Positively curved manifolds withisometric torus actions
Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)20th November 2019

Positively curved manifolds with isometric torus actions

A question and a problem.

Question
What are the topological implications of positive sectional curvature?

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)1

Positively curved manifolds with isometric torus actions

A question and a problem.

Question
What are the topological implications of positive sectional curvature?

ProblemClassify manifolds admitting metrics of positive sectional curvature.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)1

Positively curved manifolds with isometric torus actions

Examples of positively curved manifolds.

▶ There are only very few examples of manifolds admitting metrics ofpositive sectional curvature.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)2

Positively curved manifolds with isometric torus actions

Examples of positively curved manifolds.

▶ There are only very few examples of manifolds admitting metrics ofpositive sectional curvature.
▶ For dim M > 24 all known examples are diffeomorphic to Sn , ℂPn , or
ℍPn .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)2

Positively curved manifolds with isometric torus actions

Examples of positively curved manifolds.

▶ There are only very few examples of manifolds admitting metrics ofpositive sectional curvature.
▶ For dim M > 24 all known examples are diffeomorphic to Sn , ℂPn , or
ℍPn .
▶ Other examples are known in dimensions 6, 7, 12, 13 and 24▶ These are certain homogeneous spaces and biquotient spaces.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)2

Positively curved manifolds with isometric torus actions

Topological implications of positive curvature.
For closed manifolds M the following is known:

Classical results▶ Theorem of Gauß-Bonnet: sec(M2 ) > 0 ⇒ M is diffeomorphic to S2
or ℝP2 .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)3

Positively curved manifolds with isometric torus actions

Topological implications of positive curvature.
For closed manifolds M the following is known:

Classical results▶ Theorem of Gauß-Bonnet: sec(M2 ) > 0 ⇒ M is diffeomorphic to S2
or ℝP2 .
▶ Theorem of Synge: sec(M2n ) > 0 ⇒|π1 (M)| ≤ 2.
▶ Theorem of Bonnet-Myers: Ric(Mn ) > 0 ⇒ |π1 (M)| < ∞.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)3

Positively curved manifolds with isometric torus actions

Topological implications of positive curvature.
For closed manifolds M the following is known:

Classical results▶ Theorem of Gauß-Bonnet: sec(M2 ) > 0 ⇒ M is diffeomorphic to S2
or ℝP2 .
▶ Theorem of Synge: sec(M2n ) > 0 ⇒|π1 (M)| ≤ 2.
▶ Theorem of Bonnet-Myers: Ric(Mn ) > 0 ⇒ |π1 (M)| < ∞.
▶ Gromov’s Betti number Theorem: sec(Mn ) ≥ 0 ⇒ ∑ bi (M) < C(n).
iMichael Wiemeler (joint with Lee Kennard and Burkhard Wilking)3

Positively curved manifolds with isometric torus actions

Two conjectures.
▶ There are no invariants which can distinguish positively andnon-negatively curved simply connected manifolds.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)4Positively curved manifolds with isometric torus actions

Two conjectures.
▶ There are no invariants which can distinguish positively andnon-negatively curved simply connected manifolds.
▶ But there are the following conjectures:Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)4Positively curved manifolds with isometric torus actions

Two conjectures.
▶ There are no invariants which can distinguish positively andnon-negatively curved simply connected manifolds.
▶ But there are the following conjectures:

Hopf’s Conjecture I
If M is a closed, even-dimensional positively curved manifold,then the Euler characteristic of M is positive.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)4Positively curved manifolds with isometric torus actions

Two conjectures.
▶ There are no invariants which can distinguish positively andnon-negatively curved simply connected manifolds.
▶ But there are the following conjectures:

Hopf’s Conjecture I
If M is a closed, even-dimensional positively curved manifold,then the Euler characteristic of M is positive.

Hopf’s Conjecture II
S2 × S2 does not admit a positively curved metric.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)4Positively curved manifolds with isometric torus actions

Remarks.

′▶ The first conjecture would imply that S2n+1 × S2n +1 does not admit
a positively curved metric.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)5Positively curved manifolds with isometric torus actions

Remarks.

′▶ The first conjecture would imply that S2n+1 × S2n +1 does not admit
a positively curved metric.
▶ The first conjecture is true in dimensions two and four.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)5Positively curved manifolds with isometric torus actions

A programme.

