GAUSS-WEINGARTEN EQUATIONS, GAUSS-CODAZZI EQUATIONS, FUNDAMENTAL THEOREM OF SURFACES

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Gauss-Weingarten equations. The Gauss-Weingarten equations are analogous, for surfaces, to the Frenet equations for curves. The Frenet equations express the vectors , , as linear combinations of the three orthogonal unit basis vectors , basis vectors that constitute a moving trihedral located at point P on a space curve. The Gauss-Weingarten equations express the vectors with respect to a trihedral, located at point P on the surface and consisting, not of three orthogonal unit vectors as in the case of space curves, but of the three linearly independent vectors (that is, the Gauss-Weingarten equations express the vectors as linear combinations of the set of basis vectors ).

Theorem 1. Let S be a simple surface element of class defined by the one-to-one mapping

x = x(u, v)

y = y(u, v)

z = z(u, v)

of a region R of the uv-plane into xyz-space. Let

be the position vector to point P on the surface. Then the vectors and their derivatives satisfy the system

where

and E, F, G, L, M, N are the fundamental coefficients of the first and second orders.

Derivation

The first three equations are called the Gauss equations and the last two are called the Weingarten equations. The quantities Γ_{i j}^k are called the Christoffel symbols of the second kind. Note that the quantities Γ_{i j}^k depend only on the first fundamental coefficients E, G, F and their derivatives while the α and β coefficients depend on both the first and second fundamental coefficients.

Gauss-Codazzi equations. In the case of space curves the functions κ(s) and τ(s) giving curvature and torsion as a function of arc length s define the curve to within position (i.e. they define it up to, but not including, position in space — the curves are determined except for location and orientation in space; two curves with the same curvature and torsion as functions of arc length are identical except for position and orientation in space). Do we have an analogous situation for the case of surfaces? Is there a set of functions that will define a surface to within position in space? Do the first and second fundamental coefficients E, F, G, L, M, N given as functions of u and v define a surface up to position in space? The answer to this question is no, not unless certain additional conditions are met. The answer to the full question is given in the Fundamental Theorem of Surfaces which we will soon present.

If one is given the first and second fundamental coefficients E, F, G, L, M, N as functions of u and v and certain conditions are met, the answer to the question is positive. The conditions that must be met involve third order mixed partial derivatives of . The conditions are

which state that the third order mixed partial derivatives are independent of order of differentiation.

Theorem 2. Let S be a simple surface element of class defined by the one-to-one mapping

of a region R of the uv-plane into xyz-space in which the coefficients of the Gauss-Weingarten equations are of class C¹. Then the mixed derivatives exist and satisfy the conditions

if and only if the first and second fundamental coefficients E, F, G, L, M, N satisfy the following equations of Codazzi and Gauss:

Codazzi equations

Gauss equation

Equation 4) is of special interest. It shows that the expression LN - M² is dependent solely on the first fundamental coefficients E, G, F and their derivatives since the quantities Γ_{i j}^k depend only on the first fundamental coefficients E, G, F and their derivatives. This is significant since the expression LN - M² is the numerator in the expression for Gaussian curvature K = (LN - M²)/(EG - F²). It means the Gaussian curvature depends only on the first fundamental coefficients E, G, F.

Theorem 3. The total curvature (Gaussian curvature) of a surface is expressible in terms of the coefficients E, G, F of the first fundamental form and their first and second derivatives.

Fundamental Theorem of Surfaces. If E, F, G, L, M, N are given functions of u and v, sufficiently differentiable, which satisfy the Gauss-Codazzi equations and the added conditions that EG - F² > 0, E > 0, G > 0, there exists a surface, uniquely determined except for its position in space, which has respectively as its first and second fundamental forms the quadratic forms Edu² + 2 Fdudv + Gdv² and Ldu² + 2 Mdudv + Ndv².

This theorem guarantees not only the existence of a surface, provided the conditions are met, but also that if two surfaces have the same fundamental forms, they are congruent. Thus all the properties of a surface that are independent of position in space are expressible in terms of its two fundamental forms and their coefficients.

References.

1. Graustein. Differential Geometry.

2. Struik. Lectures on Classical Differential Geometry.

3. Lipschutz. Differential Geometry

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