Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces
Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group 𝐺 are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant met...
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| description | Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group 𝐺 are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact, the same result holds for a compact connected homogeneous manifold 𝐺/𝘏 with 𝐺 compact connect and semi-simple.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 068, 6 pages
Gromov Rigidity of Bi-Invariant Metrics
on Lie Groups and Homogeneous Spaces
Yukai SUN † and Xianzhe DAI ‡
† School of Mathematical Sciences, East China Normal University,
500 Dongchuan Road, Shanghai 200241, P.R. of China
E-mail: 52195500003@stu.ecnu.edu.cn
‡ Department of Mathematics, UCSB, Santa Barbara CA 93106, USA
E-mail: dai@math.ucsb.edu
Received May 04, 2020, in final form July 22, 2020; Published online July 25, 2020
https://doi.org/10.3842/SIGMA.2020.068
Abstract. Gromov asked if the bi-invariant metrics on a compact Lie group are extremal
compared to any other metrics. In this note, we prove that the bi-invariant metrics on
a compact connected semi-simple Lie group G are extremal (in fact rigid) in the sense
of Gromov when compared to the left-invariant metrics. In fact the same result holds for
a compact connected homogeneous manifold G/H with G compact connect and semi-simple.
Key words: extremal/rigid metrics; Lie groups; homogeneous spaces; scalar curvature
2020 Mathematics Subject Classification: 53C20; 53C24; 53C30
1 Introduction
In [6], Gromov asks: are bi-invariant metrics on compact Lie groups extremal? (This is already
problematic for SO(5).) Here a Riemannian metric g on a differentiable manifold M is extremal
in the sense of Gromov (not to be confused with Calabi’s extremal metrics in Kähler geometry)
if any metric g′ on M with g′ ≥ g and Rg′ ≥ Rg must have Rg′ = Rg, where Rg, Rg′ denote
the scalar curvature of g, g′ respectively. The metric g is rigid in the sense of Gromov if in fact
g′ = g from the conditions above.
The first result of this type is [10] in which Llarull showed that the standard metric on Sn
is rigid. The work gives a positive answer to an earlier question of Gromov, which is motivated
by Gromov–Lawson’s famous work on the non-existence of positive scalar curvature metrics
on the torus [7], later extended to more general class of manifolds, namely the enlargeable
manifolds. In the same spirit, Llarull in fact proved that a metric on a compact manifold
admitting a
(
1,Λ2
)
-contracting map to Sn is rigid. Min-Oo discussed the extremality/rigidity
of hermitian symmetric spaces of compact type in [12]. The extremality/rigidity of complex
and quaternionic projective spaces was established by Kramer [8]. Later, Goette and Semmel-
mann [4] proved that compact symmetric spaces of type G/K with rk(G) − rk(K) ≤ 1 are
extremal (see also [3]). Then Listing improves Goette–Semmelmann’s result in [9], by weakening
the extremality condition.
Note that a Lie group with a bi-invariant metric is a symmetric space, but not of the types
considered above. In this short note, we present a partial positive answer to Gromov’s question.
Namely, we show that the bi-invariant metrics on a compact connected semi-simple Lie group G
are rigid among the left-invariant metrics. More generally, we show that the normal metrics
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:52195500003@stu.ecnu.edu.cn
mailto:dai@math.ucsb.edu
https://doi.org/10.3842/SIGMA.2020.068
https://www.emis.de/journals/SIGMA/Gromov.html
2 Y. Sun and X. Dai
on any compact connected homogeneous space G/H without torus factor are rigid among G-
invariant metrics on G/H.
Theorem 1. Let M = G/H be a compact homogeneous space, with G a compact connected semi-
simple Lie group. Then any bi-invariant metric (also known as normal homogeneous metric) g0
on G/H is rigid among the G-invariant metrics. In other words, if g is a G-invariant metric
on G/H such that g ≥ g0 and Rg ≥ Rg0, then g = g0.
As an immediate consequence, we have
Corollary 2. Any bi-invariant metric on a compact connected semi-simple Lie group is rigid
among the left-invariant metrics.
According to [11], if a connected Lie group admits a bi-invariant metric, it is isomorphic to
the product of a compact Lie group with an abelian one. The semi-simple condition rules out
the abelian factor. On the other hand, we have the famous result of Gromov–Lawson [7] and
Schoen–Yau [13, 14] which implies that the only metrics of nonnegative scalar curvatures on the
torus are the flat ones.
