Curvature and Symmetry of Milnor Spheres
ثبت نشده
چکیده
Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi] . On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He],[Po], [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with non-negative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:
منابع مشابه
Exotic Spheres and Curvature
Since their discovery by Milnor in 1956, exotic spheres have provided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres.
متن کاملA ug 2 00 0 HIGHER ORDER INTERSECTION NUMBERS OF 2 - SPHERES IN 4 - MANIFOLDS
This is the beginning of an obstruction theory for deciding whether a map f : S → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4-manifold X . The first obstruction is Wall’s well known self-intersection number μ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is d...
متن کاملCurvature collineations on Lie algebroid structure
Considering prolongation of a Lie algebroid equipped with a spray, defining some classical tensors, we show that a Lie symmetry of a spray is a curvature collineation for these tensors.
متن کاملHigher order intersection numbers of 2-spheres in 4-manifolds
This is the beginning of an obstruction theory for deciding whether a map f : S → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall’s self-intersection number μ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if μ(f) vanishes and has formally v...
متن کاملHigher order intersection numbers of 2 - spheres in 4 - manifolds Rob Schneiderman
This is the beginning of an obstruction theory for deciding whether a map f : S → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall’s self-intersection number μ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if μ(f) vanishes and has formally v...
متن کامل