Grove’s ProgrammeClassify simply connected positively curved manifolds with large
isometry group first.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)6

Positively curved manifolds with isometric torus actions

The result of Grove and Searle.
Theorem (Grove and Searle 1992)
Let Mn be positively curved and simply connected.
Assume that there is an isometric, effective action of a torus Td with
d≥[

n+1
].
2
n

Then M is diffeomorphic to Sn or ℂP 2 .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)7

Positively curved manifolds with isometric torus actions

Wilking’s Theorem.
Theorem (Wilking 2003)
Let Mn be manifold with π1 (M) = 0, sec(M) > 0, and n ≥ 10.
Suppose that there is an effective isometric action of a d-dimensional
torus Td on Mn with1d ≥ n + 1.
4nThen Mn is homeomorphic to ℍP 4 or to Sn , or M is homotopy equivalentn
to ℂP 2 .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)8Positively curved manifolds with isometric torus actions

Main tool in the proof.

Connectedness Lemma (Wilking 2003)
Assume sec(Mn ) > 0. If Nn−k ⊂ Mn is a totally geodesic submanifold,then the inclusion Nn−k → Mn is (n − 2k + 1)-connected.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)9Positively curved manifolds with isometric torus actions

Further results until 2010.

▶ Kleiner (1990) and independently Searle and Yang (1994) classify
simply connected positively curved 4-manifolds with S1 -symmetry.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)10Positively curved manifolds with isometric torus actions

Further results until 2010.

▶ Kleiner (1990) and independently Searle and Yang (1994) classify
simply connected positively curved 4-manifolds with S1 -symmetry.
▶ Fang and Rong (2005) gave a homeomorphism classification of
positively curved n-manifolds with symmetry rank [ n−12 ].

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)10Positively curved manifolds with isometric torus actions

Further results until 2010.

▶ Kleiner (1990) and independently Searle and Yang (1994) classify
simply connected positively curved 4-manifolds with S1 -symmetry.
▶ Fang and Rong (2005) gave a homeomorphism classification of
positively curved n-manifolds with symmetry rank [ n−12 ].
▶ Dessai (2007) gives vanishing results for coefficients of the elliptic
genus of a positively curved two-connected manifold withisometric S1 -action.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)10Positively curved manifolds with isometric torus actions

Kennard’s result.

Theorem (Kennard 2013)
Assume sec(Mn ) > 0. If n ≡ 0 mod 4 and M admits an effective,
isometric Td -action withd ≥ 2log2 n − 2,then χ(M) > 0.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)11

Positively curved manifolds with isometric torus actions

Main tool in proof.
Periodicity Theorem (Kennard 2012)
Let Mn with sec(Mn ) > 0, π1 (M) = 0.
Assume there is a pair of totally geodesic, transversely intersecting
submanifolds of codimensions k1 ≥ k2 . If k1 + 3k2 ≤ n,then H∗ (M; ℚ) is 4-periodic.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)12Positively curved manifolds with isometric torus actions

Main tool in proof.
Periodicity Theorem (Kennard 2012)
Let Mn with sec(Mn ) > 0, π1 (M) = 0.
Assume there is a pair of totally geodesic, transversely intersecting
submanifolds of codimensions k1 ≥ k2 . If k1 + 3k2 ≤ n,then H∗ (M; ℚ) is 4-periodic.

Definition
Here H∗ (M) is called k-periodic, if there is a e ∈ Hk (M) such that∪e: Hi (M) → Hi+k (M)is an isomorphism for all 0 ≤ i ≤ dim M − k or H∗ (M) ≅ H∗ (Sdim M ).
Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)12Positively curved manifolds with isometric torus actions

Remarks▶ If H∗ (M; ℚ) is four-periodic and b1 (M) = b3 (M) = 0, then H∗ (M; ℚ)
is one of the following:nn−2nH∗ (Sn ; ℚ), H∗ (ℂP 2 ; ℚ), H∗ (ℍP 4 ; ℚ), or H∗ (S2 × ℍP 4 ; ℚ).

