Lecture Notes for 120 - UCLA Department of Mathematics

5.4. PRINCIPAL CURVATURES 119
its partial derivatives in some parametrization are
@f
@w = @q
@w · N (q) ,
@ 2 f
@w 1 @w 2
=
@ 2 q
@w 1 @w 2
· N (q)
So f has a critical point at q, andthesecondderivativematrixatq is simply [II] .
The second derivative test then tells us something about how the surface is placed
in relation to T q M. Specifically we see that if both principal curvatures have the
same sign, or K>0, then the surface must locally be on one side of the tangent
plane, while if the principal curvatures have opposite signs, or K < 0, then the
surface lies on both sides. In that case it’ll look like a saddle.
We can also relate the second fundamental form in general directions to the
principal curvatures.
Theorem 5.4.5. (Euler, 1760) Let X 2 T q M be a unit vector and apple 1 ,apple 2 the
principal curvatures then
II (X, X) = apple 1 cos 2 + apple 2 sin 2
where is the angle between X and the principal direction corresponding to apple 1 .
Proof. Simply select an orthonormal basis E 1 ,E 2 of principal directions and
use that
X = cos E 1 +sin E 2 ,
II (E 1 ,E 1 ) = apple 1 ,
II (E 2 ,E 2 ) = apple 2 ,
II (E 1 ,E 2 ) = 0 = II(E 2 ,E 1 ) .
As an important corollary we get a nice characterization of the principal curvatures.
Corollary 5.4.6. Assume that the principal curvatures are ordered apple 1 apple 2 ,
then
max II (X, X) = apple 1,
|X|=1
min II (X, X) = apple 2.
|X|=1
We can now give a rather surprising characterization of planes and spheres.
Theorem 5.4.7. (Meusnier, 1776) If a surface q has the property that apple 1 = apple 2
at all points, then apple = apple 1 is constant and the surface is part of a plane or sphere.
Proof. Since the principal curvatures agree at all points it follows that
✓ ◆
@N @q
@w = L = apple @q
@w @w
⇤