Abstract
A. Cayley [5] and F. Klein [12] were the first mathematicians who introduced the notion of a metric in a real projective space, by specializing a set of quadrics (called the absolute). We show in this paper, that from an algebraic point of view these projective spaces can be described by vector spaces, in which a metric structure is given by one or more symmetric bilinear forms. We study these Cayley- Klein vector spaces and their groups of automorphisms (orthogonal Cayley- Klein groups) over arbitrary commutative fields of characteristic ≠ 2.
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