisfies the Sobolevto-Lipschitz property if any f ∈ W 1,2 (X) with |Df | ≤ 1 m-a.e. in X is a 1-Lipschitz function on X (more precisely, f has a 1-Lipschitz representative). • (Infinitesimally Hilbertian) We say that (X, d, m) is infinitesimally Hilbertian if Ch is a quadratic form. This condition holds if and only if (W 1,2 (X), · W 1,2 ) is a Hilbert space. In this case, we define E : • (Laplacian) If (X, d, m) is infinitesimally Hilbertian, then we define D(∆) := f ∈ W 1,2 (X) : there exists ∆f ∈ L 2 (X) such that we have E(f, g) = − X g∆f dm for any g ∈ W 1,2 (X) .
-For each t > 0, we have that P t f ∈ D(∆) and that in L 2 (X). Moreover, we have the following properties (see [17,Subsection 5 -For each t > 0 and f ∈ L 2 (X), we have See also [17,Theorem 5.1.12]. -For each t > 0, P t : L 2 (X) → L 2 (X) is a linear map satisfying X gP t f dm = X f P t g dm for any f, g ∈ L 2 (X).
-For each s, t > 0, we have P s+t = P s • P t .
-For each f ∈ D(∆) and s > 0, we have that in L 2 (X) and that ∆P s f = P s ∆f .
In particular, we can extend the map P t : L 2 (X) ∩ L p (X) → L 2 (X) ∩ L p (X) to P t : L p (X) → L p (X). We can also show the following properties by the above properties: -For each f ∈ W 1,2 (X), we have Ch(P t f − f ) → 0 as t → 0.
-For each p ∈ [1, ∞) and f ∈ L p (X), we have P t f − f L p → 0 as t → 0.
there exists divV ∈ L 2 (X) such that we have .

2.3.
The RCD * Condition and Some Properties. In this subsection we recall the definition of the RCD * (K, N ) space and its properties.
We say that an infinitesimally Hilbertian metric measure space (X, d, m) satisfies the Bakry-Émery condition BE(K, N ) with K ∈ R and N ∈ Definition 2.4. We say that an infinitesimally Hilbertian metric measure space (X, d, m) satisfies the Bakry-Ledoux condition BL(K, N ) with K ∈ R and N ∈ [1, ∞) if for all u ∈ W 1,2 (X) and t > 0 we have Definition 2.6. We say that a compact infinitesimally Hilbertian metric measure space (X, d, m) satisfies the RCD * (K, N ) condition with K ∈ R and N ∈ [1, ∞) if (X, d, m) satisfies the BE(K, N ) condition and the Sobolev-to-Lipschitz property.
For more general metric measure space, we add the volume growth assumption to Definition 2.6. However, it is automatically satisfied in our situation because we assume that m(X) = 1. Note that Definition 2.6 implies that (X, d) is a geodesic space by [4, Theorem 3.9, Theorem 3.10] and [9, Theorem 2.5.23]. The original definition also implies this property (see [15,Remark 3.8] and [32,Remark 4.6]).
The definition of the RCD * (K, N ) condition is consistent to the smooth case. Under the RCD * (K, N ) condition, we have that Let us make a remark on the heat kernel. Let (X, d, m) be a compact infinitesimally Hilbertian metric measure space satisfying the RCD * (K, N ) condition. Then, E is a strongly local Dirichlet form on (X, m) by [2,Proposition 4.8], [3,Proposition 4.11], and we have holds for any f ∈ L 1 (X). By [24, Theorem 1.2], for any ǫ > 0, there exist constants holds for each x, y ∈ X and t > 0. By this and the Bishop-Gromov inequality [33, Theorem 2.3], we have the following: • For any f ∈ L 1 (X), we have • For any f ∈ C(X), the function For any f ∈ W 1,2 (X) and t > 0, we have P t f ∈ LIP(X) by the BL(K, N ) condition and the Sobolev-to-Lipschitz property, and so P t f ∈ TestF(X). In particular, TestF(X) ⊂ W 1,2 (X) is dense. Note that since we assumed that (X, d, m) is compact and m(X) = 1, we can skip the truncation procedure. We next recall some basic facts about the spectrum of −∆ on a compact infinitesimally Hilbertian metric measure space (X, d, m) satisfying the RCD * (K, N ) condition. By [29, Theorem 1.1], [20, Theorem 1] and [21, Theorem 8.1], the inclusion W 1,2 (X) → L 2 (X) is a compact operator. Thus, the spectrum of −∆ is discrete and positive: as the smooth case. See also the proof of [15,Theorem 4.22]. Let {φ i } ∞ i=0 be the corresponding eigenfunctions. Then, Finally, let us recall the notion of the regular Lagrangian flow, which is a flow for a vector field in a non-smooth setting. Although time dependent vector fields are considered in [6], we only deal with time independent vector fields because it is enough for our purpose. Definition 2.9 ([6]). Let (X, m) be a compact infinitesimally Hilbertian metric measure space (X, d, m) satisfying the RCD * (K, N ) condition and take T > 0. We say that Fl V : [0, T ] × X → X is a regular Laglangian flow for a vector field V ∈ L 2 (T X) if the following properties hold: (i) There exists a constant C > 0 such that is continuous and Note that the Borel measurability in (ii) can be verified under the assumption of the following existence theorem because we construct the flow using the disintegration theorem in [6,Theorem 8.4] and [7,Theorem 7.8].
