A fundamental problem in Differential Geometry is the following:
Problem. Determine the topology of a given manifold in terms of its geometry.
Of course this problem is vaguely stated, but the following result provides a beautiful example of what a good answer to this problem should be:
Theorem (Sphere theorem). Let be a compact simply-connected manifold admitting a Riemannian metric whose sectional curvatures verify
Then, is homeomorphic to the sphere .
Concerning the statement of this theorem, let us recall that a Riemannian metric whose sectional curvatures satisfy
are called –pinched (resp. strictly –pinched). Therefore, the sphere theorem can be reformulated as: ”Any compact, simply-connected, strictly -pinched manifold is homeomorphic to the sphere”.
The sphere theorem has a long history which can be summarized as follows: in 1951, Rauch showed that compact simply-connected, strictly -pinched manifold are homeo- morphic to for . After that, Klingenberg proved the same result for when the dimension of is even and Berger improved Klingenberg’s result to but still assuming that the dimension of is even. Finally, in 1961, Klingenberg extended Berger’s result to odd dimensions so that the sphere theorem was proved in full generality.
Remark 1. The sphere theorem is false (in even dimensions) if the strictly -pinching condition is replaced by -pinching: in fact, the complex projective spaces are compact, simply-connected admitting -pinched metrics but is not homeomorphic to (for ). However, Berger proved that the complex projective spaces are (morally) the ”unique” type of counter-examples: more preci- sely, any compact simply-connected -pinched manifold is homeomorphic to the sphere or isometric to a symmetric space.
Remark 2. From Berger’s theorem quoted in the previous remark, it follows that the sphere theorem is true in odd dimensions if the strictly -pinching is replaced by -pinching. However, we ignore whether the sphere theorem in odd dimensions can be improved (i.e., -pinching replaced by -pinching with ), although Berger showed that the sphere theorem in even dimensions holds assumung -pinching (for sufficiently small ).
Remark 3. In two and three dimensions, the sphere theorem holds assuming only strictly zero-pinching, i.e., any compact, simply-connected manifold () with positive sectional curvatures are diffeomorphic (in particular homeomorphic) to the sphere. In fact, the bidimensional case follows from Gauss-Bonnet theorem and the three-dimensional case was proved by R. Hamilton using the Ricci flow (we’ll come back to the application of Hamilton’s Ricci flow to the geometrization of manifolds later).
A quick sketch of the proof of the sphere theorem goes as follows: firstly, one shows that the hypothesis of the sphere theorem implies that the injectivity radius of satisfies (this is the so-called Klingenberg’s injectivity radius estimate). Secondly, one applies this injectivity radius estimate to show that the assumption implies
where and . In other words, under the hypothesis of the sphere theorem, we can write as a union of two balls. By well-known arguments in topology, this implies that is homeomorphic to the sphere.
A natural question related to the statement of the sphere theorem is: can we replace ”homeomorphic” by ”diffeomorphic”?
Remark 4. The previous argument presents as a union of two balls glued along their common boundary so that is homeomorphic to . However, it is known from the works of Milnor that there are exotic structures on , i.e., certain manifolds which are homeomorphic to but not diffeomorphic to . Thus, the ”proof” of the sphere theorem presented here does not allows to obtain a diffeomorphism between and in general.
The statement of the sphere theorem with the word ”homeomorphic” changed by ”diffeomorphic” is called differentiable sphere theorem:
Differentiable Sphere Theorem. Any compact, simply-connected, strictly -pinched manifold is diffeomorphic to the sphere .
This theorem was proved assuming -pinching for close to by Gromoll and Calabi with (in 1966). After that, in 1971, Sugimoto and Shiohama, Karcher, Ruh proved this result with independent of (in fact, ). Nevertheless, in 1991, Chen proved the differentiable sphere theorem in dimension 4 using the Ricci flow.
