LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS In studying the relation between the curvatures of a Riemannina manifold and its geometry/topology, another very useful method is the so called Bochner technique. 1. Bochner’s formula ¶ Bochner’s formula. We start with Theorem 1.1. Let (M, g) be a Riemannian manifold, and X ∈ Γ ∞(TM). (1) If ∇X is symmetric, i.e. ⟨∇uX, v⟩ = ⟨∇vX, u⟩ for all u, v ∈ TxX, then 1 2 ∆(|X| 2 ) = |∇X| 2 + ⟨X, ∇(divX)⟩ + Rc(X, X). (2) If ∇X is anti-symmetric, i.e. ⟨∇uX, v⟩=−⟨∇vX, u⟩ for all u, v ∈ TxX, then 1 2 ∆(|X| 2 ) = |∇X| 2 − Rc(X, X). Proof. (1) With Riemannian normal coordinates centered at x, we have ∇∂i∂j (x) = 0, ∀i, j. Recall at x one can write (∇2 f)x(∂i , ∂j ) = (∂i∂jf)(x) and (∆f)(x)=X(∂i∂if)(x). It follows that at x, 1 2 ∆(|X| 2 )= 1 2 X i ∂i∂i⟨X, X⟩= X i ∂i⟨∇∂iX, X⟩ ⋆ = X i ∂i⟨∇XX, ∂i⟩ = X i ⟨∇∂i∇XX, ∂i⟩ = X i ⟨∇X∇∂iX, ∂i⟩−⟨∇[X,∂i]X, ∂i⟩−⟨R(X, ∂i)X, ∂i⟩ = X i ⟨∇X∇∂iX, ∂i⟩−⟨∇[X,∂i]X, ∂i⟩+Rm(X, ∂i , X, ∂i) = X i ⟨∇X∇∂iX, ∂i⟩−⟨∇[X,∂i]X, ∂i⟩ +Rc(X, X). 1
2 LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS Note that if we write X = Xi∂i , then Tr(∇X) = X i ⟨∇∂iX, ∂i⟩ ♣= X i ∂i⟨X, ∂i⟩ = X i ∂iX i = divX. it follows X i ⟨∇X∇∂iX, ∂i⟩ = X i X⟨∇∂iX, ∂i⟩ = X(divX) = ⟨X, ∇divX⟩. On the other hand, since [∂i , X] = ∇∂i∂ − ∇X∂i = ∇∂iX, − X i ⟨∇[X,∂i]X, ∂i⟩ = X i ⟨∇∇∂iXX, ∂i⟩ ⋆ = X i ⟨∇∂iX, ∇∂iX⟩ = |∇X| 2 . So the conclusion follows. (2) If ∇X is anti-symmetric, then there will be a negative sign at the right hand side of the two ⋆ =, and we will get 0 after the ♣=. So the conclusion follows. □ ¶ Bochner’s formula for smooth functions. In particular, if u ∈ C ∞(M), then X = ∇u is smooth and ∇X = ∇2u is symmetric. Morever, divX = div∇u = ∆u. It follows Theorem 1.2. For any u ∈ C ∞(M), 1 2 ∆(|∇u| 2 ) = |∇2u| 2 + ⟨∇u, ∇(∆u)⟩ + Rc(∇u, ∇u). Sometimes one need to replace the Hessian term |∇2u| 2 by a simpler one. Note that by Cauchy-Schwartz inequality, for any A = (aij ), |A| 2 = X ij |aij | 2 ≥ X I a 2 ii ≥ 1 m ( X i aii) 2 = 1 m (TrA) 2 . As a result, we get |∇2u| 2 ≥ 1 m (∆u) 2 and thus Corollary 1.3. For any u ∈ C ∞(M), 1 2 ∆(|∇u| 2 ) ≥ 1 m (∆u) 2 + ⟨∇u, ∇(∆u)⟩ + Rc(∇u, ∇u). Remark. In particular, if Ric ≥ (m − 1)k, then for any u ∈ C ∞(M), (*) 1 2 ∆(|∇u| 2 ) ≥ 1 m (∆u) 2 + ⟨∇u, ∇(∆u)⟩ + (m − 1)k|∇u| 2
LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS 3 Conversely, if this inequality holds for any u ∈ C ∞(M), then we must have Ric ≥ (m − 1)k. To see this, given any x0 ∈ M and any X0 ∈ Tx0M, we take u ∈ C ∞(M) so that ∇u(x0) = X0 and (∇2u)x0 = cId. Then by Bochner formula and (*), |∇2u| 2 + Rc(X0, X0) = 1 2 ∆(|∇u| 2 ) − ⟨∇u, ∇(∆u)⟩ ≥ 1 m (∆u) 2 + (m − 1)k|X0| 2 , On the other hand, our choice of u implies |∇2u| 2 = 1 m (∆u) 2 , so we get Ric ≥ (m − 1)k. The condition (*) is used in ✿✿✿✿✿✿✿✿ discrete✿✿✿✿✿✿✿✿✿✿✿ geometric ✿✿✿✿✿✿✿✿✿ analysis (on graphs one can define ∇ and ∆ but not curvature tensor) as a definition of “Ric ≥ (m − 1)k”. ¶ Bochner formula for closed 1-forms. Recall from Lecture 2 that given any smooth 1-form ω ∈ Ω 1 (M), the musical isomorphism produce a smooth vector field X = ♯ω. It is not hard to check |X| = |ω|, |∇X| 2 = |∇ω| 2 . Now suppose ω ∈ Ω 1 (M) is a ✿✿✿✿✿✿ closed 1-form. Then locally ω is exact, i.e. locally of the