The Bochner Technique and Weighted Curvatures

In this note we study the Bochner formula on smooth metric measure spaces. We introduce weighted curvature conditions that imply vanishing of all Betti numbers.


Introduction
Let (M, g) be an oriented Riemannian manifold, let vol g denote its volume form and let f be a smooth function on M. The triple (M, g, e −f vol g ) is called a smooth metric measure space. Based on considerations from diffusion processes, Bakry-Émery [BE85] introduced the tensor Ric f = Ric + Hess f as a weighted Ricci curvature for a geometric measure space. In fact, this tensor appeared earlier in work of Lichnerowicz [Lic70]. Volume comparison theorems for smooth metric measure spaces with Ric f bounded from below have been established by Qian [Qia97], Lott [Lot03], Bakry-Qian [BQ05] and Wei-Wylie [WW09].
In this note we study the Bochner technique on smooth metric measure spaces. The distortion of the volume element introduces a diffusion term to the Bochner formula: where Ric is the Bochner operator on p-forms. Lott [Lot03] proved that if Ric f ≥ 0, then all ∆ f -harmonic 1-forms are parallel and, for compact manifolds, H 1 (M; R) is isomorphic to the space of all parallel 1-forms ω which satisfy ∇e −f , ω = 0. Moreover, if Ric f > 0, then all ∆ f -harmonic 1-forms vanish.
We introduce new weighted curvature conditions that imply rigidity and vanishing results for ∆ f -harmonic p-forms for p ≥ 1. We can restrict to p-forms ω for 1 ≤ p ≤ ⌊ n 2 ⌋ since ω is parallel if and only if * ω is parallel, where * denotes the Hodge star.
By convention, we will refer to the eigenvalues of the curvature operator simply as the eigenvalues of the associated curvature tensor.
Let ω be a ∆ f -harmonic p-form with |ω| ∈ L 2 (M, e −f vol g ) for 1 ≤ p < n 2 . If the eigenvalues of the weighted curvature tensor Rm +h g satisfy then ω is parallel. If the inequality is strict, then ω vanishes.
In particular, if M is compact, then H p (M) = {ω ∈ Ω p (M) | ∇ω = 0 and i ∇f ω = 0} and in case the inequality is strict, the Betti numbers b p (M) and b n−p (M) vanish for 1 ≤ p < n 2 .
For p = 1 the Ricci curvature of the modified curvature tensor is the Bakry-Émery Ricci tensor, and the assumption in the theorem implies that it is nonnegative. In this sense the theorem is a generalization of Lott's [Lot03] results for 1-forms.
A stronger curvature assumption also allows control in the middle dimension. Recall that a curvature tensor is l-nonnegative (positive) if the sum of its lowest l eigenvalues is nonnegative (positive).
Let C(V ) denote the vector space of (0, 4)-tensors with T (X, Y, Z, W ) = −T (Y, X, Z, W ) = T (Z, W, X, Y ). If T also satisfies the algebraic Bianchi identity, then T is called algebraic curvature tensor, T ∈ C B (V ).
The Kulkarni-Nomizu product of S 1 , S 2 ∈ Sym 2 (V ) is given by With this convention the algebraic curvature tensor I = 1 2 g g corresponds to the curvature tensor of the unit sphere.
Recall that the decomposition of C(V ) into O(n)-irreducible components is given by where R ic = S 2 0 (V ) g is the subspace of algebraic curvature tensors of trace-free Ricci type, S 2 0 (V ) = h ∈ Sym 2 (V ) | tr(h) = 0 , and W denotes the subspace of Weyl tensors. Explicitly, every algebraic curvature tensor decomposes as Rm = scal 2(n − 1)n g g + 1 n − 2R ic g + W.
1.2. Lichnerowicz Laplacians on smooth metric measure spaces. Let (M, g, f ) be a smooth metric measure space. The formal adjoints of the exterior and covariant derivative with respect to the measure e −f vol g are given by More generally, for a vector field U on M, we will consider The associated generalized Lichnerowicz Laplacian on (0, k)-tensors is given by where the curvature term is given by To emphasize that the curvature term is calculated with respect to the curvature tensor Rm, we will also write Ric Rm (T ) for Ric(T ).
Recall that for an endomorphism L of V and a (0, k)-tensor T we have In particular, the Ricci identity implies that the definition of the curvature term in the Lichnerowicz Laplacian naturally carries over to algebraic curvature tensors.
Proof. (a) The case U = 0 recovers the well-known Bochner formula. The generalized Hodge Laplacian satisfies In addition to the classical Lichnerowicz Laplacian we have on the right hand side and thus all diffusion terms balance out.
(b) As in (a), it suffices to consider all terms that depend on U and show that This is a straightforward calculation: Remark 1.2. The curvature tensor Rm of a Riemannian manifold satisfies A straightforward computation based on the second Bianchi identity shows that all terms that involve U cancel.
The Bochner technique with diffusion relies on the following basic observations. Firstly, the maximum principle implies: If |T | has a maximum, then T is parallel. If U = ∇f, then we can use integration to conclude: Lemma 1.5. Let (M, g, f ) be a smooth metric measure space with M e −f vol g < ∞. If T is a (0, k)-tensor with |T | ∈ L 2 (M, e −f vol g ) and then T is parallel.