We use the following form of the result of [6]. See also [7].
Remark 2.11. Let us give some comments about which assertions in [6] correspond to Theorem 2.10. We set A := LIP(X). The concept of the regular Lagrangian flow is closely related to the continuity equation We call d H,d the Hausdorff distance.
The Hausdorff distance defines a metric on the collection of compact subsets of X.
The Gromov-Hausdorff distance defines a metric on the set of isometry classes of compact metric spaces (see [28,Proposition 11.1.3]).
Definition 2.14 (ǫ-Hausdorff approximation map). Let (X, d X ), (Y, d Y ) be metric spaces. We say that a map ψ : X → Y is an ǫ-Hausdorff approximation map for ǫ > 0 if the following two conditions hold.
If there exists an ǫ-Hausdorff approximation map from X to Y , then we can show that d GH (X, Y ) ≤ 3ǫ/2. Conversely, if d GH (X, Y ) < ǫ, then there exists a 2ǫ-Hausdorff approximation map from X to Y . • We say a sequence • Let m be a Borel measure on X. We say that a sequence {(M i , g i , H n )} converges to a metric measure space (X, d, m) in the measured Gromov-Hausdorff topology if holds for any r > 0, x i ∈ M i and x ∈ X with x i GH → x. Note that taking subsequence, such a limit measure exists by [13, Theorem 1.6, Theorem 1.10]. Moreover, (X, d, m) satisfies the RCD * (K, N ) condition by [15,Theorem 3.22].
Suppose that a sequence {(M i , g i , H n )} converges to (X, d, m) in the measured Gromov-Hausdorff topology.
• We say that f i ∈ L 2 (M i ) (i ∈ Z >0 ) converges to f ∈ L 2 (X) strongly at x ∈ X ([23, Definition 3.7]) if we have that ) converges to f ∈ L 2 (X) weakly in L 2 (L 2 boundedness and weakly convergence [23, Definition 3.4, Proposition and for all r > 0, and for all r > 0, y i , z i ∈ X i and y, z ∈ X with y i , the following conditions are mutually equivalent: (i) f i → f strongly at a.e. x ∈ X.
(ii) f i → f strongly in L 2 .
Note that the implication (i) ⇒ then we have the following: (i) f i → f and ∇f i → ∇f strongly in L 2 , (ii) f ∈ D(∆ X ) and ∆f i → ∆f weakly in L 2 .

3.1.
Splitting of the Measure. In this subsection we show that there exists a Borel measure m X on X such that m = H n−p × m X holds under Assumption 3.1 below.