Finally, S. Brendle and R. Schoen (2007) proved the differentiable sphere theorem in the general case using the Ricci flow. During the rest of this post, we will describe some important points of Brendle-Schoen’s argument. Roughly speaking, the argument is based on the following two contributions. Firstly, in 1988, Micallef and Moore introduced PDE methods to the sphere theorem so that they gave a proof of this result using the (partial) Morse theory of the energy functional of maps from to . In doing so, they introduced the important notion of positive isotropic curvature (PIC) and they observed that strict -pinching implies PIC, so that they could work just with the weaker notion of PIC to get their sphere theorem. Secondly, a recent theorem of Bohm and Wilking (2006) proved that manifolds with positive sectional curvature are space forms using a new method of construction and deformation of ”pinching sets” for an ODE associated to the Ricci flow. Basically, Bohm and Wilking shows that positive -curvature is preserved under the Ricci flow and, moreover, this curvature condition can be ”deformed” to the constant sectional curvature condition via a family of pinching cones which are invariant by the action of the Ricci flow.
After this quite vague description of Brendle and Schoen scheme, let us describe the program of this post. In the next section, we recall the definition of Ricci flow and we state the result of proposition 10 of Brendle and Schoen paper, namely, (weakly) PIC is preserved by the Ricci flow. Then, we dedicate the subsequent section to a brief explanation of Bohm and Wilking construction of pinching sets from a given (Ricci flow invariant) curvature condition. Here we take the opportunity to say that we plan to write another post to clarify the powerful method of Bohm and Wilking. Finally, in the last section, we show that the previous results about PIC manifolds implies the differentiable sphere theorem.
–(Weakly) PIC is preserved by the Ricci flow–
We recall that the unnormalized Ricci flow is given by
Using the method of moving frames, Hamilton proved that the curvature operator of satisfies
where and is the adjoint representation. Here we are identifying with . In an orthonormal frame we have
Next, we recall Micallef and Moore’s notion of positive isotropic curvature (PIC): we say that the curvature tensor has PIC if and only if for any orthonormal 4-frame it holds
Also, we say that is weakly PIC if
Remark 5. This condition appears naturally when dealing with the study of the stability of minimal surfaces in Riemannian manifolds. Therefore, the PIC condition is ”two-di- mensional” analogous of the positive sectional curvature condition in the context of the stability of geodesics in Riemannian manifolds. See Micallef and Moore’s paper for mo- re details.
In this context, Brendle and Schoen shows that the weakly PIC condition is preserved by the Ricci flow. To do so, they start with Hamilton’s result saying that it suffices to show that weakly PIC is preserved by the ODE . Thus, our task is reduced to show the following fact:
Proposition 1 (proposition 10 of Brendle and Schoen). Assume that is a solution of the ODE on the time interval . If has weakly PIC then has weakly PIC for all .
Proof. Fix and denote by the solution of the ODE
with initial condition . Observe that is defined on a time interval such that
Moreover, for any , we have . Hence, in order to complete the proof of the proposition, it suffices to show that has PIC for all . The argument will proceed by contradiction. Assume that there exists such that does not have PIC and put
By definition, and has weakly PIC. Furthermore, there exists an orthonormal 4-frame such that
Using this condition in a clever way (after some calculations), Brendle and Schoen were able to prove that this last identity implies that
See the section 2 (and specially the corollary 9) of Brendle and Schoen’s paper. Thus, from the ODE , we get
Hence, we conclude that there exists a time so that
a contradiction with the fact that has PIC for all . This completes the proof of the proposition.
Once we know that the (weakly) PIC condition is invariant under the Ricci flow, we turn to a quick description of Bohm and Wilking construction of pinching sets.
–Bohm and Wilking pinching cones–
To avoid certain unpleasant analytical technicalities, we will not work directly work with the PIC condition on the manifold but instead we consider the PIC condition on . More precisely, given a curvature operator on , we denote by the curvature operator on given by
where stands for the canonical projection. We consider
Some elementary properties of are listed below:
Lemma 1. It holds
- every curvature operator has non-negative sectional curvature;
- any non-negative curvature operator lies in ;
- any positive curvature operator lies in the interior of .
