Projective Differential Geometry of Systems of Linear Partial Differential Equations

Abstract

—In S _n — over a field F which will generally be either the real or the complex one — consider a surface Φ defined as the locus of points x whose projective coordinates (x⁽⁰⁾ x⁽¹⁾,..., x⁽ⁿ⁾) are functions of two parameters u and v. These functions will all be supposed continuous with derivatives of sufficiently high order.x _u will denote the point$\left( {\frac{{\partial {x^{\left( 0 \right)}}}}{{\partial u}},\frac{{\partial {x^{\left( 1 \right)}}}}{{\partial u}},...,\frac{{\partial {x^{\left( n \right)}}}}{{\partial u}}} \right)$ and other derivatives of x will be similarly denoted; x _u and x _v are called the first derived points of x. The tangent plane at x to Φ is the plane joining the independent points x, x _u , x _v .