Mathematics – Differential Geometry
Scientific paper
1999-08-29
Mathematics
Differential Geometry
AMS-TeX, 4 pages
Scientific paper
Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 which admit a Riemannian metric with sectional curvature bounded in absolute value by C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. Thus in these dimensions the lower positive bound on volume in Cheeger's Finiteness Theorem can be replaced by a purely topological condition, simply-connectedness. In dimension 4 instead of simply-connectedness here only non-vanishing of the Euler characteristic has to be required. As a topological corollary we obtain that for k+l<7 there are over a given smooth closed l-manifold only finitely many principal $T^k$ bundles with simply connected and non-diffeomorphic total spaces. Furthermore, for any given numbers C and D and any dimension m it is shown that for each natural number i there are up to isomorphism always only finitely many possibilities for the i-th homotopy group of a simply connected closed m-manifold which admits a metric with sectional curvature bounded in absolute value by C and diameter bounded from above by D.
No associations
LandOfFree
Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-635771