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A Generalized Cross Ratio 3 A GENERALIZED CROSS RATIO MICHAEL R. PILLA [email protected] SWAIN HALL EAST 205 729 EAST 3RD ST. BLOOMINGTON, IN 47405 Abstract. In one complex variable, the cross ratio is a well-known quantity associated with four given points in the complex plane that remains invariant under linear fractional maps. In particular, if one knows where three points in the complex plane are mapped under a linear fractional map, one can use this invariance to explicitly determine the map and to show that linear frac- tional maps are 3-transitive. In this paper, we define a generalized cross ratio and determine some of its basic properties. In particular, we determine which hypotheses must be made to guarantee that our generalized cross ratio is well defined. We thus obtain a class of maps that obey similar transitivity proper- ties as in one complex dimension, under some more restrictive conditions. 1. Background. In 1872, Felix Klein famously introduced a point of view regarding what geometry should be about [4]. Known as the Erlangen program, Klein viewed geometry as a study of invariants under group transformations. An important and robust example of such an invariant quantity is given by the cross ratio. Given four finite distinct points z1, z2, z3, and z4 in the complex plane, the cross ratio is defined as (z1 − z3)(z2 − z4) (1) (z1,z2,z3,z4)= . (z1 − z4)(z2 − z3) If zi = ∞ for some i =1, 2, 3, 4 then we cancel the terms with zi. For example, if z1 = ∞, then arXiv:2106.10570v1 [math.CV] 19 Jun 2021 (z2 − z4) (z1,z2,z3,z4)= . (z2 − z3) With this definition, we may view the cross ratio as a function of z given by (z,z1,z2,z3). Recall that a linear fractional map, also known as a Mobius transfor- mation, is defined as az + b f(z)= cz + d where the coefficients a, b, c, and d are complex numbers such that ad − bc 6= 0 (otherwise f(z) is a constant function). 2010 Mathematics Subject Classification. 32A08. Key words and phrases. LINEAR FRACTIONAL MAPS, CROSS RATIO, TRANSITIVITY. 1 2MICHAEL R. PILLA [email protected] SWAIN HALL EAST 205 729 EAST 3RD ST. BLOOMINGTON, IN 47405 It is well known that the cross ratio is preserved under linear fractional maps. Thus if φ is a linear fractional map in one complex variable and φ(zi) = wi for i =1, 2, 3 then (w − w )(w − w ) (z − z )(z − z ) 2 1 3 = 2 1 3 (w − w3)(w1 − w2) (z − z3)(z1 − z2) where w = φ(z). In such a case one may solve for w to determine the map explicitly. 2. Homogeneous coordinates. Given a linear fractional map az + b φ(z)= cz + d we may define the associated matrix of φ as a b m = . φ c d Although linear fractional maps can be associated with matrices, they are clearly C 1 not linear transformations of (take f(z) = z , for example). Instead, the asso- ciated matrix acts as a linear transformation on what are known as homogeneous coordinates in C2. Homogeneous coordinates are defined as follows. We associate the point u = (u ,u ) ∈ C2 \{0} with the point z = u1 ∈ C. We will write z ∼ u 1 2 u2 if z ∈ C is the point associated with u ∈ C2. This space is equivalent to the Rie- mann sphere and is known as the complex projective line CP1. The point (1, 0) is associated with the point at infinity on the Riemann sphere. Notice if φ(z) = w 1 and the point u ∈ CP is associated with z, then v = mφu is associated with the point w and vice versa. That is, if z ∼ u, then φ(z) ∼ mφu. The reason we are introducing these homogeneous coordinates is made evident by the following discussion. A linear transformation in C2 can be represented by a complex matrix as v a b u au + bu 1 = 1 = 1 2 . v c d u cu + du 2 2 1 2 u v Let z ∼ 1 and w ∼ 1 . Then we can associate the above linear transfor- u2 v2 mation with the linear fractional transformation a u1 + b v au + bu u2 az + b w = 1 = 1 2 = = . v2 cu1 + du2 c u1 + d cz + d u2 Thus, the associated matrices are linear transformations acting on CP1! (see [4] −1 for more details) A routine calculation shows that mφ1◦φ2 = mφ1 mφ2 and mφ = −1 (mφ) as well. We next define the cross ratio in homogeneous coordinates. As we will see, this is the key to generalized the definition to higher