Assumption 3.1. 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 lim i→∞ λ n−p+1 (g i ) = n − p and either lim i→∞ λ 1 (∆ C,p , g i ) = 0 or lim i→∞ λ 1 (∆ C,n−p , g i ) = 0. Let f 1,i , . . . , f n−p+1,i denotes the first (n − p + 1)-th eigenfunctions on (M i , g i ) with f k,i 2 L 2 = 1/(n − p + 1) (k = 1, . . . , n − p + 1). Put Let X be a compact metric space and m be a Borel measure on S n−p (1) × X with unit volume. Suppose that, for each i, there exists a map b i : M i → X such that the map The goal of this section is to prove the following proposition. In this subsection, we show the splitting of the measure m = H n−p × m X . Our approach has been inspired by [18]. We first show the following easy lemma. (iv) For each k, l = 1, . . . , n − p + 1 with k = l, we have that Proof. Clearly, f k is a Lipschitz function. If we get (iii), we have ∆f k = −(n−p)f k ∈ W 1,2 (M ), and so we have f k ∈ TestF(M ).
We first show that {f k,i } strongly converges to f k as i → ∞ at each point z ∈ M . Note that by the gradient estimate for eigenfunctions [28,Theorem 7.3], there exists a constant C > 0 such that f k,i L ∞ + ∇f k,i L ∞ ≤ C holds for all i ∈ Z >0 and k = 1, . . . , n−p+1. Take arbitrary z = (u, x) ∈ S n−p (1)×X and and so lim i→∞ |f k,i (z i ) − f k (z)| = 0 by Proposition 2.18 (iii). Since f k,i and f k are Lipschitz functions whose Lipschitz constants are bounded independently of i, we have that Therefore, we get (i) by Proposition 2.16. We get (ii) by Theorem 2.17 (i). By Theorem 2.17 (ii), we have and so we get (iii). For each k = 1, . . . , n − p + 1, we have that in M for each k, l = 1, . . . , n − p + 1 with k = l. These imply (iv).
Let us apply Theorem 2.10 to vector fields generating rotations in S n−p (1).
with u · v = 0 and T > 0. Then, the vector field for a.e. t ∈ (0, T ) by Lemma 3.3. This implies that for m-a.e. z ∈ M , for any t ∈ [0, T ]. A simple calculation implies that ∇f k , ∇(g • p 2 ) = 0 m-a.e. in M for each g ∈ LIP(X) and k = 1, . . . , n − p + 1 similarly to Lemma 3.11 (iv) below. Therefore, for each g ∈ LIP(X) and m-a.e. z ∈ M , we have for a.e. t ∈ [0, T ], and so for any t ∈ [0, T ]. Let {x j } j∈Z>0 be a countable dense subset of X. Then, by considering g j := d(x j , ·), we get that for m-a.e. z ∈ M , holds for any j ∈ Z >0 and t ∈ [0, T ]. This implies for m-a.e. z ∈ M , p 2 Fl Vuv (t, z) = p 2 (z) for any t ∈ [0, T ]. We have that Fl Vuv This implies the final assertion.
Corollary 3.5. For any T ∈ SO(n − p + 1), the transformation preserves the measure m.
Proof. Modifying on m-negligible subset, we have that Fl Vuv t ∈ SO(n−p+1) for each u, v ∈ S n−p (1) with u · v = 0 and t ∈ [0, 2π]. Conversely, any T ∈ SO(n − p + 1) can be expressed as a composition of several transformations of the form Fl Vuv t . Thus, we get the corollary.
The following proposition is the goal of this subsection. Proof. We immediately have the claim by Corollary 3.5.
Proof. By the volume estimate relative to H n−p on S n−p (1), there exists a constant C > 0 such that max k ∈ Z >0 : there exist x 1 , . . . , x k ∈ S n−p (1) such that B r (x i ) ∩ B r (x j ) = ∅ holds for each i = j ≥ r −(n−p) /C holds for all r > 0. Take r > 0. We can choose k ∈ Z >0 with k ≥ r −(n−p) /C and x 1 , . . . , x k ∈ S n−p (1) such that B r (x i ) ∩ B r (x j ) = ∅ holds for each i = j. By Claim 3.7, we have that µ B (B r (x i )) = µ B (B r (x j )) = µ B (B r (x)) for all i, j = 1, . . . , k and x ∈ S n−p (1). Therefore, we get that µ B (B r (x)) ≤ Cr n−p µ B (S n−p (1)) for all x ∈ S n−p (1).
Take arbitrary subset A ⊂ S n−p (1) with H n−p (A) = 0 and ǫ > 0. Then, by the definition of the Hausdorff measure, there exist a sequence of subsets Choose x j ∈ S j for each j. Then, we have S j ⊂ B 2diamSj (x j ), and so (diamS j ) n−p µ B (S n−p (1)) ≤ Cǫµ B (S n−p (1)).
