Previous: Characterization of the Sphere

(G11h21-G21h11)(du1/dt)2 + (G11h22-G22h11)(du1/dt)(du2/dt) + (G12h22-G22h12)(du2/dt)2 = 0.
(g11h21-g21h11)(du1)2 + (g11h22-g22h11)du1du2 + (g12h22g22h12)(du2)2 = 0.

2.4. Principal Curvatures, Principal Directions, and Lines of Curvature

We now wish to study the normal curvature kN as a function of the tangent vector V = aixi 0. As remarked earlier, kN depends only on the direction of the vector V and not on its length, so we consider only unit vectors V, for which
gijaiaj = 1.
Then kN is a continuous function of the variables (a1, a2) on the unit circle in the tangent plane so it must have a maximum and a minimum. We set I = I(a1, a2) and II = II(a1, a2). Then I/ai = 2gijaj and II/ai = 2hijaj so kN/ai = [(2gijaj) II- (2hijaj) I]/I2 = (2/I)(kNgijaj - hijaj). At a maximum or minimum of kN, we therefore have
hijaj - kNgijaj = 0
Thus
(hi1a1 + hi2a2) = kN (gi1a1 + gi2a2)
so, by eliminating kN, we obtain
(h11a1 + h12a2) (g21a1 + g22a2) = (h21a1 + h22a2)(g11a1 + g12a2),
Expanding this out, we obtain the condition
(g11h21-g21h11)(a1)2 + (g11h22-g22h11)(a1a2) +(g12h22-g22h12)(a2)2 = 0.
At an umbilic this expression is satisfied identically, and at a non-umbilic point, there must be two solutions since the function must have at least one maximum and one minimum. The solutions of this quadratic equation give the directions for which kN attains its maximum and minimum, and these are called principal directions. By our previous computation in section 1.2,, it follows that the principal directions at a non-umbilic point are orthogonal.
The maximum and minimum values of kN are called principal curvatures. To determine them, we rewrite
(hi1a1 + hi2a2) = kN (gi1a1 + gi2a2)
as
(hi1-kNgi1)a1 + (hi2 - kNgi2)a2 = 0.
Since this system of equations has non-trivial solutions, we must have (h11-kNg11)(h22 - kNg22) - (h21-kNg21)(h12 - kNg12) so kN satisfies the quadratic equation
g (kN)2 - m kN + h = 0
where m = g11h22 - 2g12h12 + g22h11. The roots are the principal curvatures, denoted k1 and k2.
The average of the roots of this equation is called the mean curvature of the surface at the point, denoted by H, so H = (1/2)(k1+ k2) = (1/2)(m/g) = (1/2g)(g11h22 - 2g12h12 + g22h11).
The product of the roots of this equation is called the total curvature or Gaussian curvature of the surface at the point, so
K = k1k2 = h/g
The principal curvatures, mean curvature, and Gaussian curvatures are scalar functions defined on the parametrized surface. They play an important role in the description of the geometric properties of the surface. For example a point is elliptic, parabolic, or hyperbolic according as K is positive, zero, or negative. Furthermore H2 - K = (1/4)(k1 - k2)2 0, with equality only at an umbilic.
A curve for which all unit tangent vectors are principal directions is called a line of curvature on the surface. For such a curve, X'(t) = xi(dui/dt) and the condition that each such vector determine a principal direction is
In differential form, this can be written
This is a quadratic equation in the first derivatives of the component functions u1(t) and u2(t), and by a standard existence theorem in the theory of ordinary differential equations, it follows that in any portion of the domain where there are no umbilics, it is possible to find two families of solution curves, the lines of curvature. These form an orthogonal net, so that at each point there are two lines of curvature with perpendicular tangent vectors.

Exercises

  1. Find conditions in terms of gij and hij for the parametric curves to be lines of curvature.
  1. For which of the six surfaces in the exercises 1.-6. of section 1.1 are the parametric curves lines of curvature?
  1. Find the lines of curvature for the right helicoid.
Next: Gauss Mapping and the Third Fundamental Form