Remark 3. The extremal/rigid metrics discussed here have positive scalar curvature. On the
other hand, we would like to point out a related but different scalar curvature (local) extremality
for Kähler–Einstein metrics with negative scalar curvature [2]. It is an immediate consequence
of Theorem 1.5 in [2] that for a Kähler–Einstein metric g0 with negative scalar curvature on
a compact complex manifold with integrable infinitesimal complex deformations, any metric g
sufficiently close to g0 satisfying Rg ≥ Rg0 and Vol(g) ≤ Vol(g0) must have Rg = Rg0 (and g is
also Kähler–Einstein).
2 Preliminaries
Given a Riemannian manifold (M, g), we denote by Rg the scalar curvature of g. We recall
Gromov’s notion of extremal/rigid metrics.
Definition 4. A metric g0 on M is extremal (in the sense of Gromov), if any metric g on M
satisfying g ≥ g0 and Rg ≥ Rg0 must have identical scalar curvature, Rg = Rg0 ; g0 is said to be
rigid (in the sense of Gromov) if the conditions above imply that g = g0.
For a Lie group G, we denote by Ad(a) (a ∈ G) the adjoint action of G on its Lie algebra g,
and by ad(X) (X ∈ g) the induced adjoint action of g on itself. In particular,
ad(X)Y = [X,Y ], X, Y ∈ g.
A Lie group G is semi-simple if its Lie algebra g is semi-simple, i.e., its Killing form
K(X,Y ) = Tr(ad(X) ad(Y )), X, Y ∈ g
is nondegenerate. Clearly, if g is semi-simple, it has a trivial center. For a compact Lie group,
the semi-simple condition is equivalent to its Lie algebra having trivial center.
If a metric on G is both left-invariant and right-invariant, then it is called bi-invariant.
When G is compact, bi-invariant metrics always exist. Left-invariant metrics on G are in one-
to-one correspondence with inner products on its Lie algebra g. The following well known result
plays a crucial role in the proof of our main result.
Theorem 5 ([11, Lemma 7.2]). In the case of a connected group G, a left-invariant metric is
actually bi-invariant if and only if the linear transformation ad(X) is skew-adjoint with respect
to the corresponding inner product, for every X in the Lie algebra g of G.
Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces 3
Now let M = G/H be a compact connected homogeneous space, where G is a compact
connected Lie group, H a closed subgroup, and the action of G on G/H is effective. Let h ⊂ g
be the Lie algebra of H. Denote by AdG the adjoint action of G on g and AdH = AdG |H
its restriction to H. Since AdH preserves h, it induces an action on g/h, which is equivalent
to the isotropy representation of H. A metric g on M = G/H is called G-invariant if it is
invariant under the left action of G. G-invariant metrics on G/H are naturally identified with
inner products on g/h which are invariant under the AdH action, see Proposition 3.16 in [1].
In particular, a bi-invariant metric on G gives rise to a G-invariant metric on G/H. The
corresponding metric on G/H, usually referred as a normal homogeneous metric on G/H in the
literature, will still be called bi-invariant here.
3 Proof of the theorem
Our proof relies crucially on a simple elegant formula for the scalar curvature for G-invariant
metrics, as well as another lemma, in [15]. We first recall this formula and the setup.
Let g0 be a bi-invariant metric on G; still denote by g0 the induced metric on G/H. Let
g = h + m be an AdH invariant decomposition orthogonal with respect to g0. Then G-invariant
metrics on G/H are identified with AdH -invariant inner products on m.
Let 〈·, ·〉0 be the AdH -invariant inner product on m corresponding to g0. Let 〈·, ·〉 be an
AdH -invariant inner product on m inducing a G-invariant metric g on G/H. Then, there is
a positive self-adjoint operator S on (m, 〈X,Y 〉0) commuting with the AdH -action such that
〈X,Y 〉 = 〈S(X), Y 〉0
for all X,Y ∈ m.
Since any eigenspace of S is AdH -invariant, there are AdH -invariant subspaces m1, . . . ,ms
of m such that
m = m1 ⊕ · · · ⊕ms (1)
in orthogonal decomposition with respect to 〈·, ·〉0; the action of AdH on each mi is irreducible,
and S(X) = λiX for all X ∈ mi, for some λ1, . . . , λs > 0. Consequently,
〈X,Y 〉 = λ1〈X1, Y1〉0 + · · ·+ λs〈Xs, Ys〉0,
for X = X1 + · · ·+Xs, Y = Y1 + · · ·+ Ys ∈ m decomposed with respect to (1). The metric g is
called diagonal with respect to the decomposition in (1).
For such metrics, there is a simple elegant formula for the scalar curvature; we refer the reader
to [15] for a more general discussion. Before we state this formula, let us point out the simplified
situation when M = G. Each mi in (1) is spanned by a basis vector whenever one chooses an
orthonormal basis of m = g consisting of eigenvectors of S. (Thus, such decompositions are by
no means unique.)