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)13

Positively curved manifolds with isometric torus actions

Remarks▶ If H∗ (M; ℚ) is four-periodic and b1 (M) = b3 (M) = 0, then H∗ (M; ℚ)
is one of the following:nn−2nH∗ (Sn ; ℚ), H∗ (ℂP 2 ; ℚ), H∗ (ℍP 4 ; ℚ), or H∗ (S2 × ℍP 4 ; ℚ).
▶ If H∗ (M; ℚ) is four-periodic, b1 (M) = 0 and n ≡ 0 mod 4,then b3 (M) = 0.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)13

Positively curved manifolds with isometric torus actions

Further results since 2013.

▶ Amann and Kennard (2014/2015/2017) further improved thebound in Kennard’s original result.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)14

Positively curved manifolds with isometric torus actions

Further results since 2013.

▶ Amann and Kennard (2014/2015/2017) further improved thebound in Kennard’s original result.
▶ Weisskopf (2017) combined Kennard’s result with the methods of
Dessai.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)14

Positively curved manifolds with isometric torus actions

Further results since 2013.

▶ Amann and Kennard (2014/2015/2017) further improved thebound in Kennard’s original result.
▶ Weisskopf (2017) combined Kennard’s result with the methods of
Dessai.
▶ Goertsches and W. (2015) studied positively curved
GKM-manifolds.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)14

Positively curved manifolds with isometric torus actionsFirst main result.
Theorem (Kennard, W., Wilking 2019)
Assume sec(Mn ) > 0 and that there is an isometric effective action of atorus Td of dimensiond ≥ 5.
Then every component F of MT has the rational cohomology of Sm , ℂPm
or ℍPm .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)15

Positively curved manifolds with isometric torus actionsFirst main result.
Theorem (Kennard, W., Wilking 2019)
Assume sec(Mn ) > 0 and that there is an isometric effective action of atorus Td of dimensiond ≥ 5.
Then every component F of MT has the rational cohomology of Sm , ℂPm
or ℍPm .
▶ This is first result in this direction where the dimension of theacting torus does not grow with the dimension of the manifold.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)15

Positively curved manifolds with isometric torus actions

Some corollaries.

Corollary
Hopf’s Conjecture I holds for manifolds with isometric T5 -actions.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)16

Positively curved manifolds with isometric torus actions

Some corollaries.

Corollary
Hopf’s Conjecture I holds for manifolds with isometric T5 -actions.

Corollary
The isometry group of a potential positively curved metric on′S2n+1 × S2n +1 has rank at most four.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)16

Positively curved manifolds with isometric torus actions

New tool in proof.

T-Splitting Theorem
If T2d+1 → SO(V) is a faithful representation,

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)17

Positively curved manifolds with isometric torus actions

New tool in proof.

T-Splitting Theorem
If T2d+1 → SO(V) is a faithful representation,then there exists a d-dimensional subgroup H ⊂ T2d+1 such that theinduced representation T2d+1 /H → SO(VH ) is faithful and has exactly
d + 1 non-trivial, pairwise inequivalent, irreducible subrepresentations.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)17

Positively curved manifolds with isometric torus actions

New tool in proof.

T-Splitting Theorem
If T2d+1 → SO(V) is a faithful representation,then there exists a d-dimensional subgroup H ⊂ T2d+1 such that theinduced representation T2d+1 /H → SO(VH ) is faithful and has exactly
d + 1 non-trivial, pairwise inequivalent, irreducible subrepresentations.
In particular, there exist d + 1 circles in T2d+1 /H whose fixed point sets in
VH intersect pairwise transversely.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)17

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Localization in equivariant cohomology, leads to relations
between H∗ (M) and H∗ (MT ).

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)18

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Localization in equivariant cohomology, leads to relations
between H∗ (M) and H∗ (MT ).
▶ Therefore it suffices to find for each component F ⊂ MT , aninvariant submanifold F ⊂ P ⊂ M with controlled cohomology ring.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)18

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Localization in equivariant cohomology, leads to relations
between H∗ (M) and H∗ (MT ).
▶ Therefore it suffices to find for each component F ⊂ MT , aninvariant submanifold F ⊂ P ⊂ M with controlled cohomology ring.
▶ To find P do the following:▶ For simplicity assume d ≥ 7.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)18

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Localization in equivariant cohomology, leads to relations
between H∗ (M) and H∗ (MT ).
▶ Therefore it suffices to find for each component F ⊂ MT , aninvariant submanifold F ⊂ P ⊂ M with controlled cohomology ring.
▶ To find P do the following:▶ For simplicity assume d ≥ 7.
▶ Let P be a component of MH , where H is as in the T-Splitting
Theorem (with V = Tx M for x ∈ F).

