Annales de l’institut Fourier (AIF)

Gilbert, John

Rzeszotnik, Ziemowit

The norm of the Fourier transform on finite abelian groups

Anglès, Bruno

Ranieri, Gabriele

On the linear independence of $p$-adic $L$-functions modulo $p$

Erschler, Anna

Karlsson, Anders

Homomorphisms to $\mathbb{R}$ constructed from random walks

Thomsen, Klaus

Semi-étale groupoids and applications

Asaoka, Masayuki

Regular projectively Anosov flows on three-dimensional manifolds

Miebach, Christian

Stötzel, Henrik

Spherical gradient manifolds

Besson, Gérard

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Gérard BESSON
Pierre and Sylvestre: some recollections
Tome 60, no 7 (2010), p. 2303-2305.
<http://aif.cedram.org/item?id=AIF_2010__60_7_2303_0>
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Ann. Inst. Fourier, Grenoble
60, 7 (2010) 2303-2305
PIERRE AND SYLVESTRE: SOME RECOLLECTIONS
by Gérard BESSON
My first encounters with Pierre Bérard and Sylvestre Gallot were at
the Geometry seminar in Paris VII. The seminar was called le Séminaire
Arthur Besse and Marcel Berger was in charge of things. At that time
I was looking for an advisor so I turned up, hoping to find one, for the
latest talk. I sat there in the lecture room and, needless to say, did not
understand a single word. However, I do remember clearly that a theorem
on harmonic manifolds due to Alamigeon was cited. Just then, a young
fellow at the back, who was taking notes enthusiastically, asked a question.
Marcel Berger replied ”Well, this is a question for you !”. The young guy
was Sylvestre who had not as yet defended his ” thèse d’état ” but was
very close to doing so. Pierre, on the other hand, was not at the talk that
day. He was visiting IMPA as part of his military service and so I first met
him a couple of years later, again at the seminar Besse, when he returned
to France. I had started to work on the spectrum of the Laplacian under
the supervision of Marcel Berger, and after a lecture that I gave he kindly
came up to me and we discussed my project. In those days Pierre had a
beard, the one we see on the picture.
In 1978, Sylvestre defended his thesis. It was a tradition to give a second
lecture after the defense, called ” deuxième thèse ”, which was more like a
masters’ thesis. Sylvestre’s thesis was about inequalities for the eigenvalues
of the Laplacian and lower bound on the Ricci curvature; some striking
results were described which put Sylvestre’s works in light. His ” second ”
thesis was about Nash’s embedding theorem and all I can remember is a
slide full of Sobolev spaces and maps between them.
I really started to discuss mathematics with Pierre after his beautiful pa-
per on the spectrum of certain domains in the Euclidean space related to
2304 Gérard BESSON
crystallographic groups. To my knowledge it was the first systematic com-
putation of the spectrum of the Laplacian for both Dirichlet and Neumann
boundary conditions for a wide class of Euclidean domains. One striking
example is one half of an equilateral triangle for which the spectrum was
not know at all before!
It was not so easy to get a position in France at that time; I remember
one year with no assistant position and just around ten professor positions.
Nevertheless, Sylvestre got hired in Chambéry in 1980 and Pierre got a
job there too a year later. They started to work on a wonderful program
on isoperimetric inequalities on Riemannian manifolds satisfying a lower
bound on the Ricci curvature. Yves Colin de Verdière was in Grenoble
and naturally there was a lot of activity on spectral theory of Riemannian
manifold concentrated in the two towns which are only 60km apart. And
so I moved to Grenoble in 1981. We had a joint seminar which would
meet every week alternately in Chambéry or Grenoble. In 1983, we began
publishing the proceedings of our seminar in which various works of Pierre
and Sylvestre may be found. Their research culminated with the result on
the estimate on the heat kernel which is an analytical translation of some
isoperimetric inequalities à la Gromov. In 1984, I had just finished some
work and was wondering what I should do next. They kindly invited me
to join them in their project and what followed was an amazingly exciting
time.
In 1986, Pierre moved to Grenoble. Sylvestre got jobs at the École Poly-
technique and at the École Normale Supérieure de Lyon before finally mov-
ing to Grenoble in 1998. Roughly at the same time, in 1987, Sylvestre, Gilles
Courtois and myself started working together on the entropy of symmet-
ric spaces of negative curvature. Pierre began studying isospectrality of
Riemannian manifolds, became one of the few renowned specialist in the
world before turning to spectral problems in the theory of minimal sur-
faces theory. During this time, we all shared the same passion for Italy and
developed fruitful exchanges with our colleagues there.
I omitted some fundamental aspects of Pierre and Sylvestre activities.
Pierre has been involved in the development of mathematics at the highest
levels, International Mathematical Union, French ministry of research, the
"IST infrastructure cellule Math-Doc", University Joseph Fourier as Vice-
President in charge of the Research, etc.. Sylvestre has devoted himself to
a large number of graduate students, who have almost invariably turned
out to be very successful.
ANNALES DE L’INSTITUT FOURIER
PIERRE AND SYLVESTRE: SOME RECOLLECTIONS 2305
This short historical account describes an exceptional and passionate
journey through Mathematics and Riemannian Geometry in particular.
The influence of both Pierre and Sylvestre’s works is testified by the quality
of the participant at the workshop. I wish to thank all of them as well as
those who were eager to come but could not for their contribution to the
success of the workshop. But above all it is the story of our friendship; no
workshop nor proceedings could ever show how precious this friendship has
been.
Gérard BESSON
Institut Fourier UMR 5582CNRS-UJF
BP 74
38402 Saint Martin d’Hères Cedex (France)
gbesson@ujf-grenoble.fr
TOME 60 (2010), FASCICULE 7