Weighted Lichnerowicz Laplacians
The idea of this section is to define a weighted curvature tensor Rm so that for a given symmetric tensor S the curvature term of the Lichnerowicz Laplacian satisfies This will be achieved by adding a weight to the Ricci tensor of Rm, leaving the Weyl curvature unchanged. The specific weight will depend on the irreducible components of the tensors of type T , e.g. it is different for forms and symmetric tensors.
Let T be a (0, k)-tensor. For τ ij ∈ S k let T • τ ij denote the transposition of the i-th and j-th entries of T and for h ∈ Sym 2 (V ) let c ij (h ⊗ T ) denote the contraction of h with the i-th and j-th entries of T .
Proposition 2.2. Let (V, g) be an n-dimensional Euclidean vector space and h ∈ Sym 2 (V ).
This completes the proof. It is worth noting that there are trace-free symmetric (0, 2)-tensors h 1 , h 2 such that the curvature tensor h 1 h 2 is Weyl.
The main Theorem follows as in proposition 2.4 below by using lemma 1.5 instead of lemma 1.3. The description of the de Rham cohomology groups follows from remark 1.4. Proposition 2.4. Let (M, g) be a Riemannian manifold and let U be a vector field on M. Set S = ∇U and for 1 ≤ p < n 2 set Suppose that the eigenvalues of the weighted curvature tensor Rm +h g satisfy n−p α=1 λ α (Rm +h g) ≥ 0 and let ω be a U-harmonic p-form for 1 ≤ p < n 2 . If |ω| achieves a maximum, then ω is parallel. If in addition the inequality is strict, then ω vanishes.
If the inequality is strict, then the same argument shows that Ric Rm +h g (ω) > 0 unless ω = 0.
The above approach only works for p = n 2 if S is a multiple of the identity. However, we have Proposition 2.5. Let (M, g) be an n-dimensional Riemannian manifold and let U be a vector field on M. Set S = ∇U and fix 1 ≤ p ≤ ⌊ n 2 ⌋. Let µ 1 ≤ . . . ≤ µ n denote the eigenvalues of S. Suppose that the weighted curvature tensor If ω is a U-harmonic p-form ω such that |ω| has a maximum, then ω is parallel. If in addition the weighted curvature tensor is (n − p)-positive, then ω vanishes.
Proof. Calculating with respect to an orthonormal eigenbasis for S it follows that Let {λ α } denote the eigenvalues of (the curvature operator associated to) Rm and let {Ξ α } be an orthonormal eigenbasis. It follows from [PW19, Proposition 1.6] that g(Ric Rm (ω), ω) − g(Sω, ω) ≥ The proof can now be completed as in proposition 2.4.
This principle can also be applied to (0, 2)-tensors.
If T is U-harmonic and |T | has a maximum, then T is parallel. If in addition the inequality is strict, then T vanishes.
As in [PW19, Lemma 2.1 and proposition 2.9] we conclude that Ric Rm +h g (T ) ≥ 0. When the inequality is strict, the argument shows moreover Ric Rm +h g (T ) > 0 unless T = 0. This uses again that T is trace-less. An application of lemma 1.5 as before implies the claim.