Let {Eα} be an orthonormal basis of (m, 〈 , 〉0) adapted to the decomposition (1). We write
[Eα, Eβ]m =
∑
γ C
γ
αβEγ for some real numbers
{
Cγαβ
}
that we call structural constants. Here
[ , ]m is the m-component of [ , ]. Set
Akij =
∑
α,β,γ
(
Cγαβ
)2
,
where the summation runs over Eα ∈ mi, Eβ ∈ mj , Eγ ∈ mk.
Let di = dimmi. Let B be the negative of the Killing form: B(X,Y ) = −K(X,Y ). Then
B(X,X) ≥ 0, with equality if and only if X is central. We define the real number bi by
B(X,Y ) = bi〈X,Y 〉0 for all X,Y ∈ mi. Note that bi = 0 if and only if mi is included in the
center of g. The following formula is equation (1.3) in [15].
4 Y. Sun and X. Dai
Lemma 6 ([15, equation (1.3)]). Let g be a G-invariant metric on G/H with a corresponding
decomposition (1) as described above. Then the scalar curvature of g is
Rg =
1
2
s∑
i=1
bidi
λi
− 1
4
s∑
i,j,k=1
Akij
λk
λiλj
.
The following lemma from [15] relates bidi to the structural constants. Let
Cmi,g0|h = −
h∑
i=1
ad(Zi) ◦ ad(Zi)
be the Casimir operator of the representation of h on mi, where {Z1, . . . , Zh} is an orthonormal
basis of (h, g0|h) and ad(Zi) should be interpreted as its restriction on mi. Since mi is AdH -
irreducible, Cmi,g0|h = ci Id for some constant ci ≥ 0. Moreover, ci = 0 if and only if AdH acts
trivially on mi.
Lemma 7 ([3, Lemma 1.5]). One has, for i = 1, . . . , s,
s∑
j,k=1
Akij = bidi − 2cidi.
Remark 8. Again let us look at the situation when M = G. In this case we choose an
orthonormal basis {Ei}ni=1 of g consisting of eigenvectors of S. Then [Ei, Ej ] =
n∑
k=1
CkijEk via the
structure constants Ckij . The decomposition (1) is given by mi = Span{Ei}, hence Akij =
(
Ckij
)2
.
Moreover ci = 0 for all i. Therefore, the two lemmas above yield
Rg =
1
4
n∑
i,j,k=1
(
Ckij
)2 [ 2
λi
− λk
λiλj
]
. (2)
This formula can also be deduced from Koszul’s formula via a direct computation.
Proof of Theorem 1. Since {Eα} is an orthonormal basis for (m, 〈·, ·〉0), and 〈·, ·〉0 is bi-
invariant, Cγαβ = 〈[Eα, Eβ], Eγ〉0 is skew-symmetric in all three indices by Theorem 5. Hence Akij
is symmetric in all three indices.
Now the extremal conditions 〈X,Y 〉 ≥ 〈X,Y 〉0 and Rg ≥ Rg0 yield λi ≥ 1 (i = 1, . . . , s) as
well as Rg −Rg0 ≥ 0. Lemmas 6 and 7 give
0 ≤ Rg −Rg0 =
1
2
∑
i
bidi
λi
(1− λi)−
1
4
∑
i,j,k
Akij
(
λk
λiλj
− 1
)
=
∑
i
cidi
λi
(1− λi)−
1
4
∑
i,j,k
Akij
[
λk
λiλj
+ 1− 2
λi
]
.
Since ci ≥ 0 and di > 0, each term in the first summation is less than or equal to zero, with
equality if and only if either ci = 0 or λi = 1.
For the second summation, we use the symmetry to rewrite it as
− 1
12
∑
i,j,k
Akij
[
λk
λiλj
+
λi
λjλk
+
λj
λkλi
− 2
λj
− 2
λi
− 2
λk
+ 3
]
= − 1
12
∑
i,j,k
Akij
λ2i + λ2j + λ2k − 2λiλj − 2λiλk − 2λkλj + 3λiλjλk
λiλjλk
.
Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces 5
For a fixed triple i, j, k, we consider the order of λi, λj , λk. Without loss of generality we can
assume that λk ≥ λj ≥ λi ≥ 1. Then the summand in the sum above can be re-organized as
λ2i + λ2j + λ2k − 2λiλj − 2λiλk − 2λkλj + 3λiλjλk
= (λi − λj)2 + (λk − λj)2 + λjλk(λi − 1) + λj(λk − λj) + 2λiλk(λj − 1) ≥ 0
with equality if and only if λk = λj = λi = 1.
