[FOM] Re: Soare's article in the BSL
vladik at cs.utep.edu
Mon Jan 31 13:27:59 EST 2005
This part is indeed written somewhat confusingly.
The correct definition is given, e.g., in the 2003 NW paper in Geometrica
Dedicata: A1 is the closure of the set of all the metrics from
Met(M)=Riem(M)/Diff(M) for which the curvature is always bounded (in absoluyte
value) by 1 (the actual notation in [NW] is A1 (A one) not Al (A el) but of
course 1 and l are easily confused in printed English :-)
The confusion can be traced back to Definition 7.1 of the same section (p.20 of
the web-posted softcopy), which erroneously incorporates sectional curvature in
the definition of manifold. A manifold itself does not have a matric yet, so we
cannot talk about its curvature. Riem(M) is the set fo all Riemann metrics on M
9whatever their curvature), and then we go to A1 to cut to those Riemann
metrics for which curvature is bounded by 1.
> From: "Timothy Y. Chow" <tchow at alum.mit.edu>
> One point where I got lost was Definition 7.3, the definition of Al(M).
> I don't understand the definition:
> Define Al(M) to be the subset of Met(M) consisting of the
> closure in the Gromov-Hausdorff metric of isometry classes
> of metrics (in Riem(M)/Diff(M).
> Isn't everything in Met(M) an isometry class of a metric?
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