ISSN:
1432-1785
Keywords:
Mathematics Subject Classification (1991):Primary 51K99; Secondary 51M10, 51F20
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s002290050055
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