But then all the inequalities become equalities. Hence, either ci = 0 or λi = 1 for each i, and,
at the same time, either Akij = 0 or λk = λj = λi = 1 for each (i, j, k). If λi > 1 for some i, then
ci = 0, and Akij = 0 for all j, k. Thus bi = 0 by Lemma 7. Therefore mi is in the center of g,
which contradicts the hypotheses. We conclude that λi = 1 for all i, and the result follows. �
We end with a couple of remarks.
Remark 9. From the proof, we see that if a bi-invariant metric g0 on G/H is not rigid among
the G-invariant metrics, then G/H must have a torus factor. Indeed, let z ⊂ g be the center.
If for some i, λi > 1, then mi ⊂ z. Decompose g = z + g′ and z = mi + k. Then h ⊂ k + g′. It
follows that G/H = T di × (K ×G′)/H.
Remark 10. It is interesting to note that the extremal conditions g ≥ g0 and Rg ≥ Rg0 can
not be changed to the opposite inequalities. In fact, there exist G-invariant metrics g such that
g < g0 and Rg < Rg0 . We illustrate the situation for M = G = SU(2).
The basis E1 =
√
−1σ1, E2 =
√
−1σ2, E3 =
√
−1σ3 of g in terms of the Pauli spin matrices
σ1, σ2, σ3 satisfies [E1, E2] = 2E3 as well as its cyclic permutations. We take g0 so that
〈X,Y 〉0 = 1
8B(X,Y ), with respect to which {E1, E2, E3} is orthonormal. Following the notations
in Remark 8, we choose g so that E1, E2, E3 are the eigenvectors with eigenvalues λ1 = λ2 =
λ < 1, and λ3 = 1/2, respectively. Then g < g0. On the other hand, by (2),
Rg −Rg0 =
1
4
3∑
i,j,k=1
(
Ckij
)2 [ 2
λi
− λk
λiλj
− 1
]
= − 1
λ2
+O
(
1
λ
)
,
as λ→ 0+. Thus, for λ sufficiently small, we have Rg < Rg0 .
Note that this represents the opposite rescaling of the standard sphere as compared to the
example of Berger’s sphere mentioned in [5, p. 34].
Acknowledgements
We are deeply grateful to Wolfgang Ziller for suggesting the more general result for the homoge-
neous space as well as bringing the work [15] to our attention, which considerably simplifies our
previous computation as well as generalizes to the more general case of homogeneous spaces. We
thank Wolfgang for many helpful discussions. Thanks are also due to Professor Yurii Nikonorov
for similar remarks and useful comments. Finally we thank the referee for the careful reading
of the multiple versions of the paper and for many constructive suggestions which have helped
improve the exposition. This research is partially supported by NSFC (Y.S.) and the Simons
Foundation (X.D.).
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6 Y. Sun and X. Dai
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1 Introduction
2 Preliminaries
3 Proof of the theorem
References
|
| id | nasplib_isofts_kiev_ua-123456789-210780 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T21:46:51Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sun, Yukai Dai, Xianzhe 2025-12-17T14:37:50Z 2020 Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces. Yukai Sun and Xianzhe Dai. SIGMA 16 (2020), 068, 6 pages 1815-0659 2020 Mathematics Subject Classification: 53C20; 53C24; 53C30 arXiv:2005.00161 https://nasplib.isofts.kiev.ua/handle/123456789/210780 https://doi.org/10.3842/SIGMA.2020.068 Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group 𝐺 are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact, the same result holds for a compact connected homogeneous manifold 𝐺/𝘏 with 𝐺 compact connect and semi-simple. We are deeply grateful to Wolfgang Ziller for suggesting the more general result for the homogeneous space as well as bringing the work [15] to our attention, which considerably simplifies our previous computation as well as generalizes to the more general case of homogeneous spaces. We thank Wolfgang for many helpful discussions. Thanks are also due to Professor Yurii Nikonorov for similar remarks and useful comments. Finally, we thank the referee for the careful reading of the multiple versions of the paper and for many constructive suggestions, which have helped improve the exposition. This research is partially supported by NSFC (Y.S.) and the Simons Foundation (X.D.). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces Article published earlier |
| spellingShingle | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces Sun, Yukai Dai, Xianzhe |
| title | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_full | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_fullStr | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_full_unstemmed | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_short | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_sort | gromov rigidity of bi-invariant metrics on lie groups and homogeneous spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210780 |
| work_keys_str_mv | AT sunyukai gromovrigidityofbiinvariantmetricsonliegroupsandhomogeneousspaces AT daixianzhe gromovrigidityofbiinvariantmetricsonliegroupsandhomogeneousspaces |