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 the vanishing of all Betti numbers.
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| description | In this note, we study the Bochner formula on smooth metric measure spaces. We introduce weighted curvature conditions that imply the vanishing of all Betti numbers.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 064, 10 pages
The Bochner Technique and Weighted Curvatures
Peter PETERSEN and Matthias WINK
Department of Mathematics, University of California,
520 Portola Plaza, Los Angeles, CA, 90095, USA
E-mail: petersen@math.ucla.edu, wink@math.ucla.edu
Received May 22, 2020, in final form June 29, 2020; Published online July 09, 2020
https://doi.org/10.3842/SIGMA.2020.064
Abstract. 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.
Key words: Bochner technique; smooth metric measure spaces; Hodge theory
2020 Mathematics Subject Classification: 53B20; 53C20; 53C21; 53C23; 58A14
1 Introduction
Let (M, g) be an oriented Riemannian manifold, let volg denote its volume form and let f be
a smooth function on M . The triple
(
M, g, e−f volg
)
is called a smooth metric measure space.
Based on considerations from diffusion processes, Bakry–Émery [1] introduced the tensor
Ricf = Ric + Hess f
as a weighted Ricci curvature for a geometric measure space. In fact, this tensor appeared earlier
in work of Lichnerowicz [3]. Volume comparison theorems for smooth metric measure spaces
with Ricf bounded from below have been established by Qian [7], Lott [4], Bakry–Qian [2] and
Wei–Wylie [8].
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
∆fω = (dd∗f + d∗fd)ω = ∇∗f∇ω + Ric(ω)− (Hess f)ω,
where Ric is the Bochner operator on p-forms. Lott [4] proved that if Ricf ≥ 0, then all
∆f -harmonic 1-forms are parallel and, for compact manifolds, H1(M ;R) is isomorphic to the
space of all parallel 1-forms ω which satisfy
〈
∇e−f , ω
〉
= 0. Moreover, if Ricf > 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 eigen-
values of the associated curvature tensor.
Theorem. Let
(
Mn, g, e−f volg
)
be a smooth metric measure space. For 1 ≤ p < n
2 set
h =
1
n− 2p
Hess f − ∆f
2(n− p)(n− 2p)
g.
Let ω be a ∆f -harmonic p-form with |ω| ∈ L2
(
M, e−f volg
)
for 1 ≤ p < n
2 . Let λ1 ≤ · · · ≤ λ(n2)
denote the eigenvalues of the weighted curvature tensor Rm +h? g.
This paper is a contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov
on his 75th Birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Gromov.html
mailto:petersen@math.ucla.edu
mailto:wink@math.ucla.edu
https://doi.org/10.3842/SIGMA.2020.064
https://www.emis.de/journals/SIGMA/Gromov.html
2 P. Petersen and M. Wink
If λ1 + · · ·+ λn−p ≥ 0, then ω is parallel. If in addition M is compact, then Hp(M) =
{
ω ∈
Ωp(M) | ∇ω = 0 and i∇fω = 0
}
.
If λ1 + · · ·+λn−p > 0, then ω vanishes. If in addition M is compact, then the Betti numbers
bp(M) and bn−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 [4] results for 1-forms.
A stronger curvature assumption also allows control in the middle dimension p = n
2 . Re-
call that a curvature tensor is l-nonnegative (positive) if the sum of its lowest l eigenvalues is
nonnegative (positive).
Proposition. Let
(
Mn, g, e−f volg
)
be a smooth metric measure space. Let µ1 ≤ · · · ≤ µn
denote the eigenvalues of Hess f and let 1 ≤ p ≤
⌊
n
2
⌋
.
Let ω be a ∆f -harmonic p-form with |ω| ∈ L2
(
M, e−f volg
)
. If the weighted curvature tensor
Rm +
p∑
i=1
µi
2p(n− p)
g ? g
is (n− p)-nonnegative, then ω is parallel. If it is (n− p)-positive, then ω vanishes.
In particular, if M is compact, then Hp(M) =
{
ω ∈ Ωp(M) | ∇ω = 0 and i∇fω = 0
}
and
in case the weighted curvature tensor is (n− p)-positive, the Betti numbers bp(M) and bn−p(M)
vanish.
The notation in this paper builds up on the presentation in [5, Chapter 9] and [6].
2 Preliminaries
2.1 Algebraic curvature tensors
For an n-dimensional Euclidean vector space (V, g) let T (0,k)(V ) denote the vector space of
(0, k)-tensors and Sym2(V ) the vector space of symmetric (0, 2)-tensors on V .