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)18

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Localization in equivariant cohomology, leads to relations
between H∗ (M) and H∗ (MT ).
▶ Therefore it suffices to find for each component F ⊂ MT , aninvariant submanifold F ⊂ P ⊂ M with controlled cohomology ring.
▶ To find P do the following:▶ For simplicity assume d ≥ 7.
▶ Let P be a component of MH , where H is as in the T-Splitting
Theorem (with V = Tx M for x ∈ F).
▶ Then there are four totally geodesic submanifolds of P intersecting
pairwise transversely.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)18

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Localization in equivariant cohomology, leads to relations
between H∗ (M) and H∗ (MT ).
▶ Therefore it suffices to find for each component F ⊂ MT , aninvariant submanifold F ⊂ P ⊂ M with controlled cohomology ring.
▶ To find P do the following:▶ For simplicity assume d ≥ 7.
▶ Let P be a component of MH , where H is as in the T-Splitting
Theorem (with V = Tx M for x ∈ F).
▶ Then there are four totally geodesic submanifolds of P intersecting
pairwise transversely.
▶ Applying the Periodicity Theorem to the two submanifolds ofsmallest codimension, implies that P has four-periodic cohomology.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)18

Positively curved manifolds with isometric torus actions

Outline of proof II.

▶ Computations in equivariant cohomology then show b3 (P) = 0

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)19

Positively curved manifolds with isometric torus actions

Outline of proof II.

▶ Computations in equivariant cohomology then show b3 (P) = 0▶ Therefore by classical results one only has to deal with the casethat H∗ (P; ℚ) = H∗ (S2 × ℍPn ; ℚ).

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)19

Positively curved manifolds with isometric torus actions

Outline of proof II.

▶ Computations in equivariant cohomology then show b3 (P) = 0▶ Therefore by classical results one only has to deal with the casethat H∗ (P; ℚ) = H∗ (S2 × ℍPn ; ℚ).
▶ In this case computations in equivariant cohomology and theconnectedness lemma lead to the desired result.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)19

Positively curved manifolds with isometric torus actions

Second main result.
Theorem (Kennard, W., Wilking 2019)
Assume sec(Mn ) > 0, π1 (M) = 0. If M admits an isometric effective
equivariantly formal action of Td , withd ≥ 8,
nnthen M has the rational cohomology of Sn , ℂP 2 , or ℍP 4 .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)20

Positively curved manifolds with isometric torus actions

Second main result.
Theorem (Kennard, W., Wilking 2019)
Assume sec(Mn ) > 0, π1 (M) = 0. If M admits an isometric effective
equivariantly formal action of Td , withd ≥ 8,
nnthen M has the rational cohomology of Sn , ℂP 2 , or ℍP 4 .
▶ Bott’s Conjecture asks whether a positively curved manifold isrationally elliptic.
▶ Together with Hopf’s Conjecture it would imply that Hodd (M; ℚ) = 0
in even-dimensions. This in turn implies equivariant formality.
Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)20

Positively curved manifolds with isometric torus actions

A corollary.

Corollary′

The isometry group of a potential positively curved metric on S2n × S2n′
and S2n × S2n −1 , n′ ≤ n has rank at most seven.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)21

Positively curved manifolds with isometric torus actions

A corollary.

Corollary′

The isometry group of a potential positively curved metric on S2n × S2n′
and S2n × S2n −1 , n′ ≤ n has rank at most seven.
▶ For the proof note that all torus actions on these manifolds areequivariantly formal.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)21

Positively curved manifolds with isometric torus actions

A corollary.