Letting ǫ → 0, we obtain µ B (A) = 0 and get the claim. By Claim 3.8 and the Radon-Nikodym theorem, we have the representation µ B = ρH n−p , where ρ : S n−p (1) → [0, ∞] is some Borel function. By Claim 3.7, we have that for each T ∈ SO(n − p + 1) ρ • T = ρ H n−p -a.e. in S n−p (1). This implies that ρ is constant H n−p -a.e. in S n−p (1). We have that For each Borel sets A ⊂ S n−p (1) and B ⊂ X, we get that This implies the proposition.

3.2.
Product Metric Measure Spaces and the RCD * Condition. In the previous subsection we showed that there exists a Borel measure m X on X such that m = H n−p × m X holds under Assumption 3.1. In this subsection we show that (X, m X ) satisfies the RCD(n − p − 1, p) condition. More generally, we consider the following assumption.
. Moreover, we assume the following: For each i = 1, 2, let p i : M → X i denotes the projection.
The goal of this subsection is to prove the following proposition: Proposition 3.10. In addition to Assumption 3.9, we assume that (X 1 , d 1 , m 1 ) is an n-dimensional closed Riemannian manifold with the Riemannian distance and m 1 = H n . Then, (X 2 , d 2 , m 2 ) satisfies the RCD * (K, N − n) condition if N − n ≥ 1.
We first show the following easy lemma.
Lemma 3.11. Under Assumption 3.9, we have the following properties: (i) For any f ∈ LIP(M ) and x = (x 1 , x 2 ) ∈ M , we have that and that (ii) For any f ∈ LIP(X i ) (i = 1, 2), we have that (iii) For each i = 1, 2, the map p * i : , and we have that |∇(f • p i )| = |∇f | • p i m-a.e. in M for any f ∈ W 1,2 (X i ) (i = 1, 2).
(v) For any f i ∈ W 1,2 (X i ) (i = 1, 2), we have that Proof. We get (i) and (ii) straightforward by the definition.
We show (iii) for i = 1. Take arbitrary f ∈ W 1,2 (X 1 ). For any sequence {f n } ⊂ LIP(X 1 ) with lim n→∞ f n − f L 2 = 0, we have lim n→∞ f n • p 1 − f • p 1 L 2 = 0, and so We can assume ǫ n < 1 for each n. We have that by (i), and so Since we have 1 2 we can take a sequence {x 2 (n)} ⊂ X 2 such that for each n. Put g n := f n (·, x 2 (n)) ∈ LIP(X 1 ). Then, we have g n − f L 2 → 0 as n → ∞ and This implies |∇(f • p 1 )| = |∇f | • p 1 m-a.e. in M .
Let us prove (iv). We first consider the case f ∈ Lip(X 1 ). Then, we have for m-a.e. (x 1 , x 2 ) ∈ M . By considering −h instead of h, we also get and so for m-a.e. (x 1 , x 2 ) ∈ M . For general f ∈ W 1,2 (X), approximating f by Lipschitz functions, we get (iv). Finally we show (v). We have (v) for each f i ∈ LIP(X i ) (i = 1, 2) by (iv). For general f i ∈ W 1,2 (X i ), approximating f i by Lipschitz functions, we get (v).
We immediately get the following corollary by Lemma 3.11 (iii). For any f ∈ L 2 (X i ), we shall denote f • p i ∈ L 2 (M ) by f briefly if there is no confusion.
Lemma 3.14. Under Assumption 3.9, we have the following properties: for any f ∈ D(∆ Xi ) (i = 1, 2). Thus, we use the same notation ∆ for ∆ M and ∆ Xi (i = 1, 2). For any f ∈ D(∆ Xi ), we shall denote (∆ Xi f ) • p i by ∆f briefly if there is no confusion.