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 ∈ CB(V ).
The Kulkarni–Nomizu product of S1, S2 ∈ Sym2(V ) is given by
(S1 ? S2)(X,Y, Z,W ) = S1(X,Z)S2(Y,W )− S1(X,W )S2(Y,Z)
+ S1(Y,W )S2(X,Z)− S1(Y, Z)S2(X,W ).
With this convention the algebraic curvature tensor I = 1
2g ? 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
C(V ) = 〈I〉 ⊕ 〈R̊ic〉 ⊕ 〈W 〉 ⊕ Λ4V,
where 〈R̊ic〉 = S2
0(V ) ? g is the subspace of algebraic curvature tensors of trace-free Ricci type,
S2
0(V ) =
{
h ∈ Sym2(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− 2
R̊ic ? g +W.
The Bochner Technique and Weighted Curvatures 3
2.2 Lichnerowicz Laplacians on smooth metric measure spaces
Let (M, g, f) be a smooth metric measure space. The formal adjoints of the exterior and covari-
ant derivative with respect to the measure e−f volg are given by
d∗f = d∗ + i∇f and ∇∗f = ∇∗ + i∇f .
More generally, for a vector field U on M , we will consider
d∗U = d∗ + iU and ∇∗U = ∇∗ + iU .
The associated generalized Lichnerowicz Laplacian on (0, k)-tensors is given by
∆UT = ∇∗U∇T + Ric(T )− (∇U)T,
where the curvature term is given by
Ric(T )(X1, . . . , Xk) =
k∑
i=1
n∑
j=1
(R(Xi, ej)T )(X1, . . . , ej , . . . , Xk).
A tensor T is called U -harmonic if ∆UT = 0.
To emphasize that the curvature term is calculated with respect to the curvature tensor Rm,
we will also write RicRm(T ) for Ric(T ).
Recall that for an endomorphism L of V and a (0, k)-tensor T we have
(LT )(X1, . . . , Xk) = −
k∑
i=1
T (X1, . . . , L(Xi), . . . , Xk).
In particular, the Ricci identity implies that the definition of the curvature term in the Lich-
nerowicz Laplacian naturally carries over to algebraic curvature tensors.
Proposition 2.1. Let (M, g) be a Riemannian manifold and U a vector field on M . For
a (0, k)-tensor T on M set RicU (T ) = Ric(T )− (∇U)T .
(a) Every p-form satisfies
(dd∗U + d∗Ud)ω = ∇∗U∇ω + RicU (ω).
(b) Every symmetric (0, 2)-tensor satisfies
(∇X∇∗UT )(X) +
(
∇∗Ud∇T
)
(X,X) = (∇∗U∇T )(X,X) +
1
2
(RicU T )(X,X),
where d∇T (Z,X, Y ) = (∇XT )(Y,Z)− (∇Y T )(X,Z).
Proof. (a) The case U = 0 recovers the well-known Bochner formula. The generalized Hodge
Laplacian satisfies
dd∗U + d∗Ud = dd∗ + d∗d+ diU + iUd = ∆ + LU .
In addition to the classical Lichnerowicz Laplacian we have on the right hand side
∇U − (∇U) = LU
and thus all diffusion terms balance out.
4 P. Petersen and M. Wink
(b) As in (a), it suffices to consider all terms that depend on U and show that
(∇XiUh)(X) +
(
iUd
∇h
)
(X,X) = (∇Uh)(X,X)− 1
2
((∇U)h)(X,X).
This is a straightforward calculation
(∇XiUh)(X) +
(
iUd
∇h
)
(X,X)
= (∇Xh)(U,X) + h(∇XU,X) + (∇Uh)(X,X)− (∇Xh)(U,X)
= (∇Uh)(X,X) + h(∇XU,X)
= (∇Uh)(X,X)− 1
2
((∇U)h)(X,X). �
Remark 2.2. The curvature tensor Rm of a Riemannian manifold satisfies
∇∗U∇Rm +
1
2
RicU (Rm) =
1
2
(∇X∇∗U Rm)(Y,Z,W )− 1
2
(∇Y∇∗U Rm)(X,Z,W )
+
1
2
(∇Z∇∗U Rm)(W,X, Y )− 1
2
(∇W∇∗U Rm)(Z,X, Y ).
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:
Lemma 2.3. Let (M, g) be a Riemannian manifold, U a vector field on M . Let T be a tensor
such that
g(∇∗U∇T, T ) ≤ 0.