Pierre and Sylvestre: some recollections

Colin de Verdière, Yves

Truc, Françoise

Confining quantum particles with a purely magnetic field

Seshadri, Harish

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Harish SESHADRI
Almost-Einstein manifolds with nonnegative isotropic curvature
Tome 60, no 7 (2010), p. 2493-2501.
<http://aif.cedram.org/item?id=AIF_2010__60_7_2493_0>
© Association des Annales de l’institut Fourier, 2010, tous droits
réservés.
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions
générales d’utilisation (http://aif.cedram.org/legal/). Toute re-
production en tout ou partie cet article sous quelque forme que ce
soit pour tout usage autre que l’utilisation à fin strictement per-
sonnelle du copiste est constitutive d’une infraction pénale. Toute
copie ou impression de ce fichier doit contenir la présente mention
de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
60, 7 (2010) 2493-2501
ALMOST-EINSTEIN MANIFOLDS WITH
NONNEGATIVE ISOTROPIC CURVATURE
by Harish SESHADRI (*)
Abstract. — Let (M, g), n > 4, be a compact simply-connected Riemannian
n-manifold with nonnegative isotropic curvature. Given 0 < l 6 L, we prove that
there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g
satisfies
l 6 s 6 L
and the Einstein tensor satisfies ∣∣∣Ric − s
n
g
∣∣∣ 6 ε
then M is diffeomorphic to a symmetric space of compact type.
This is related to the result of S. Brendle on the metric rigidity of Einstein
manifolds with nonnegative isotropic curvature.
Résumé. — Soit (M, g), une variété riemannienne compacte simplement connexe
de dimension n > 4, à courbure isotrope positive ou nulle. Nous montrons que pour
tout 0 < l < L, il existe un ε = ε(l, L, n) qui satisfait la propriété suivante : si la
courbure scalaire s de g satisfait
l 6 s 6 L
et que le tenseur d’Einstein satisfait∣∣∣Ric − s
n
g
∣∣∣ 6 ε
alors M est difféomorphe à un espace symétrique de type compact.
Ceci est lié au résultat de S. Brendle sur la rigidité métrique des variétés d’Ein-
stein à courbure isotrope positive ou nulle.
1. Introduction
A Riemannian manifold (M, g) is said to have nonnegative isotropic cur-
vature if
R1313 +R1414 +R2323 +R2424 − 2R1234 > 0
Keywords: Almost-Einstein manifolds, non-negative isotropic curvature.
Math. classification: 53C21.
(*) This work was supported by DST Grant No. SR/S4/MS-283/05.
2494 Harish SESHADRI
for every orthonormal 4-frame {e1, e2, e3, e4}.
In the case of strict inequality above we say that the manifold has positive
isotropic curvature. Recently S. Brendle [1] proved that a compact Einstein
manifold with nonnegative isotropic curvature has to be a locally symmetric
space of compact type. In this note we relax the restriction that the metric
is Einstein to the condition that the Einstein tensor is small in norm and
obtain the following smooth rigidity result:
Theorem 1.1. — Let (Mn, g), n > 4, be a compact simply-connected
Riemannian manifold with nonnegative isotropic curvature. Given 0 < l 6
L, there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature
s of g satisfies
l 6 s 6 L
and the Einstein tensor satisfies∣∣∣Ric− s
n
g
∣∣∣ 6 ε
then M is diffeomorphic to a symmetric space of compact type.
This result was inspired by the paper of P. Petersen and T. Tao [7]
where it is proved that “almost” quarter-pinching of sectional curvatures
again leads to smooth rigidity as above. The main difference between their
conclusion and ours is that symmetric spaces of rank > 2 are allowed in