form ω = du. As a result, X = ♯ω = ∇u, and ∇X is symmetric1 . So we may apply part (1) of Theorem 1.1 to get 1 2 ∆(|ω| 2 ) = |∇ω| 2 + ⟨♯ω, ∇div♯ω⟩ + Rc(♯ω, ♯ω). To proceed let’s do some local computation. Suppose ω = ωidxi . Then ♯ω = X i ωi∂i , div♯ω = X i ∂iωi , ∇div♯ω = X i,j ∂j∂iωi∂j . and thus ⟨♯ω, ∇div♯ω⟩ = X i,j ωj∂j∂iωi . On the other hand, as we have seen in Lecture 4, for any smooth function f, div(f ♯ω) = fdiv♯ω + ⟨∇f, ♯ω⟩. Now we assume M✿✿✿✿✿ is ✿✿✿✿✿✿✿✿✿ compact. After integration we get 0 = Z M div(f ♯ω)dVg = Z M (fdiv♯ω + ⟨∇f, ♯ω⟩)dVg = Z M (fdiv♯ω + ⟨df, ω⟩)dVg. It follows that ⟨df, ω⟩L2 = Z M ⟨df, ω⟩dVg = Z M f(−div♯ω)dVg = ⟨f, −div♯ω⟩L2 . So if we define δω = −div♯ω. Then δ : Ω1 (M) → C ∞(M) is the L 2 -dual of d, ⟨df, ω⟩L2 = ⟨f, δω⟩L2 , ∀f ∈ C ∞(M), ω ∈ Ω 1 (M). 1 In fact one can show that ∇X is symmetric if and only if ω = ♭X is closed
4 LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS For any closed 1-form ω we define ∆ω = dδω. Then locally ∆ω = d(− X i ∂iωi) = − X i,j ∂j∂iωi and thus ⟨ω, ∆ω⟩ = − X i,j ωj∂j∂iωi = −⟨♯ω, ∇div♯ω⟩. So we end with Theorem 1.4 (Bochner’s formula for closed 1-form). Let (M, g) be a compact Riemannian manifold, then for any closed 1-form ω ∈ Ω 1 (M), 1 2 ∆(|ω| 2 ) = |∇ω| 2 − ⟨ω, ∆ω⟩ + Rc(♯ω, ♯ω). ¶ Harmonic k-form. More generally, one can define δ : Ωk (M) → Ω k−1 (M) so that ⟨ω, dη⟩L2 = ⟨δω, η⟩L2 , ∀ω ∈ Ω k (M), η ∈ Ω k−1 (M), and define the Hodge Laplacian on all smooth k-forms to be ∆ := dδ + δd : Ωk (M) → Ω k (M). One can check that when k = 0 this definition coincides with the Laplace-Beltrami operator ∆ on smooth functions (and thus differed with ∆ = tr∇2 by a ✿✿✿✿✿✿✿✿ negative sign). A differential form ω ∈ Ω k (M) is called a harmonic k-form if ∆ω = 0. In view of the fact ⟨ω, ∆ω⟩L2 = ⟨ω, dδω⟩L2 + ⟨ω, δdω⟩L2 = ⟨dω, dω⟩L2 + ⟨δω, δω⟩L2 and the definition of ∆, we have Proposition 1.5. ω ∈ Ω k (M) is harmonic if and only if dω = 0 and δω = 0. Now suppose ω is a harmonic 1-form on compact Riemannian manifold (M, g). Then ω is closed, and thus by Theorem 1.4, 0 = Z M 1 2 ∆(|ω| 2 ) = Z M |∇ω| 2 + Z M Rc(♯ω, ♯ω). So we get Theorem 1.6 (Bochner). Let (M, g) be a compact Riemannian manifold, then (1) Suppose Ric ≥ 0. If ∆ω = 0, then ∇ω = 0. (2) Suppose Ric ≥ 0, and Ric > 0 at one point. If ∆ω = 0, then ω = 0. According to the famous Hodge theory, the space of harmonic k-forms is isomorphic to the de Rham cohomology group Hk dR(M). So we conclude Corollary 1.7. Let (M, g) be a closed oriented Riemannian manifold, Ric ≥ 0, and Ric > 0 at one point. Then b1(M) = 0
LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS 5 Remarks. There is a Bochner-Weitzenb¨ock formula that generalize the Bochner formula above to k-forms using which one can prove: If (M, g) is a closed Riemannian manifold with ✿✿✿✿✿✿✿✿✿✿✿✿ nonnegative✿✿✿✿✿✿✿✿✿✿✿ curvature ✿✿✿✿✿✿✿✿✿✿ operator, then all harmonic forms of order 1 ≤ k ≤ m − 1 on M are parallel. ¶ Bochner formula for Killing forms. Now let’s turn to part (2) in Theorem 1.1. As we have seen in PSet 1, ∇X is anti-symmetric if and only if X is a Killing field on (M, g). So we get Corollary 1.8. For any Killing vector field on M, 1 2 ∆(|X| 2 ) = |∇X| 2 − Rc(X, X). As a result, Theorem 1.9 (Bochner, 1946). Any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Since the space of all Killing vector fields on (M, g) is the Lie algebra of the isometry group Iso(M, g)[which is a Lie group], and the isometry group of any compact Riemannian manifold is compact, we conclude that the isometry group of any compact Riemannian manifold with negative Ricci curvature must be a finite group. 