On the Riemann’s relation on symmetric open Riemann surfaces

L et W be an arbitrary Riemann surface and {F n } its exhaustio n b y regu lar regio n s, th en th ere exists o n W a canonical homology basis {A, B ,} of A-type with respect to { F " } such that A, ,B1 , A le n ) , 13, ( ") form a canonical homology basis of F" mod OF (A h lfo rs (3 )).W e say that such a basis belongs to C .H .B .(F ) A and we denote it by {A i , Bi } E C .H .B .(Fn ) A .Let F,, be the Hilbert space of square integrable harmonic differentials defined on W and F2 be its subspaces.Definition 1 (Accola (1)).W e say th at th e special bilinear relation holds between (01 an d (02 i f w e have (1 , 1 ) ( c o i , (1 ) 2 * ) = lim E ( k " W 2 (0,) (a f inite sum) k----1 A Bk A k Bk f or (02 and (0,((0, with only a finite number of non vanishing Periods) where (02 * denotes the conjugate differential o f (02 .In the same way w e say that the special bilinear relations hold between F , an d F 2 i f the special bilinear relations hold betw een all (01 E r an d (02 E 1' 2 .Definition 2 (Accola (1)).W e say th at the bilinear relations in Accola's sense hold between 1 -", and (02 w ith respect t o { F " } and IA " B i l E C .H .B .(F")A if w e have koo (1,2) ( 0 ) , ( 0 2 * ) lirn r E ( ,Ç (7,2- i l k (7)2 B k W 1 ) ' -*°" At Bk ,