If |T | has a maximum, then T is parallel.
Remark 2.4. Note that a p-form ω satisfies (dd∗U + d∗Ud)ω = 0 if and only if dω = 0 and
d∗Uω = 0.
As in [4], if M is compact and oriented, standard elliptic theory implies that
Hp(M) =
{
ω ∈ Ωp(M) | dω = 0 and d∗Uω = 0
}
.
Suppose that RicU ≥ 0 on p-forms. It follows that a p-form ω is U -harmonic if and only if ω is
parallel and iUω = 0. Thus,
Hp(M) =
{
ω ∈ Ωp(M) | ∇ω = 0 and iUω = 0
}
.
If U = ∇f , then we can use integration to conclude:
Lemma 2.5. Let (M, g, f) be a smooth metric measure space with
∫
M e−f volg < ∞. If T is
a (0, k)-tensor with |T | ∈ L2
(
M, e−f volg
)
and
g(∇∗f∇T, T ) ≤ 0,
then T is parallel.
The Bochner Technique and Weighted Curvatures 5
3 Weighted Lichnerowicz Laplacians
The idea of this section is to define a weighted curvature tensor R̃m so that for a given symmetric
tensor S the curvature term of the Lichnerowicz Laplacian satisfies
g(RicRm(T )− (S)T, T ) = g
(
Ric
R̃m
(T ), T
)
.
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 τij ∈ Sk let T ◦ τij denote the transposition of the i-th and j-th
entries of T and for h ∈ Sym2(V ) let cij(h ⊗ T ) denote the contraction of h with the i-th and
j-th entries of T .
Proposition 3.1. For h ∈ Sym2(V ) let H : V → V denote the associated symmetric operator.
If T ∈ T (0,k)(V ), then
Rich?g(T )(X1, . . . , Xk) = 2
∑
i 6=j
(T ◦ τij)(X1, . . . ,H(Xi), . . . , Xk)
−
∑
i 6=j
g(Xi, Xj)cij(h⊗ T )(X1, . . . , X̂i, . . . , X̂j , . . . , Xk)
−
∑
i 6=j
h(Xi, Xj)cij(g ⊗ T )(X1, . . . , X̂i, . . . , X̂j , . . . , Xk)
− (n− 2)(HT )(X1, . . . , Xk) + k · tr(h)T (X1, . . . , Xk).
Proof. The algebraic curvature tensor R = h? g satisfies
R(X,Y, Z,W ) = g(H(X), Z)g(Y,W )− g(Y,Z)g(H(X),W )
+ g(X,Z)g(H(Y ),W )− g(H(Y ), Z)g(X,W )
and hence
R(X,Y )Z = (H(X) ∧ Y +X ∧H(Y ))Z
is the corresponding (1, 3)-tensor. It follows that
Rich?g(T )(X1, . . . , Xk) =
k∑
i=1
n∑
a=1
(R(Xi, ea)T )(X1, . . . , ea, . . . , Xk)
=
k∑
i=1
n∑
a=1
((H(Xi) ∧ ea)T )(X1, . . . , ea, . . . , Xk)
+
k∑
i=1
n∑
a=1
((Xi ∧H(ea))T )(X1, . . . , ea, . . . , Xk).
It is straightforward to calculate
k∑
i=1
n∑
a=1
((Xi ∧H(ea))T )(X1, . . . , ea, . . . , Xk)
=
∑
i 6=j
n∑
a=1
T (X1, . . . , (H(ea) ∧Xi)Xj , . . . , ea, . . . , Xk)
6 P. Petersen and M. Wink
+
k∑
i=1
n∑
a=1
T (X1, . . . , (H(ea) ∧Xi)ea, . . . , Xk)
=
∑
i 6=j
n∑
a=1
T (X1, . . . , g(H(ea), Xj)Xi − g(Xi, Xj)H(ea), . . . , ea, . . . , Xk)
+
k∑
i=1
n∑
a=1
T (X1, . . . , g(H(ea), ea)Xi − g(ea, Xi)H(ea), . . . , Xk)
=
∑
i 6=j
n∑
a=1
T (X1, . . . , g(ea, H(Xj))Xi, . . . , ea, . . . , Xk)
−
∑
i 6=j
n∑
a=1
g(Xi, Xj)T (X1, . . . ,H(ea), . . . , ea, . . . , Xk)
+
k∑
i=1
n∑
a=1
h(ea, ea)T (X1, . . . , Xk)−
k∑
i=1
n∑
a=1
T (X1, . . . ,H(g(ea, Xi)ea), . . . , Xk)
=
∑
i 6=j
T (X1, . . . , Xi, . . . ,H(Xj), . . . , Xk) [here Xi is in the j-th position]
−
∑
i 6=j
n∑
a,b=1
g(Xi, Xj)h(ea, eb)T (X1, . . . , eb, . . . , ea, . . . , Xk) + k · tr(h)T (X1, . . . , Xk)
−
k∑
i=1
T (X1, . . . ,H(Xi), . . . , Xk)
=
∑
i 6=j
(T ◦ τij)(X1, . . . ,H(Xj), . . . , Xi, . . . , Xk) [here H(Xj) is in the j-th position]
−
∑
i 6=j
g(Xi, Xj)cij(h⊗ T )(X1, . . . , X̂i, . . . , X̂j , . . . , Xk)
+ k · tr(h)T (X1, . . . , Xk) + (HT )(X1, . . . , Xk).