dimensions. First we define the 1 following quantity. If u = (u1,u2) and v = (v1, v2) are points in CP , then we define A GENERALIZED CROSS RATIO 3 u v (2) [u, v] = det 1 1 . u v 2 2 Since we have reserved subscripts for the components of a point in several vari- ables, for the remainder of our discussion we choose to designate superscripts for distinct points. As we will only consider linear fractional maps, there should be no confusion about notation. Just remember, superscripts will not designate exponents but rather distinct points. We may then define the cross ratio as follows. Definition. Let u1, u2, u3, and u4 be four distinct points in CP1, then we define the cross ratio to be [u1,u3][u2,u4] (u1,u2,u3,u4)= . [u1,u4][u2,u3] Amazingly, this definition agrees with the traditional definition of the cross ratio. For finite points we can see this as follows. Given four finite distinct points z1, z2, z3, and z4 in C, take a moment to convince yourself that we may write the associated points as ui = (zi, 1) for i =1, 2, 3, 4 and find z1 z3 z2 z4 det det (z1 − z3)(z2 − z4) 1 1 1 1 (z1,z2,z3,z4)= = (z1 − z4)(z2 − z3) z1 z4 z2 z3 det det 1 1 1 1 [u1,u3][u2,u4] = = (u1,u2,u3,u4). [u1,u4][u2,u3] If one of our points is associated with the point at infinity, then without loss of generality, we let z1 = ∞ and the point associated with z1 may, without loss of generality, be written as u1 = (1, 0) which gives us z2 z4 det (z2 − z4) 1 1 (z1,z2,z3,z4)= = (z2 − z3) z2 z3 det 1 1 1 z3 z2 z4 det det 0 1 1 1 [u1,u3][u2,u4] = = = (u1,u2,u3,u4). 1 z4 z2 z3 [u1,u4][u2,u3] det det 0 1 1 1 Thus, the cross ratio is not only invariant under φ but also mφ. In addition to this, the definition of the cross ratio in terms of homogeneous coordinates is more unifying in the sense that the case of “infinity” is included in the definition. This resonates with Klein’s Erlangen program and its focus on projective geometry (geometry done on projective spaces) as a more unifying geometry. In particular, we utilize this point of view to define a cross ratio in several complex variables. 4MICHAEL R. PILLA [email protected] SWAIN HALL EAST 205 729 EAST 3RD ST. BLOOMINGTON, IN 47405 3. Linear Fractional Maps in Several Complex Variables. To define the cross ratio in several complex variables, we must first say what it means to be a linear fractional map in CN for N > 1. We saw that in one variable the linear fractional maps were not linear transformations on C but rather on the complex projective line CP1. In this paper, we take the perspective that we would like our linear fractional maps in CN to be linear transformations on the complex projective space CPN . ′ ′ N Thus we associate the point z = (z ,z2) where z ∈ C and z2 ∈ C, z 6= 0, with ′ 1 1 the point z1 ∈ CN . This associated space is known as the complex projective space z2 CPN . We now consider a linear transformation in CPN which can be represented by a complex matrix as A B C∗ D where A is an N × N matrix, B and C are column vectors in CN , D ∈ C, and C∗ represents the conjugate transpose of C. Denote the rows of A by ai for i =1, ..., N T and B = b1 ··· bN and let h·, ·i be the standard inner product. For a point z′ 1 in CP N , we have z 2 ′ ha1, z1i + b1z2 ′ ′ . w1 A B z1 . = ∗ = . w2 C D z2 ha , z′ i + b z N 1 N 2 hz′ , Ci + Dz 1 2 z′ w′ Let z ∼ 1 and w ∼ 1 . Then we can associate the above linear transfor- z w 2 2 mation in CPN with the not necessarily linear transformation in CN given by ′ ′ ′ w1 ha1, z1i + b1z2 haN , z1i + bN z2 w = = ′ , ..., ′ w2 hz1, Ci + Dz2 hz1, Ci + Dz2 ! z′ z′ ha , 1 i + b ha , 1 i + b 1 z2 1 N z2 N = z′ , ..., z′ h 1 , Ci + D h 1 , Ci + D z2 z2 ha , zi + b ha , zi + b = 1 1 , ..., N N hz, Ci + D hz, Ci + D Az + B = . hz, Ci + D This is precisely the definition given by Cowen and MacCluer [1] and is the one we will adopt. Definition. We say φ is a linear fractional map in CN if Az + B (3) φ(z)= . hz, Ci + D A GENERALIZED CROSS RATIO 5 where A is an N × N matrix, B and C are column vectors in CN , D ∈ C, and h·, ·i is the standard inner product. This class of maps has been studied in more generality by others [3], [6], [7], [8].
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