Encyclopedia of Distances Review

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Encyclopedia of Distances ReviewAt first quite impressed by the scope of this work, I was soon disappointed by a certain lack of rigor and attention to detail. Well but rather unevenly written, the text intermixes usually clear definitions with slightly confused explanatory passages (e.g., p.148 where the complex dimension of a complex manifold is reused as its real dimension, confusing the discussion of the complex tangent bundle). Occasionally there are also blatantly false statements with which the reader must contend. For example, I spent some time puzzling over this sentence on p.155: "In fact, a manifold is Kobayashi hyperbolic if and only if it is biholomorphic to a bounded homogeneous domain." I still can't imagine what result the authors managed to twist into this patently absurd statement.
Some bad definitions seem to have crept in based on incorrect readings of abstracts available on the web. For example, a Sasakian metric is normally defined as a normal contact metric, i.e., a Riemannian metric on a contact manifold for which a certain cone construction yields a Kahler manifold with Kahler form related to the lift of the contact form. Yet on p.160 the authors define such a metric to be "a [Riemannian] metric of positive scalar curvature on a contact manifold, naturally adapted to the contact structure." If a Sasaki metric is also Einstein (in which case it's called "Sasakian-Einstein"), then its scalar curvature is positive, but this condition does not hold in general. (Ignore the grammatical error in the following sentence.)
While Hilbert's projective distance for convex bounded Euclidean domains makes an appearance, no mention is made of Kobayashi's generalization to manifolds with a projective connection or its holomorphic projective analog. Similarly in the new section on Cosmology and Relativity, redshift plays a prominent role, but the natural generalization of the usual redshift-based distance on Einstein-de Sitter space to an intrinsic pseudodistance on conformal Lorentz manifolds is not mentioned. The definition of the Kobayashi holomorphic distance and the discussion of related concepts is limited to complex manifolds; recent work on its extension to almost complex manifolds and complex spaces is omitted.
To become truly "encyclopedic," the next edition should correct these deficiencies, go into a little more depth on the most important topics, and possibly undergo peer review on a subject-by-subject basis to minimize inaccuracies.
Still, the current edition does appear useful as a large collection of definitions and terms that may be used to launch fruitful web searches. I also found its summaries of fields outside my immediate areas of interest quite entertaining.Encyclopedia of Distances Overview

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