2. Cheeger-Gromoll splitting theorem ¶ Cheeger-Gromoll splitting theorem. Let (M, g) be a complete non-compact connected Riemannian manifold. Recall that a ✿✿✿✿ line in M is a normal geodesic γ : R → M so that d(γ(a), γ(b)) = |a − b|, ∀a, b ∈ R. Unlike the case of rays, it is possible that there is no ray in a complete non-compact Riemannian manifold. For example, a cylinder R × S 1 admits many lines, while the paraboloid z = x 2 + y 2 admits no line at all. As another application of Bochner formula, we prove the following structure theorem for Riemannian manifolds with positive Ricci curvature that admit lines: Theorem 2.1 (Cheeger-Gromoll, 1971). Let (M, g) be a complete non-compact Riemannian manifold with Ric ≥ 0. Suppose there exists a line in M. Then (M, g) is isometric to R × N, where N is an (m − 1)-dimensional complete Riemannian manifold with Ric ≥ 0. The idea is to construct a function on M which behaves like the function f(x, r) = r on N × R, so that the level sets of f gives the desired component N. So what is the speciality of the function f(x, r) = r? It is smooth, with gradient ∇f = ∂r which has length 1, and has Hessian ∇2 f = 0 (so that ∇f is parallel). It is in the proof of “∇2 f = 0” that we need Bochner’s formula
6 LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS To apply Bochner’s formula one need the function to be smooth. However, the construction below uses the distance, and thus the function is only Lipschitz. To solve this problem we will apply theories on PDEs in the barrier sense. More precisely, we need the Hopf-Calabi strong maximum principle, Theorem 2.2 (Hopf-Calabi strong maximum principle). Let Ω ⊂ M be a connected open set. Suppose ∆f ≤ 0 in M in the barrier sense, and f attains an interior minimum, then f is constant on Ω. We also need the well-known Weyl lemma to increase regularity: Theorem 2.3 (Weyl Lemma). If ∆f = 0 in the barrier sense, then f is smooth. ¶ Busemann function. The function that we need is the so-called Busemann function (introduced by Busemann in 1955), defined as follows. Since (M, g) is complete and non-compact, for any ray γ : [0, +∞) → M, let b t γ : M → R, bt γ (x) = t − d(x, γ(t)). By triangle inequality, it is easy to see • b t γ (x) ≤ d(γ(0), x), • for any t < s, one has b s γ (x) − b t γ (x) = (s − t) + d(x, γ(t)) − d(x, γ(s)) ≥ 0, • |b t γ (x) − b t γ (y)| ≤ d(x, y). As a result, the limit bγ(x) := lim t→+∞ (t − d(x, γ(t))) is well-defined and is Lipschitz with Lipschitz constant 1. We call the function bγ : M → R the Busemann function associated with γ. By Laplacian comparison theorem, ✿✿✿✿✿✿✿✿ formally we have ∆bγ(x) ≥ − lim t→+∞ m − 1 d(x, γ(t)) = 0. This can be proved rigorously by constructing a lower barrier. Let’s admit this: Proposition 2.4. Let bγ be the Busemann function associated with a ray γ, then ∆bγ ≥ 0 in the barrie sense. Now let l : (−∞,∞) → M be a line in M, and let γ+, γ− : [0, ∞) → M be the two rays in l defined by γ+(t) = l(t) and γ−(t) = l(−t). Then by Proposition 2.4, ∆(bγ+ (x) + bγ− (x)) ≥ 0. On the other hand, by definition we have 2t = d(γ−(t), γ+(t)) ≤ d(γ−(t), x) + d(γ+(t), x), which implies bγ+ (x) + bγ− (x) ≤ 0
LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS 7 Note that on the line l, if we denote x = l(s), then bγ+ (x) + bγ− (x) = limt→∞ (2t − d(x, l(t)) − d(x, l(−t))) = limt→∞ (2t − t − s − t + s) = 0. So we conclude that bγ+ + bγ− is a subharmonic function that achieves its maximum at an interior point. By Theorem 2.2, bγ+ (x) + bγ− (x) = 0. In other words, bγ+ (x) = −bγ− (x), and thus ∆bγ+ (x) = −∆bγ− (x) ≤ 0. So we arrive at ∆bγ+ (x) = −∆bγ− (x) = 0. By Theorem 2.3, bγ+, bγ− ∈ C ∞(M). Moreover, since bγ+ has Lipschitz constant 1, |∇bγ+ | ≤ 1. Also note that by definition, on the line l we have bγ+(l(s)) = s and thus |∇bγ+ | = 1. ¶ The proof of Cheeger-Gromoll splitting theorem. Now we apply Bochner formula to the function bγ+ , to get 1 2 ∆(|∇bγ+| 2 ) = |∇2 bγ+| 2 + Ric(∇bγ+ , ∇bγ+ ) ≥ 0. So again, |∇bγ+ | 2 is a sub-harmonic function that achieves its interior maximum. So by the Hopf strong maximum principle again, |∇bγ+| 2 = 1. It follows that |∇bγ+| = 1 and in particular, ∇bγ+ is a complete vector field. Moreover, it follows that ∆(|∇bγ+ |) = 0, and thus |∇2 bγ+ | 2 = 0, i.e. ∇2 bγ+ = 0. Finally we construct the splitting. Let Mt = b −1 γ+ (t). Since |∇bγ+| = 1, any t ∈ R is a regular value of bγ+ . So Mt is a smooth submanifold of M of dimension m − 1. Denote N = M0. Let φs : M → M be the flow of the vector field ∇bγ+ . Then φs is a diffeomorphism. Moreover, for any s ∈ R and any x ∈ N = M0, we have φs(x) ∈ Ms. So we get a smooth map Φ : R × N → M, Φ(s, p) := φs(p) which is bijective, and whose inverse Φ −1 : M → R → N, x 7→ (bγ+(x) , φ−bγ+(x) (x)) is smooth. So Φ is a diffeomorphism
8 LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS It remains to prove that Φ is an isometry. Note that if we let γp(s) = φs(p) be the integral curve passing p, then γ˙ p = ∇bγ+ =⇒ ∇γ˙ p = ∇2 bγ+ = 0 which implies that γp is the geodesic γp(s) = expp (sXp), where Xp = ∇bγ+ (p). As a result, we have • Φ is a radial isometry: we have |∂s| = 1 and |dΦ(s,p)(∂s)| = |γ˙ p| = |∇bγ+ | = 1. • Φ maps “vectors orthogonal to radial direction ∂s” to “vectors orthogonal to radial direction dΦ(s,p)(∂s)”: For any X0 ∈ TpN = TpM0, we have (dΦ)(s,p)(0, X0) = (dφs)p(X0) ∈ Tφs(p)Ms ⊥ ∇bγ+ (φs(p)) = dΦ(s,p)(∂s). • Φ preserves the length (and thus the inner product by polarization) of all vectors orthogonal to ∂s: For any X0 ∈ Tp, we may extend X0 to a local coordinate vector field Xe0 on T N such that [∂s, Xe0] = 0. Then ∇γ˙p(s)((dφs)p(X0)) = ∇dφs(Xe0) (∇bγ+ ) − [(∇bγ+ )(φs(p)),(dφs)pXe0]. The first term vanishes since ∇(∇bγ+ ) = 0, while the second term vanishes since it equals dφs([∂s, Xe0]) = 0. So we conclude that (dφs)p(X0) is parallel along γp(s), and thus |dΦ(s,p)(0, Xp)| = |dφs(X0)| = |X0|. So we conclude that (M, g) is isometric to R × N. Finally since (M, g) has nonnegative Ricci curvature, and N is a Riemannian submanifold, and K(∂s, X0) = 0, we conclude that N has non-negative Ricci curvature