Similarly one computes
k∑
i=1
n∑
a=1
((H(Xi) ∧ ea)T )(X1, . . . , ea, . . . , Xk)
=
∑
i 6=j
(T ◦ τij)(X1, . . . , Xj , . . . ,H(Xi), . . . , Xk) [here Xj is in the j-th position]
−
∑
i 6=j
h(Xi, Xj)cij(g ⊗ T )
(
X1, . . ., X̂i, . . ., X̂j , . . ., Xk
)
− (n− 1)(HT )(X1, . . ., Xk).
Adding up both terms yields Rich?g(T ) as claimed. �
Proposition 3.2. Let (V, g) be an n-dimensional Euclidean vector space and h ∈ Sym2(V ).
The following hold:
1. Every T ∈ Sym2(V ) satisfies
Rich?g(T ) = −nHT − 2〈T, h〉g − 2 tr(T )h+ 2 tr(h)T,
g(Rich?g(T ), T ) = −ng(HT, T )− 4 tr(T )〈T, h〉+ 2 tr(h)|T |2.
The Bochner Technique and Weighted Curvatures 7
2. Every p-form ω satisfies
Rich?g(ω) = −(n− 2p)Hω + p tr(h)ω,
g(Rich?g(ω), ω) = −(n− 2p)g(Hω,ω) + p tr(h)|ω|2.
3. Every algebraic (0, 4)-curvature tensor Rm satisfies
Rich?g(Rm) = −2(h? Ric)− 2g ? (c24(h⊗ Rm))− (n− 2)H Rm +4 tr(h) Rm .
Proof. (a) Due to the symmetry of T it follows that
Rich?g(T )(X1, X2) = 2{T (H(X1), X2) + T (X1, H(X2)}
− 2{g(X1, X2)〈h, T 〉+ h(X1, X2) tr(T )}
− (n− 2)(HT )(X1, X2) + 2 tr(h)T (X1, X2).
(b) Since ω ◦ τij = −ω for every transposition τij it follows that∑
i 6=j
(ω ◦ τij)(X1, . . . ,H(Xi), . . . , Xp) = −
∑
i 6=j
ω(X1, . . . ,H(Xi), . . . , Xp)
= −(p− 1)
p∑
i=1
ω(X1, . . . ,H(Xi), . . . , Xp)
= (p− 1)(Hω)(X1, . . . , Xp)
and furthermore cij(g ⊗ ω) = cij(h⊗ ω) = 0 for all i 6= j. This implies the claim.
(c) The symmetries of the curvature tensor imply that∑
i 6=j
(Rm ◦τij)(X1, . . . ,H(Xi), . . . , X4)
= (H Rm)(X1, X2, X3, X4) + (H Rm)(X2, X3, X1, X4) + (H Rm)(X3, X1, X2, X4) = 0
due to the first Bianchi identity.
Computing with respect to an orthonormal eigenbasis of H it follows that
(g(·, ·)c12(h⊗ Rm))(X,Y, Z,W ) = 0,
(g(·, ·)c13(h⊗ Rm))(X,Y, Z,W ) =
n∑
a,b=1
g(X,Z) Rm(g(H(ea), eb)eb, Y, ea,W )
=
n∑
a=1
g(X,Z) Rm(H(ea), Y, ea,W )
=
n∑
a=1
g(Z,X) Rm(ea, Y,H(ea),W )
= (g(·, ·)c31(h⊗ Rm))(X,Y, Z,W ).