our case, while almost quarter-pinching gives only rank-1 spaces.
We remark that for any L, ε the conditions s 6 L and
∣∣Ric− sng∣∣ 6
ε can be achieved just by rescaling the metric by a large constant. In
particular, consider the connected sum Sn−1 × S1 # Sn−1 × S1 which
admits a metric with positive isotropic curvature by [5]. Rescaling this
metric gives the two bounds above. However this manifold does not support
a locally symmetric metric (irreducible or reducible) of compact type. This
is seen by observing that the fundamental group of the latter space has to
contain an abelian subgroup of finite index. Hence the lower bound on scalar
curvature is necessary. On the other hand it is not known if just positive
Ricci curvature and nonnegative isotropic curvature already imply that the
underlying compact manifold is diffeomorphic to a locally symmetric space,
even without the assumption of simple-connectivity.
We make a few remarks about the proofs. The proof of Theorem 1.1
proceeds as follows: Let (M, g) be a n-manifold satisfying the hypotheses
of Theorem 1.1. By choosing ε 6 l2n , we can ensure that the Ricci curvature
is uniformly (i.e. depending only on n, l and L) positive. An elementary
argument (Lemma 3.1) shows that we have an uniform upper bound on the
norm of the curvature tensor. One can improve this by applying the Ricci
ANNALES DE L’INSTITUT FOURIER
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flow for a short time: This gives a nearby metric with uniform bounds on
the higher covariant derivatives of the curvature tensor (Lemma 2.1). One
also has an uniform upper bound on diameter since the Ricci curvature
is uniformly positive. A theorem of Petrunin-Tuschmann also guarantees
an uniform lower bound on injectivity radius. To apply their result one
needs finite second homotopy group which is the case if (M, g) has positive
isotropic curvature. All these ingredients enable one to prove Theorem 1.1
by contradiction when (M, g) has positive isotropic curvature.
To deal with nonnegative isotropic curvature we use the results of the
author [8] and S. Brendle [1] which allow us to reduce the nonnegative case
to the positive case.
2. A smoothing lemma
Lemma 2.1. — Let (M, g) be a compact Riemannian manifold of di-
mension n. Let A > 0 be such that |R| 6 A, where R is the curvature
tensor of (M, g).
(i) There exists δ = δ(A,n) > 0 and E = E(A,n) such that the Ricci
flow starting at g is defined on the time interval [0, δ]. On this
interval
|R(t)| 6 E.
(ii) For each k = 1, 2, . . . there exists C(n, k) > 0 such that
|∇kR(t)| 6 C(n, k)
for t ∈ [0, δ].
(iii) If the injectivity radius inj(M, g) is bounded below by a constant
α > 0, we can assume that there is a β = β(α,A, n) > 0 such that
inj(M, g(δ)) > β.
(iv) If the scalar curvature of (M, g) is bounded below by a > 0 then
the scalar curvature of (M, g(δ)) is still bounded below by a.
(v) If g has positive isotropic curvature so does g(δ).
(vi) There is a D = D(A,n) such that if the norm of the Einstein tensor
Z of (M, g) satisfies |Z| 6 ε, then |Z(δ)| 6 εD
Proof. — We refer the reader to the notes of P. Topping [9] for details
in the proof below.
(i) and (ii) follow from the evolution equation for the curvature tensor
and its covariant derivatives under Ricci flow. Note that one just needs an
uniform bound on |R| for the validity of (i) and (ii).
TOME 60 (2010), FASCICULE 7
2496 Harish SESHADRI
Since |R(t)| is uniformly bounded for [0, δ] by (i), it follows directly
