Congruence criteria for finite subsets of complex
projective and complex hyperbolic spaces |
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Authors: | Ulrich Brehm Boumediene Et-Taoui |
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Institution: | Institut für Geometrie, Technische Universit?t Dresden, D-01062 Dresden, Germany.?e-mail: brehm@math.tu-dresden.de, DE Laboratoire de Mathématiques, Université de Haute Alsace, 4, rue des Fréres Lumière, F-68093 Mulhouse Cédex, France, FR
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Abstract: | We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP
n
. For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish
a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP
n
and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of
the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by
congruence classes or if distances π/2 are allowed. Finally we do the same
kind of investigation also for the complex hyperbolic space ℂH
n
. Most of the results are completely analogous, however, there are also some interesting differences.
Received: 15 January 1996 |
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Keywords: | Mathematics Subject Classification (1991):Primary 51K99 Secondary 51M10 51F20 |
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