[FOM] Re: Soare's article in the BSL

Vladik Kreinovich 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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