from the Ricci flow equation that the metrics g(t) are uniformly bi-Lipshitz
equivalent to g. More precisely, one knows that if the Ricci curvature sat-
isfies |Ric(t)| 6 c along a Ricci flow then
e−2ctg(0) 6 g(t) 6 e2ctg(0).
Hence we have (iii).
(iv) is a consequence of the fact that the infimum of scalar curvature
increases along Ricci flow.
Since Ricci flow preserves positive isotropic curvature, by the work of
Brendle and Schoen [2], we have (v).
Finally we prove (vi): Let Z := Ric− sng denote the Einstein tensor. We
have, using ∂g
ij
∂t = 2 Ric
ij ,
∂|Z|2
∂t
= ∂
∂r
(
gijgklZikZjl
)
(2.1)
= 2
(
Ricij gkl + gij Rickl
)
ZikZjl + 2
〈
Z,
∂Z
∂t
〉
.
The evolution of Z is given by
∂Z
∂t
(X,W ) = 4Z(X,W )− 2Z(W,Ric(X)) + 2
〈
R(X, .,W, .),Ric
〉
− 2
n
|Ric |2〈X,W 〉,
where Ric(X) (in the second term on the right hand side) denotes Ricci
curvature regarded as an operator. Note that the bilinear form
T (X,Y ) := −2Z(W,Ric(X)) + 2
〈
R(X, .,W, .),Ric
〉
− 2
n
|Ric |2〈X,W 〉
is identically zero if the metric is Einstein. In general, we claim that there
is a constant C1 = C1(A,n) such that
|T (X,Y )| 6 C1|Z||X||W |.
This can be seen as follows: Since Ric = Z + sng, we have
T (X,Y ) = −2Z(W,Ric(X)) + 2
〈
R(X, .,W, .), Z + s
n
g
〉
− 2
n
〈
Z + s
n
g, Z + s
n
g
〉
〈X,W 〉
= −2Z(W,Ric(X)) + 2
〈
R(X, .,W, .), Z
〉
− 2
n
|Z|2〈X,W 〉
+ 2
〈
R(X, .,W, .), s
n
g
〉
− 2s
2
n2
〈X,W 〉
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It is clear there is a constant C ′ = C ′(A,n) such that the absolute value of
the first three terms in the last expression above is bounded by C ′|Z||X||W |
for all X,W . As for the last two terms, note that
〈
R(X, .,W, .), g
〉
=
Ric(X,W ) and hence
2
〈
R(X, .,W, .), s
n
g
〉
− 2s
2
n2
〈X,W 〉 = 2s
n
Z(X,W ).
This completes the proof of the claim.
Therefore, for some C2 = C2(A,n), we have
2
〈
Z,
∂Z
∂t
〉
6 2
〈
4Z,Z
〉
+ C2|Z|2
= 4|Z|2 − 2|∇Z|2 + C2|Z|2(2.2)
6 4|Z|2 + C2|Z|2
Plugging (2.2) in (2.1) and again using the upper bound on the Riemann
curvature tensor, we get, for some C3 = C3(A,n),
∂|Z|2
∂t
6 4|Z|2 + C3|Z|2.
It follows from the maximum principle that |Z|2(t) 6 |Z|2(0)eC3t, yield-
ing the required estimate. 
3. Proof of Theorem 1.1
We begin with a simple but useful lemma. Let c ∈ R. By
K iso > c
we mean that
K iso(ei, ej , ek, el) := Rikik +Rilil +Rjkjk +Rjljl − 2Rijkl > c
for every orthonormal 4-frame {ei, ej , ek, el}.
Lemma 3.1. — Given c, C ∈ R, there exists b = b(c, C, n) such that if
(Mn, g) is a Riemannian manifold with
K iso > c, s 6 C,
then the norm of the Weyl tensor W is bounded by b:
|W | 6 b.
TOME 60 (2010), FASCICULE 7
2498 Harish SESHADRI
Proof. — The proof is similar to that of Proposition 2.5 of [5]. Note that
K iso(ei, ej , ek, el) +K iso(ei, ej , el, ek) := 2(Rikik +Rilil +Rjkjk +Rjljl).
From this it follows that 4s can be expressed as a sum of n(n−1) isotropic
curvatures as in [5]. Since we have an upper bound on s and a lower bound
on K iso, we get an upper bound b1 = b1(c, C, n) for K iso. Since
4Wijkl = 4Rijkl = K iso(ei, ej , el, ek)−K iso(ei, ej , ek, el)
the two-sided bounds on K iso give an upper bound b2 = b2(c, C, n) on
|Wijkl| = |Rijkl|. Since this holds for an arbitrary orthonormal 4-frame, we
can apply the above bound to the 4-frame{
ei,
1√
2
(ej − el), ek,
1√
2
(ej + el)
}
to get
2b2 >
∣∣W (ei, ej − el, ek, ej + el)∣∣
=
∣∣Wijkj −Wilkl +Wijkl −Wilkj∣∣
>
∣∣Wijkj −Wilkl∣∣− ∣∣Wijkl −Wilkj∣∣
Hence ∣∣Wijkj −Wilkl∣∣ 6 2b2 + ∣∣Wijkl −Wilkj∣∣ 6 4b2.
