Appendix G. Elementary Aspects of Metric Geometry
I'm still at it, After-mathematics.- From "Still D.R.E." by Dr. Dre featuring Snoop Dogg
When studying a class of objects, even if one is only interested in the
properties of objects in this class, it is often useful to enlarge the class.
In metric geometry, enlarging the class of Riemannian manifolds, we shall
consider the notions of metric spaces, length spaces, and Aleksandrov spaces.
One property which we usually wish the enlarged class of geometric objects
to have is that of (pre) compactness. In addition, we wish to study the
geometric properties of objects in the enlarged class. In this appendix, we
discuss enlargements of the class of smooth manifolds and their properties.
In contrast, most of the other topics in this book series may be classified as
parts of geometric analysis (i.e., analysis on smooth manifolds).
With this in mind, we remark that many of Perelman's (ingenious)
arguments bring the synthesis of traditional geometric analysis and met-
ric/ comparison geometry to a higher level. As in other areas of mathemat-
ics, one may anticipate a further broadening and deepening of this synthesis.
We provide two quotations which reflect some motivations for the inclusion
of the metric geometry in this appendix.
On p. xiii of [18] it is written:
'... it was a common belief that "geometry of manifolds"
basically boiled down to "analysis on manifolds".. .. It is now
understood that a tremendous part of geometry essentially
belongs to metric geometry, ... '
Cheeger and Grove have written (seep. vi of [34]):
'Thus, it seems that distinctions such as "metric geometry"
versus "geometric analysis" are to some extent artificial and
if pressed too far, are genuinely destructive. To reiterate,
increasingly, the solution of specific geometric problems re-
quires a mixture of synthetic, analytic and topological argu-
ments ... the work of Perelman (on the program originated by
Hamilton) being just one, albeit spectacular, example. This
circumstance can only make the subject more interesting.'
Metric geometry is now an integral part of the study of Ricci fl.ow. Here
we only provide a cursory introduction to this subject. Excellent reference
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