This implies∑
i 6=j
(g(·, ·)cij(h⊗ Rm))(X,Y, Z,W )
= 2
n∑
i=1
{g(X,Z) Rm(H(ei), Y, ei,W ) + g(X,W ) Rm(H(ei), Y, Z, ei)
8 P. Petersen and M. Wink
+ g(Y, Z) Rm(X,H(ei), ei,W ) + g(Y,W ) Rm(X,H(ei), Z, ei)}
= 2
n∑
i=1
{g(X,Z) Rm(Y,H(ei),W, ei)− g(X,W ) Rm(Y,H(ei), Z, ei)
− g(Y, Z) Rm(X,H(ei),W, ei) + g(Y,W ) Rm(X,H(ei), Z, ei)}
= 2
(
g ?
[
n∑
i=1
Rm(·, H(ei), ·, ei)
])
(X,Y, Z,W )
= 2 (g ? c24(h⊗ Rm)) (X,Y, Z,W ).
Similarly it follows that∑
i 6=j
(h(·, ·)cij(g ⊗ Rm)) = 2 (h? c24(g ⊗ Rm)) = 2 (h? Ric) .
This completes the proof. �
Remark 3.3. For a Weyl tensor W and h a symmetric (0, 2)-tensor it is not hard to check that
Rich?g(W ) satisfies
g(Rich?g(W ),W ) = −(n− 2)g(HW,W ) + 4 tr(h)|W |2,
g
(
Rich?g(W ), g ? R̊ic
)
= −8(n− 2)〈c24(h⊗W ),Ric〉 = −8(n− 2)
〈
c24(̊h⊗W ), R̊ic
〉
,
g(Rich?g(W ), g ? g) = 0.
It is worth noting that there are trace-free symmetric (0, 2)-tensors h1, h2 such that the curvature
tensor h1 ? h2 is Weyl.
The main Theorem follows as in Proposition 3.4 below by using Lemma 2.5 instead of
Lemma 2.3. The description of the de Rham cohomology groups follows from Remark 2.4.
Proposition 3.4. 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
H =
1
n− 2p
S − 1
2(n− p)(n− 2p)
tr(S)I,
where I : TM → TM denotes the identity operator.
Suppose that the eigenvalues λ1 ≤ · · · ≤ λ(n2)
of the weighted curvature tensor Rm +h ? g
satisfy
λ1 + · · ·+ λn−p ≥ 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.
Proof. Proposition 3.2 (b) and −Iω = pω imply that
g(Rich?g ω, ω) = −(n− 2p)g(Hω,ω) + p tr(h)|ω|2 = −g(((n− 2p)H + tr(h)I)ω, ω)
= −g
((
S − tr(S)
2(n− p)
I +
tr(S)
2(n− p)
I
)
ω, ω
)
= −g(Sω, ω).
Thus the Bochner formula takes the form
∆Uω = ∇∗U∇ω + Ric(ω)− (∇U)ω = ∇∗U∇ω + RicRm+h?g(ω).
The Bochner Technique and Weighted Curvatures 9
The argument in [6, proof of Theorem A] shows that RicRm+h?g(ω) ≥ 0. Lemma 2.3 implies
the claim.
If the inequality is strict, then the same argument shows that RicRm+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 3.5. 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 µ1 ≤ · · · ≤ µn denote the eigenvalues of S.
Suppose that the weighted curvature tensor
Rm +
p∑
i=1
µi
2p(n− p)
g ? g
is (n− p)-nonnegative. 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
−g((Sω), ω) = −
∑
i1<···<ip
(Sω)i1...ipωi1...ip =
∑
i1<···<ip
p∑
j=1
µij
(ωi1...ip)2 ≥
(
p∑
i=1
µi
)
|ω|2.
Let {λα} denote the eigenvalues of (the curvature operator associated to) Rm and let {Ξα} be
an orthonormal eigenbasis. It follows from [6, Proposition 1.6] that
g(RicRm(ω), ω)− g(Sω, ω) ≥
∑
α
λα|Ξαω|2 +
(
p∑
i=1
µi
)
|ω|2 =
∑
α
λα +
p∑
i=1
µi
p(n− p)
|Ξαω|2.
The proof can now be completed as in Proposition 3.4. �
This principle can also be applied to (0, 2)-tensors.
Proposition 3.6. Let T ∈ Sym2(V ) with tr(T ) = 0, let S = ∇U and set
H =
S
n
− tr(S)
2n2
I.
Let λ1 ≤ · · · ≤ λ(n2)
denote the eigenvalues of the weighted curvature tensor Rm +h? g and
suppose that
λ1 + · · ·+ λbn
2
c ≥ 0.