Proof. For the proof of item 1, take and an orthonormal 2-frame of . We can extend this 2-frame to a 4-frame of such that and . The weakly PIC condition on applied to this 4-frame means that
It follows that has non-negative sectional curvature, so that the item 1 is proved.
Next, we show the item 2: given a non-negative curvature operator on , we have that is non-negative curvature operator. In particular, has weakly PIC, i.e., .
Finally, item 3 is a direct consequence of item 2.
The properties of the cone stated in lemma 1 allows to the application of the powerful method of Bohm and Wilking. To explain how this technique works, we need the concept of pinching families of cones of curvature operators:
Definition 1. A continuous family of closed convex -invariant cones (with non-empty interior) of curvature operators is called a pinching family whenever:
- any has positive scalar curvature,
- is contained in the interior of the tangent cone of at for all and ,
- converges (in the Hausdorff topology) to the one-dimensional cone containing the sphere curvature operator .
The basic motivation for the introduction of this notion becomes more clear from the following theorem:
Theorem 1 (theorem 5.1 of Bohm and Wilking). Let be a pinching family and be a Riemannian manifold whose curvature operator belongs to the cone . Then, the normalized Ricci flow starting at exists for all and converges to a metric of constant sectional curvature.
Let us point out that the statement of this theorem is somewhat natural (although it is far from trivial): indeed, from the results of R. Hamilton, we know that it suffices to study the ODE in order to get good properties of the Ricci flow. On the other hand, in the definition of pinching family, we require that the vector field (associated to this ODE) stays inside the interior of our family of cones. In particular, if one start with a curvature operator inside the initial cone , it is reasonable to expect that the evolution of the ODE points towards the interior of the cones , so that we are going to see the Ricci flow converging to the one-dimensional cone (by the third item of definition 1) which is exactly the cone of curvature operators with constant sectional curvature. Of course, there are plenty of details to be checked in this program (of Bohm and Wilking). We plan to discuss this beautiful theorem in a future post. For the moment being, we will assume this result and go back to the discussion of the differentiable sphere theorem.
In view of the theorem 1 of Bohm and Wilking and the properties of the cone stated in lemma 1 and proposition 1, a natural strategy to settle the differentiable sphere theorem involves the construction of a pinching family , starting at (since we know that any manifold carrying a metric of positive sectional curvature is diffeomorphic to the sphere). It turns out that this is the situtation as the following result shows:
Theorem 2 (Brendle and Schoen, section 3). There exists a pinching family , starting at .
This result corresponds to the propositions 12, 13 and 14 of Brendle and Schoen paper. The idea of the proof of this theorem is essentially contained in Bohm and Wilking article and it can be quickly described as follows: firstly one consider a suitably defined family of linear transformations on the space of curvature operators connecting a given curvature operator to the sphere curvature operator . More precisely,
where is the Ricci tensor and is the scalar curvature of . The form of is designed so that the pullback of the vector field by differs from only by a certain operator depending only on the Ricci tensor. Since the Ricci tensor is known to be ”well-behaved” under the Ricci flow, there is some hope that a well-chosen subfamily of is a pinching family. Due to limitations of space, we will not detail this idea but we refer to section 3 of Brendle and Schoen paper for a more elucidative discussion (and also our future post about the results of Bohm and Wilking). Anyway, we will indicate below the explicit pinching family of cones containing in terms of :
Again, this specific choice of parameters can be explained by the analysis of the eigenvalues of the difference of the -pullback of and itself (in order to eliminate the Weyl part of this operator) but we will postpone this discussion for a future post.
