Sectional Curvature - Manifolds With Positive Sectional Curvature

Manifolds With Positive Sectional Curvature

Little is known about the structure of positively curved manifolds. The soul theorem (Cheeger & Gromoll 1972; Gromoll & Meyer 1969) implies that a complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold. As for compact positively curved manifolds, there are two classical results:

• It follows from the Myers theorem that the fundamental group of such manifold is finite.

• It follows from the Synge theorem that the fundamental group of such manifold in even dimensions is 0, if orientable and otherwise. In odd dimensions a positively curved manifold is always orientable.

Moreover, there are relatively few examples of compact positively curved manifolds, leaving a lot of conjectures (e.g., the Hopf conjecture on whether there is a metric of positive sectional curvature on ). The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if is a Riemannian manifold admitting a free isometric action of a Lie group G, and M has positive sectional curvature on all 2-planes orthogonal to the orbits of G, then the manifold with the quotient metric has positive sectional curvature. This fact allows one to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples (Ziller 2007):

• The Berger spaces and .

• The Wallach spaces (or the homogeneous flag manifolds):, and .

• The Aloff–Wallach spaces .

• The Eschenburg spaces

• The Bazaikin spaces, where .