Asian Journal of Mathematics
Volume 12 (2008)
Impossible Metric Conditions on Exotic R4's
Pages: 285 – 288
There are many theorems in the differential geometry literature of the following sort.
Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary.
At first glance it seems unlikely that such theorems have anything to say about smooth manifolds homeomorphic to R4. However, there is a common theme to all the proofs which forbids the existence of such metrics on most (and possibly all) exotic R4's.
Exotic 4-spaces; vector fields
2010 Mathematics Subject Classification
Primary 57N13. Secondary 57R70.