Chapter IV: Fundamental Equations of Surface Theory, Congruence Theorem

In curve theory, the fundamental invariants are the curvature and torsion, and the fundamental equations are the Frenet formulas. These express the derivatives of the vectors T, N, and B of the Frenet frames of a curve in terms of these vectors T, N, and B themselves, with the coefficients essentially being the curvature and torsion.

The situation in surface theory is more complicated, but conceptually no different. The fundamental invariants are now the coefficients of the first and second fundamental forms, namely the six functions g_ij and h_ij of the parameters (u¹,u²). These functions are not independent, and the relationship between them is contained in the theorems of Gauss and Codazzi, to be established in section 3 of this chapter. Taking the place of the Frenet frame is the frame X₁,X₂,, where X₁ and X₂ are not orthogonal unit vectors in general. The counterpart of the Frenet equations are the Weingarten and Gauss equations, expressing the partial derivatives of X₁,X₂, in terms of the vectors X₁,X₂, themselves.

From the Weingarten equations, we immediately derive the theorem, proved earlier: A surface consisting entirely of planar points is a plane. This follows since the hypothesis implies that h_ij = 0 for all i, j, so _i = 0 = _j and is constant. This gives X_i· =(X·)_i = 0 and (X·)_j = 0 so X· = c, a constant, the condition for the surface X to lie in a plane.

More subtle is another consequence of the Weingarten equations:

Proposition 1: A surface with Gaussian curvature K = 0 is, in the neighborhood of a non-planar point, generated by a family of lines such that the tangent plane remains the same along any generator.

Also, we have (X·)₂ = X₂· + X·₂ = 0, so X· = p(u¹) is a function depending only on u¹. Differentiating with respect to u¹ gives X·₁ = p'(u¹). Note that ₁ = -h₁^jX_j = -h_1kg^kjX_j 0 since h₁₁ 0 g₁₁. Therefore the asymptotic curve lies in this plane X·₁ = p'(u¹). But it already lies in the plane X · = c and the vectors and ₁ are distinct, so it follows that the asymptotic curve lies in the intersection of two distinct plane, and therefore that it is a straight line. As remarked earlier, the normal vector is constant along this line, so the theorem is proved.

A surface with K = 0 and no planar points is called developable. The reason for this is that any such surface is isometric to a portion of the plane. We may approach this fact in a different way by concentrating on ruled surfaces and singling out those for which the Gaussian curvature is everywhere 0. A ruled surface is of the form X(u¹,u²) = Y(u¹) + u²Z(u¹) for a regular curve Y(u¹) and a unit vector field Z(u¹) . Then X₁(u¹,u²) = Y'(u¹) + u²Z'(u¹) while X₂(u¹,u²) = Z(u¹). Thus X₁(u¹,u²) x X₂(u¹,u²) = Y'(u¹) x Z(u¹) + u²Z'(u¹) x Z(u¹). When we normalize this vector, the result will be independent of u² only when Y'(u¹) x Z(u¹) = 0 identically, i.e. the vector Z(u¹) is a multiple of the tangent vector Y'(u¹) so we may assume Z(u¹) = T(u¹), the unit tangent. Thus the ruled surface is the tangential surface of the curve Y(u¹) and we have already seen that the metric coefficients of that surface depend only on the curvature of the curve, not on its torsion. There is therefor a curve in the plane with the same curvature and torsion zero, and the tangential surface of this curve, a portion of the plane, will be isometric to the original surface.

where _i^j_k = _k^j_i, c_ik =c_ki. Taking the dot product with gives c_ik = X_ik· = h_ik by the definition of the second fundamental form. To determine the coefficients _i^j_k, we take the dot product with X_m to obtain X_ik·X_m = _i^j_kX_j·X_m = _i^j_kg_jm. We set _imk = _i^j_kg_jm(being careful about the order of the subscripts). Note that _imk= _kmi but in general there is no symmetry in the other subscripts.

Using the inverse of the first fundamental form, we may write _i^j_k = _imkg^jm. This process is called "raising or lowering the indices" and the sets of coefficients _i^j_k and _imk (or h_i^j and h_ij) are called "associated"; one set completely determines the other.

It follows that the coefficients _ikm, and hence also the _i^j_k, can be expressed completely in terms of the coefficients g_ik of the first fundamental form and their partial derivatives. The coefficients _ikm and _i^j_k are called the Christoffel symbols of the first and second type. It is remarkable that they are intrinsic, depending only on the first fundamental form of the surface.