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Exotic spheres and curvature. (English) Zbl 1149.53020
Manifolds which are homeomorphic but not diffeomorphic to a standard sphere are called exotic spheres. In dimensions one, two, and three it is well known that every topological manifold admits a unique smooth structure. The lowest dimension in which exotic smooth structures exist is dimension four. Many four-dimensional manifolds are known to admit infinitely many distinct smooth structures. However, this phenomenon cannot occur in higher dimensions. In dimensions at least five, any manifold has at most a finite number of smooth structures, and in many cases this number is one.
In [Ann. Math. (2) 64, 399–405 (1956; Zbl 0072.18402)], J. Milnor showed that spheres, specifically \(S^ 7\), were the first manifolds admitting exotic smooth structures. M. Kervaire and J. Milnor [Ann. Math. (2) 77, 504–537 (1963; Zbl 0115.40505)] were able to show how large the family of exotic spheres is in each dimension greater than four. Although in each case this number is finite, there are exotic spheres in infinitely many dimensions, and in dimensions \(4n + 3\) the number of exotic spheres grows very rapidly with \(n\). The question of whether there is an exotic \(S^ 4\), and if so, how many there are is still open today. This is essentially the smooth Poincaré conjecture in dimension four.
A general approach to describing exotic spheres is provided by the twisted sphere construction. A twisted sphere is a manifold constructed by gluing two discs \(D^ n\) along their boundaries using an orientation preserving diffeomorphism \(S^{n-1}\to S^{n-1}\). Such a manifold is homeomorphic to \(S^ n\). In dimensions other than four, every exotic sphere is a twisted sphere. In dimension four the only twisted sphere is the standard sphere, so if an exotic \(4\)-sphere exists, it cannot be a twisted sphere.
Another term used when discussing exotic spheres is a homotopy sphere that is a smooth manifold with the same homotopy type as \(S^ n\). Thus every exotic sphere is a homotopy sphere. In dimensions at least five, any smooth manifold with the homotopy type of a sphere must be homeomorphic to a sphere. This is the Generalised Poincaré Conjecture, proved by S. Smale in [Ann. Math. (2) 74, 391–406 (1961; Zbl 0099.39202)]. Thus in these dimensions the set of diffeomorphism classes of homotopy spheres is precisely the union of the diffeomorphism class of the standard sphere with the diffeomorphism classes of the exotic spheres.
In dimensions one and two, homotopy spheres are always diffeomorphic to the standard spheres. In dimension three, the differentiable structure on \(S^ 3\) is unique which means that there is no exotic \(S^ 3\). Moreover, every \(3\)-manifold has a unique differentiable structure, and any two homeomorphic \(3\)-manifolds are necessarily diffeomorphic. On the other hand, it was not known if a homotopy \(S^ 3\) was necessarily homeomorphic, and therefore diffeomorphic, to \(S^ 3\). This is the famous Poincaré conjecture, which has recently been affirmatively resolved. Thus any homotopy \(3\)-sphere is diffeomorphic to \(S^ 3\). In dimension four, any homotopy \(4\)-sphere is homeomorphic to \(S^ 4\) but the question of whether such a manifold is necessarily diffeomorphic to \(S^ 4\) still remains open.
In this paper, the authors survey what is known about the curvature of exotic spheres. Any smooth manifold can be equipped with a smooth Riemannian metric. A Riemannian metric endows the manifold with shape, so it makes sense to try and quantify the curvature of such an object. The most widely known measure of curvature is the Gaussian curvature in dimension two. If the Gaussian curvature at a point is positive, then the geometry in a neighborhood of the point resembles that of a sphere, if the Gaussian curvature is zero, the local geometry is that of a Euclidean plane, and if the Gaussian curvature is negative, the local geometry resembles that of a saddle.
Every closed compact surface admits a metric with constant Gaussian curvature. This, combined with the Gauss-Bonnet Theorem shows that the only closed compact surfaces admitting a metric of strictly positive Gaussian curvature are the \(2\)-sphere and the real projective plane. The only closed compact surfaces admitting a metric of identically zero Gaussian curvature are the torus and Klein bottle. The remaining closed compact surfaces admit a metric of strictly negative Gaussian curvature.
The simplicity of this topology-curvature relationship in dimension two is not mirrored in higher dimensions. The first natural generalization of the Gaussian curvature is the notion of sectional curvature. The next curvature to be introduced is the Ricci curvature, and finally the scalar curvature. The standard \(n\)-dimensional sphere of radius \(R\) has constant sectional curvature \(1/R^ 2\), Ricci curvature \((n-1)/R^ 2\), and scalar curvature \(n(n-1)/R^ 2\). By studying the curvature of exotic spheres, the authors gain some insight into the extent to which the geometry of these objects resembles that of the standard sphere.

MSC:
53C20 Global Riemannian geometry, including pinching
57R60 Homotopy spheres, Poincaré conjecture
57R55 Differentiable structures in differential topology
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