Differential geometry notes 7

by Emin Orhan

Parametric representation of a point set M in three-dimensional Euclidean space:

\mathbf{x}(u^1,u^2) = \big( x_1(u^1,u^2), x_2(u^1,u^2),x_3(u^1,u^2) \big)

where u^1, u^2 are real-valued variables defined in a simply-connected, bounded domain B.

We make the following assumptions about the parametric representations:

(1) \mathbf{x}(u^1,u^2) is of class r \geq 1 in B. Each point of M corresponds to just one ordered pair (u^1,u^2) in B.

(2) The Jacobian is of rank 2 in B:

J = \Large \begin{pmatrix}    \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\    \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \\    \frac{\partial x_3}{\partial u_1} & \frac{\partial x_3}{\partial u_2}    \end{pmatrix}

A representation satisfying these assumptions is called an allowable representation.

Partial derivatives will be denoted with the following notation:

\Large \mathbf{x}_{\alpha} = \frac{\partial \mathbf{x}}{\partial u^{\alpha}} and \Large \mathbf{x}_{\alpha \beta} = \frac{\partial^2 \mathbf{x}}{\partial u^{\alpha} \partial u^{\beta}}.

If J is of rank R<2 for a particular (u^1,u^2), the corresponding point is called a singular point with respect to the representation and it is called singular if it is singular with respect to every allowable representation.

By imposing a transformation u^{\alpha} = u^{\alpha}(\bar{u}^1,\bar{u}^2) (with \alpha = 1, 2), we can obtain a new parametric representation \mathbf{x}(\bar{u}^1,\bar{u}^2) of M. The transformation has to satisfy the following assumptions:

(1) The domain \bar{B} of the transformation includes B.

(2) u^{\alpha} are of class r\geq 1 everywhere in \bar{B} and are one-to-one.

(3) The Jacobian D is different from zero everywhere in \bar{B}:

D = \Large \frac{\partial (u^1,u^2)}{\partial (\bar{u}^1,\bar{u}^2)} = \begin{vmatrix}    \frac{\partial u^1}{\partial \bar{u}^1} & \frac{\partial u^1}{\partial \bar{u}^2} \\    \frac{\partial u^2}{\partial \bar{u}^1} & \frac{\partial u^2}{\partial \bar{u}^2}    \end{vmatrix}

A transformation satisfying these assumptions is called an allowable coordinate transformation. As in the theory of curves, allowable coordinate transformations define equivalence classes of allowable parametric representations.

Definition 24.1: A point set S in \mathbb{R}^3 which can be represented by the allowable representations of an equivalence class is called a portion of a surface. u^1 and u^2 are called the coordinates on S. The curves u^1 = const and u^2 = const are called the coordinate curves of the u^1 u^2 coordinate system.

Definition 24.2: A union U of portions of surfaces is called a surface if every two portions S, S^{\prime} of U can be joined by finitely many portions S = S_1, \ldots, S_n = S^{\prime} such that the intersection of two subsequent portions S_i, S_{i+1} is a portion of a surface.