Corollary′

The isometry group of a potential positively curved metric on S2n × S2n′
and S2n × S2n −1 , n′ ≤ n has rank at most seven.
▶ For the proof note that all torus actions on these manifolds areequivariantly formal.
▶ There exist equivariantly non-formal actions on S2 × S3 .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)21

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Equivariant formality, implies thatH∗ (M; ℚ) ≅ H∗T (M; ℚ)/(H>0 (BT)),

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)22

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Equivariant formality, implies thatH∗ (M; ℚ) ≅ H∗T (M; ℚ)/(H>0 (BT)),and thatι∗ : H∗T (M; ℚ) → H∗T (MT ; ℚ)
is injective.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)22

Positively curved manifolds with isometric torus actions

Outline of proof I.
▶ Equivariant formality, implies thatH∗ (M; ℚ) ≅ H∗T (M; ℚ)/(H>0 (BT)),and thatι∗ : H∗T (M; ℚ) → H∗T (MT ; ℚ)
is injective.
▶ Hence it suffices to determine the image of ι∗ .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)22

Positively curved manifolds with isometric torus actions

Outline of proof II.
▶ Let M1 = {x ∈ M; dim Tx ≤ 1} = ⋃ d−1
TMichael Wiemeler (joint with Lee Kennard and Burkhard Wilking)

d−1

⊂Td

MT .

23

Positively curved manifolds with isometric torus actions

Outline of proof II.
▶ Let M1 = {x ∈ M; dim Tx ≤ 1} = ⋃ d−1
T

d−1

⊂Td

MT .

Lemma (Chang, Skjelbred 1974)
Assume that the T-action on M is equivariantly formal. Then for every
closed invariant subspace M1 ⊂ X ⊂ M,
ι∗ H∗T (X; ℚ) = ι∗ H∗T (M; ℚ) ⊂ H∗T (MT ; ℚ).

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)23

Positively curved manifolds with isometric torus actions

Outline of proof II.
▶ Let M1 = {x ∈ M; dim Tx ≤ 1} = ⋃ d−1
T

d−1

⊂Td

MT .

Lemma (Chang, Skjelbred 1974)
Assume that the T-action on M is equivariantly formal. Then for every
closed invariant subspace M1 ⊂ X ⊂ M,
ι∗ H∗T (X; ℚ) = ι∗ H∗T (M; ℚ) ⊂ H∗T (MT ; ℚ).
▶ Hence, it suffices to determine the combinatorial structure of M1 .

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)23

Positively curved manifolds with isometric torus actions

Outline of proof III.

In even dimensions n, we now have to consider two cases:7▶ ∃T7 ⊂ T8 and F0 ⊂ MT , dim F0 ≥ 4.
▶ The T8 -action is GKM. (This case is similar to the discussion in
Goertsches-W. 2015)

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)24

Positively curved manifolds with isometric torus actions

Outline of proof IV.
In the first case. Then:▶ Using work of Smith, Bredon, Hsiang-Su,
M1 ⊂ ⋃ NT5 =: X,
T5 ⊂T7

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)25

Positively curved manifolds with isometric torus actions

Outline of proof IV.
In the first case. Then:▶ Using work of Smith, Bredon, Hsiang-Su,
M1 ⊂ ⋃ NT5 =: X,
T5 ⊂T7

where NT5 denotes the fixed point component of T5 ⊂ T7
containing F0▶ Using a Mayer-Vietoris argument, we show that7ι∗ H∗T7 (X; ℚ) ⊂ H∗ (MT ; ℚ)
is isomorphic to a similar algebra constructed for a linear action onsome Sn , ℍPn , or ℂPn , respectively.

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)25

Positively curved manifolds with isometric torus actions

Outline of proof IV.
In the first case. Then:▶ Using work of Smith, Bredon, Hsiang-Su,
M1 ⊂ ⋃ NT5 =: X,
T5 ⊂T7

where NT5 denotes the fixed point component of T5 ⊂ T7
containing F0▶ Using a Mayer-Vietoris argument, we show that7ι∗ H∗T7 (X; ℚ) ⊂ H∗ (MT ; ℚ)
is isomorphic to a similar algebra constructed for a linear action onsome Sn , ℍPn , or ℂPn , respectively.
▶ The Chang–Skjelbred Lemma now implies the claim.
Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)25

Positively curved manifolds with isometric torus actions

Thank you!

Michael Wiemeler (joint with Lee Kennard and Burkhard Wilking)26