If T is U -harmonic and |T | has a maximum, then T is parallel. If in addition the inequality is
strict, then T vanishes.
Proof. Proposition 3.2(a) implies that
g(Rich?g(T ), T ) = −ng
((
H +
tr(h)
n
I
)
T, T
)
= −ng
((
S
n
− tr(S)
2n2
I +
tr(S)
2n2
I
)
T, T
)
= −g(ST, T ).
10 P. Petersen and M. Wink
It follows from Proposition 2.1(b) that
(∇X∇∗UT ) (X) +
(
∇∗Ud∇T
)
(X,X) = (∇∗U∇T ) (X,X) +
1
2
(RicRm+h?g T ) (X,X).
As in [6, Lemma 2.1 and Proposition 2.9] we conclude that RicRm+h?g(T ) ≥ 0. When the
inequality is strict, the argument shows moreover RicRm+h?g(T ) > 0 unless T = 0. This uses
again that T is trace-less.
An application of Lemma 2.5 as before implies the claim. �
Acknowledgements
We would like to thank the referees for useful comments.
References
[1] Bakry D., Émery M., Diffusions hypercontractives, in Séminaire de probabilités, XIX, 1983/84, Lecture
Notes in Math., Vol. 1123, Springer, Berlin, 1985, 177–206.
[2] Bakry D., Qian Z., Volume comparison theorems without Jacobi fields, in Current Trends in Potential
Theory, Theta Ser. Adv. Math., Vol. 4, Theta, Bucharest, 2005, 115–122.
[3] Lichnerowicz A., Variétés riemanniennes à tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271
(1970), A650–A653.
[4] Lott J., Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv. 78 (2003),
865–883, arXiv:math.DG/0211065.
[5] Petersen P., Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, Vol. 171, Springer, Cham, 2016.
[6] Petersen P., Wink M., New curvature conditions for the Bochner technique, arXiv:1908.09958v3.
[7] Qian Z., Estimates for weighted volumes and applications, Quart. J. Math. Oxford 48 (1997), 235–242.
[8] Wei G., Wylie W., Comparison geometry for the Bakry–Émery Ricci tensor, J. Differential Geom. 83 (2009),
377–405, arXiv:0706.1120.
https://doi.org/10.1007/BFb0075847
https://doi.org/10.1007/s00014-003-0775-8
https://arxiv.org/abs/math.DG/0211065
https://doi.org/10.1007/978-3-319-26654-1
https://arxiv.org/abs/1908.09958v3
https://doi.org/10.1093/qmath/48.2.235
https://doi.org/10.4310/jdg/1261495336
https://arxiv.org/abs/0706.1120
1 Introduction
2 Preliminaries
2.1 Algebraic curvature tensors
2.2 Lichnerowicz Laplacians on smooth metric measure spaces
3 Weighted Lichnerowicz Laplacians
References
|
| id | nasplib_isofts_kiev_ua-123456789-210784 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T21:32:07Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Petersen, Peter Wink, Matthias 2025-12-17T14:38:35Z 2020 The Bochner Technique and Weighted Curvatures. Peter Petersen and Matthias Wink. SIGMA 16 (2020), 064, 10 pages 1815-0659 2020 Mathematics Subject Classification: 53B20; 53C20; 53C21; 53C23; 58A14 arXiv:2005.02604 https://nasplib.isofts.kiev.ua/handle/123456789/210784 https://doi.org/10.3842/SIGMA.2020.064 In this note, we study the Bochner formula on smooth metric measure spaces. We introduce weighted curvature conditions that imply the vanishing of all Betti numbers. We would like to thank the referees for their useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Bochner Technique and Weighted Curvatures Article published earlier |
| spellingShingle | The Bochner Technique and Weighted Curvatures Petersen, Peter Wink, Matthias |
| title | The Bochner Technique and Weighted Curvatures |
| title_full | The Bochner Technique and Weighted Curvatures |
| title_fullStr | The Bochner Technique and Weighted Curvatures |
| title_full_unstemmed | The Bochner Technique and Weighted Curvatures |
| title_short | The Bochner Technique and Weighted Curvatures |
| title_sort | bochner technique and weighted curvatures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210784 |
| work_keys_str_mv | AT petersenpeter thebochnertechniqueandweightedcurvatures AT winkmatthias thebochnertechniqueandweightedcurvatures AT petersenpeter bochnertechniqueandweightedcurvatures AT winkmatthias bochnertechniqueandweightedcurvatures |