Invariant Distances Related to the Bergman Function

Let D be a bounded domain in C". The invariant distance in D is given by I I v I \ is l \\ '/2\ l/2 Kp(z,i)K„(w,w) It is shown that one half of the length of a piecewise C1 curve y: [a, b] -* D with respect to the Bergman metric is equal to the length of y measured by pD, which implies that the associated inner distance p*D coincides (up to the factor {) with the Bergman-distance. Also it was proved that pD is not an inner distance. Introduction. Let D be a bounded domain in C". A distance function p: D X D [0, oo) can be used to define the length of a curve y: [a, b] —> D by the formula n-l iP{y) = sup L p(y('/).y('/+i))> i-O where a = t0 < tY • • ■ < r„ = b are arbitrary points on the segment [a, b]. For p, q g D let us define p*( p, q) as the greatest lower bound of the length of all curves in D which join p and q. Obviously p* 3* p. The distance p is called inner if p* = p. The Bergman function KD(z, w), z,w e D (see [1]), is related to the distance f [KD(z,W)KD(W,zy Pd(z>w) = 1 TT-7-T7T-7-T \ \^d(2, 2)A:d(w,w) which induces the euclidean topology in £), and is invariant under biholomorphic mappings [6, 7]. We shall also consider the Bergman distance dD{p, q), p, q g D, defined as the greatest lower bound of the Bergman length of all piecewise C1 curves in D which join p and q. The Bergman length /ß(y) of y: [a, b] -* D is defined using the Bergman metric tensor N ds2= £ y./f-l (32log KD(z,z) d2log KD(z,z) -r-^—-dz, ® dzk -\-^^-dz: ® rfzA 3zy.3zt ' dzßzk i Received by the editors January 20, 1984. 1980 Mathematics Subject Classification. Primary 32H15. f'1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 72 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use INVARIANT DISTANCES RELATED TO THE BERGMAN FUNCTION 73 Namely y* ds2 = g(t)dt ® dt with g(t) > 0, and 1/2 (i) iB(y) = f g(')l/i Ja ^ 92log/Vg(Y(Q,Y(0)T^TT(A