Combining the theorem 1 of Bohm and Wilking with the theorem 2 of Brendle and Schoen, we obtain
Theorem 3 (theorem 17 of Brendle and Schoen). Let be a compact Riemannian manifold such that the curvature tensor of belongs to the interior of the cone . Then, the normalized Ricci flow starting at converges to a metric of (positive) constant sectional curvature when .
At this point, we are able to apply this theorem (after some linear algebra) to complete the proof of the differentiable sphere theorem. This will be performed in the next section.
–End of the proof of the Differentiable Sphere Theorem–
We start with a linear algebra lemma:
Lemma 2 (lemma 19 of Brendle and Schoen). Let be an orthonormal 4-frame in . Then, there exists an orthonormal 4-frame in and real numbers with such that
We refer the reader to the Brendle and Schoen article for the short proof of this result. Using this lemma, we can charactize the weakly PIC condition for in the following way:
Proposition 3 (proposition 20 of Brendle and Schoen). Let be a curvature operator on and denote by be the induced curvature operator on . The following two properties are equivalent:
- has weakly PIC.
- for any two bi-vectors of the form , where is an orthornomal 4-frame and $a_1a_2=b_1b_2$.
Proof. Assume that has weakly PIC. Given of the form described in item 2, we may suppose (without loss of generality) that and . We define
It is not hard to check that is an orthonormal 4-frame in . Using the assumptions $a_1a_2=b_1b_2=b_1$ one gets
Since has weakly PIC, we obtain
This proves that item 1 implies item 2.
Conversely, assuming the validity of item 2, we take an orthonormal 4-frame in . By the lemma 2, one can find a orthonormal 4-frame in and real numbers such that ,
Hence, if we put
we conclude that (from the assumption of item 2)
This shows that item 2 implies item 1 so that the proof is complete.
This particular characterization of weakly PIC for allows us to show that strictly -pinched metrics belongs to the interior of the cone (of curvature operators such that has weakly PIC):
Corollary 1. Assume that the sectional curvatures of are -pinched (resp. strictly -pinched). Then, has weakly PIC (resp. PIC).
Proof. From the previous proposition, it suffices to prove that for any two bi-vectors of the form described in item 2 of this proposition. Using the fact and the first Bianchi identity, we get
At this point, we recall Berger’s inequality for pinched curvature operator:
For a short proof of this inequality, see the paper of Karcher. Applying Berger’s estimate to the previous identity, we get
This ends the proof of the corollary.
At this stage, the reader surely noticed that the proof of the differentiable sphere theorem is complete. Indeed, from the corollary 1, any strictly -pinched metric possesses a curvature operator satisfying . Thus, from theorem 3, we know that the Ricci flow of any such metric converges to a round metric (i.e., a metric with (positive) constant sectional curvature). In particular, since we are assuming that is simply-connected, it follows that is diffeomorphic to the sphere.
Closing this post, let me point out Brendle and Schoen refined their technique to include the classification of the topology of -pinched metrics (and not just strictly -pinched spaces). Namely, they strenght Berger’s result (quoted in remark 2 above) from ”homeomorphic” to ”diffeomorphic”: any -pinched manifold is diffeomorphic to the sphere or it is isometric to a symmetric space (e.g., complex projective spaces). The curious reader can find more information about this result in the Brendle and Schoen paper ”Classification of manifolds with -pinched curvatures”.
That’s all folks! See you again soon (perhaps in the discussion of Bohm and Wilking paper)!
[Update – July 14, 2008: Peter Petersen and Terence Tao recently posted in the arxiv the paper ”Classification of Almost Quarter-Pinched Manifolds” where they employ the techniques of S. Brendle and R. Schoen to show that, for any dimension , there exists a constant such that every compact simply-connected -manifold with a -pinched metric is diffeomorphic to a sphere or a (rank 1) symmetric space. This theorem improves some previous results of Berger, Abresch and Meyer where the same statement is proved with the word ”diffeomorphic” replaced by ”homeomorphic” and it covers some cases not studied by Brendle and Schoen.]