We next show (ii). Take arbitrary f i ∈ D(∆ Xi ) (i = 1, 2). Then, for any φ ∈ LIP(M ), we have Our goal is to show (X 2 , d 2 , m 2 ) satisfies the RCD * (K, N − n) condition under the assumption of Proposition 3.10. However, we can show the following weaker assertion under Assumption 3.9. Proof. We only need to show that (X i , d i , m i ) satisfies the BE(K, N ) condition by Corollary 3.12. For any u i ∈ D(∆ Xi ) with ∆u i ∈ W 1,2 (X i ) and φ i ∈ D(∆ Xi ) ∩ L ∞ (X i ) with φ i ≥ 0 and ∆ Xi φ i ∈ L ∞ (X i ), applying the BE(K, N ) condition for (M, d, m) to u i • p i , φ i • p i ∈ D(∆ M ), we get the BE(K, N ) condition for (X i , d i , m i ).
The following proposition is crucial to show Proposition 3.10. We show the BE(K, N − n) condition with an error term. Proposition 3.16. In addition to Assumption 3.9, we assume that n is an integer with N − n ≥ 1 and that (X 1 , d 1 , m 1 ) is an n-dimensional closed Riemannian manifold with the Riemannian distance and m 1 = H n . Then, for all u ∈ D(∆ X2 ) with ∆u ∈ W 1,2 (X 2 ) and all φ ∈ D(∆ X2 ) ∩ L ∞ (X 2 ) with φ ≥ 0 and ∆φ ∈ L ∞ (X 2 ), we have Proof. Take ψ ∈ C ∞ (R) such that and ψ ≥ 0. Fix p ∈ X 1 . Take sufficiently small ǫ > 0 so that we can take ψ ǫ and f ǫ below as smooth functions. Define ψ ǫ ∈ C ∞ (X 1 ) by for each x 1 ∈ X 1 , and take f ǫ ∈ C ∞ (X 1 ) such that Then, there exists a constant C > 0 such that for all x 1 ∈ B ǫ (p). Note that we can take such a constant independently of ǫ.
Letting ǫ → 0 in Claim 3.17, we get the proposition.
Let us complete the proof of Proposition 3.10. Since we have already showed Corollary 3.12, we only need to check the BL(K, N − n) condition for (X 2 , d 2 , m 2 ). The proof of the following proposition has been inspired by the proof of [25, Theorem 1.2]. Proposition 3.18. In addition to Assumption 3.9, we assume that n is an integer with N − n ≥ 1 and that (X 1 , d 1 , m 1 ) is an n-dimensional closed Riemannian manifold with the Riemannian distance and m 1 = H n . Then, the metric measure space (X 2 , d 2 , m 2 ) satisfies the BL(K, N − n) condition.
Proof. Similarly to the proof of the assertion that the BE(K, N ) condition implies the BL(K, N ) condition ([15, Proposition 4.9]), we have the following claim: Claim 3.19. For any u ∈ D(∆ X2 ) and t > 0, we have Then, for each 0 < s < t, we have ∂ ∂s h(s) Here, we used ∂ ∂s P s φ = ∆P s φ, ∂ ∂s P t−s u = −∆P t−s u in W 1,2 , |∇P t−s u| 2 ≤ e −2K(t−s) P t−s (|∇u| 2 ) (by the BL(K, N ) condition), Proposition 3.16 and the Jensen inequality m-a.e. in X 2 . Combining this and This implies the claim.
By Corollary 3.12 and Proposition 3.18, we get Proposition 3.10. By Proposition 3.6 and Proposition 3.10, we get Proposition 3.2.

Proof of the Main Theorem
In this section we complete the proof of our main theorem.
By the Lichnerowicz estimate for the first eigenvalue of the Laplacian acting on functions for metric measure spaces satisfying the RCD * (n − p − 1, p) condition [15,Theorem 4.22]: we get Theorem 4.1 similarly to Theorem 4.2 below. Thus, we only give the proof of Theorem 4.2.
The following theorem is the main result of this article.
By the Obata Rigidity theorem for metric measure spaces satisfying the RCD * condition [26,Theorem 1.4] with scaling, we have that (X, m X ) is isomorphic to either (S p (r n,p ), H p ) or (S p + (r n,p ), H p ), where r n,p := (p − 1)/(n − p − 1) and S p + (r n,p ) denotes the p-dimensional hemisphere with radius r n,p . In particular, {(M i , g i , H n )} is a non-collapsing sequence. Thus, we get (X, m X ) is isomorphic to (S p (r n,p ), H p ) by [13,Theorem 6.2]. This contradicts to the assumption, and so we get the theorem.