Since
∑
p Wipkp = 0, this implies that |Wipkp| 6 b3(c, C, n). Finally, to
get a bound on terms of the form |Wipip|, we repeat the above argument
i.e. observe that the bound on |Wipkp| holds for any orthonormal triple
{ep, ei, ek}. In particular it applies to
{
ep,
1√
2 (ei − ek),
1√
2 (ei + ek)
}
. Re-
peating the above steps and using
∑
i Wipip = 0, we get an upper bound
b4(c, C, n) on any term of the form Wipip. Hence |W | 6 b5(c, C, n). 
An immediate corollary of Lemma 3.1 is that an upper bound on scalar
curvature, a lower bound on isotropic curvature and an upper bound on
the norm of the Ricci tensor gives a bound on the norm of the Riemann
curvature tensor. This applies, in particular, to a metric satisfying the
hypotheses of Theorem 1.1.
Turning to the proof of Theorem 1.1, the first restriction we impose on ε
is that ε 6 l2n . This implies that the Ricci curvature is uniformly positive:
(3.1) Ric > l
2
√
n
g.
We first claim that Theorem 1.1 is true if we assume that (M, g) has
positive isotropic curvature. Then, by Micallef-Moore [4], π2(M) = 0.
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Theorem 0.4 of [7] states that the injectivity radius of a compact simply-
connected Riemannian n-manifold with finite second homotopy group,
bounded sectional curvature |K| 6 a and positive Ricci curvature Ric > bg
has a positive lower bound on injectivity radius dependent only on a, b and
n. The comment following Lemma 3.1 and (3.1) give us the required bounds
on curvature.
Remark. — If M is even-dimensional, then one has the following alter-
native proof for a lower bound on injectivity radius inj. If inj → 0, then
all the characteristic numbers of M , in particular the Euler characteristic ,
would have to vanish. On the other hand, a simply-connected Riemannian
n-manifold with positive isotropic curvature has to be homeomorphic to
the n-sphere [4]. This contradiction shows that collapse cannot occur in
even-dimensions.
Now suppose that there is no ε for which the conclusion of Theorem 1.1
holds. Then we get a sequence of (Mi, gi) of Riemannian n-manifolds, none
of which is diffeomorphic to a symmetric space of compact type, with uni-
formly bounded sectional curvatures and diameter (by Myers-Bonnet, since
(3.1) holds) and injectivity radius bounded below.
We now apply the smoothing Lemma 2.1 of the previous section to each of
the manifolds (Mi, gi). We obtain the “smoothed” Riemannian manifolds
(Mi, gi(δ)). By (ii) and (iii) of Lemma 2.1we can assume that a subse-
quence of (Mi, gi(δ)) converges in the C∞ topology to a smooth complete
Riemannian n-manifold (M, g). This manifold will have to be positive Ein-
stein (hence compact) by (vi) and (iv) of Lemma 2.1, and of nonnegative
isotropic curvature by (v). By [1], (M, g) is isometric to a symmetric space
of compact type. Hence Mi is diffeomorphic to a symmetric space of com-
pact type for large i, which is a contradiction.
Hence we have established the existence of
(3.2) εp = εp(l, L, n)
which yields the conclusion in the presence of positive isotropic curvature.
Next consider the general case of nonnegative isotropic curvature.
Lemma 3.2. — Let (Mn, g), n > 4, be a compact simply-connected
Riemannian manifold with nonnegative isotropic curvature. Suppose that
0 < l 6 sg 6 L,
∣∣∣Ricg −sg
n
g
∣∣∣
g
6 ε
for some 0 < l 6 L and ε 6 2ln .
TOME 60 (2010), FASCICULE 7
2500 Harish SESHADRI
Let (Nk, h) be an irreducible factor in the de Rham decomposition of
