Equiangular lines in Cr

A subset S of a complex projective space is F-regular provided each two points of S have the same non zero distance and each subset of three points of S has the same shape invariant. An important problem in the distance geometry of a complex projective space CPr-1 is the study of its F-regular subsets, and the determination of the largest integer n(r) such that CPr-1 contains an F-regular subset of n(r) points. In this paper, it is established that r + 1 less than or equal to n(r) less than or equal to 2r for any r and n(2(k)) = 2(k+1) for any integer k. In particular, we get that the maximum number of equi-isoclinic planes in the Euclidean space R2k+1 with the parameter 1/(2k+1-1), is equal to 2(k+1).