1.1. Conjugate nets 3 data define a well-posed Goursat problem for this system and determine a conjugate net f uniquely: (Q1) the values of f on the coordinate axes Bi for 1 i m, i.e., m smooth curves f Bi with a common intersection point f(0) (Q2) the values of cij, cji on the coordinate planes Bij for all 1 i j m, i.e., m(m 1) smooth real-valued functions cij Bij of two variables. It is important to note that Definition 1.1, as well as Definition 1.2 below, may be reformulated so as to deal with projectively invariant notions only, and thus they belong to projective differential geometry. In this setting the ambient space RN of a conjugate net should be interpreted as an affine part of the projective space RPN = P(RN+1), with RN+1 being the space of homogeneous coordinates. Equations (1.1) hold then for the standard lift (f, 1) RN+1 of the conjugate net f [f : 1] RPN , while an arbitrary lift ˜ = λ(f, 1) RN+1 is characterized by a more general linear system (1.3) ∂i∂j ˜ = ˜ji∂i ˜+ ˜ij∂j ˜+ ˜ij ˜ i = j (with the corresponding compatibility conditions for the coefficients ˜ij, ˜ij, which generalize equations (1.2)). We will not pursue this description fur- ther. 1.1.2. Alternative analytic description of conjugate nets. A classical description of conjugate nets makes use of the following construction. Given the functions cij, define functions gi : Rm R∗ as solutions of the system of differential equations (1.4) ∂igj = cijgj , i = j. Compatibility of this system is expressed as ∂icjk = ∂jcik and is a conse- quence of equations (1.2) (whose right-hand sides are symmetric with re- spect to the flip i j). Solutions gi can be specified by prescribing their values arbitrarily on the corresponding coordinate axes Bi. Define vectors wi = g−1∂if. i It follows from (1.1) and (1.4) that these vectors satisfy the following differential equations: (1.5) ∂iwj = gi gj cjiwi , i = j. Thus, defining the rotation coefficients as (1.6) γji = gi gj cji , we end up with the following system: ∂if = giwi , (1.7) ∂iwj = γjiwi , i = j, (1.8) ∂igj = giγij , i = j. (1.9)
Previous Page Next Page