N . If k = 2 or 3, N is diffeomorphic to S2 or S3. If k > 4, then (N,h) has
nonnegative isotropic curvature and
0 < 2l
n
< sh 6 L,
∣∣∣Rich−sh
k
h
∣∣∣
h
6 ε.
Proof. — The statement about k = 2 or 3 follows from the description of
reducible manifolds with nonnegative isotropic curvature given by M. Mi-
callef and M. Wang (Theorem 3.1, [5]). If k > 4, note that∣∣∣Ricg −sg
n
g
∣∣∣2
g
>
∣∣∣Rich−sg
n
h
∣∣∣2
h
=
∣∣∣Rich−sh
k
∣∣∣2
h
+ k
∣∣∣sh
k
− sg
n
∣∣∣2.
Hence ∣∣∣Rich−sh
k
h
∣∣∣
h
6 ε.
and
sh >
k
n
sg −
√
kε >
l
n
√
k
(√
k − 12
)
>
2l
n
.
Moreover, since (M, g) has positive Ricci curvature, so does each irre-
ducible component and hence sh 6 sg 6 L. 
We can now complete the proof of the theorem by induction. The proof
for the first nontrivial dimension n = 4 is the same as that for the inductive
step, so we assume that the result is true in all dimensions less than n. Let
(Mn, g) be a manifold as in Theorem 1.1 with the norm of the Einstein
tensor being smaller than
εr(l, L, n) := min
{
2l
n
, ε
(2l
n
, L, 4
)
, . . . , ε
(2l
n
, L, n− 1
)}
.
If (M, g) is reducible, it is enough to prove that each irreducible com-
ponent of (M, g) is diffeomorphic to a symmetric space of compact type.
Let (Nk, h), 1 6 k 6 n − 1 be such a component. By Lemma 3.2 and the
inductive hypothesis we are done.
Suppose (M, g) is irreducible. We claim that if norm of the Einstein
tensor of g is 12εp(
l
2 , 2L, n)-small (where εp is defined by (3.2)) then we
have the desired conclusion.
By the results of [8] and [1] the following holds: A compact orientable
locally irreducible Riemannian manifold (X,h) with nonnegative isotropic
curvature is either diffeomorphic to a locally symmetric space of compact
type or we can find a metric h̄ with positive isotropic curvature as close (in
the C∞ topology) to h as we want.
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Hence if (M, g) is not diffeomorphic to a symmetric space of compact
type, we can find ḡ with positive isotropic curvature so close to g that
0 < l2 6 sḡ 6 2L,
∣∣∣Ricḡ −sḡ
n
ḡ
∣∣∣
ḡ
6 εp.
Since (M, ḡ) has positive isotropic curvature and satisfies the above
bounds, it is diffeomorphic to a symmetric space by our earlier result.
Finally we choose
ε(l, L, n) = min
{
1
2 εp
( l
2 , 2L, n
)
, εr(l, L, n)
}
.

BIBLIOGRAPHY
[1] S. Brendle, “Einstein manifolds with nonnegative isotropic curvature are locally
symmetric”, to appear in Duke Mathematical Journal.
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[6] P. Petersen & T. Tao, “Classification of almost quarter-pinched manifolds”, Proc.
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[7] A. Petrunin & W. Tuschmann, “Diffeomorphism finiteness, positive pinching, and
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[8] H. Seshadri, “Manifolds with nonnegative isotropic curvature”, To appear in Com-
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[9] P. Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note
Series, vol. 325, Cambridge University Press, Cambridge, 2006, x+113 pages.
Manuscrit reçu le 3 juillet 2009,
accepté le 27 avril 2010.
Harish SESHADRI
Indian Institute of Science
Department of Mathematics
Bangalore 560012 (India)
harish@math.iisc.ernet.in
TOME 60 (2010), FASCICULE 7


Almost-Einstein manifolds with nonnegative isotropic curvature

Heu, Viktoria

Universal isomonodromic deformations of meromorphic rank 2 connections on curves

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