Blaga P. Lectures on the differential geometry of - tiera.ru

4.16. Principal directions and curvatures 157
As the length of e is not important, we shall assume that e is a unit vector: �e� = 1.
Then e = e1 · cos θ + e2 · sin θ, therefore
kn(e) = ϕ2(e, e)
ϕ1(e, e)
Thus, we obtained:
= −A(e) · e
e · e
����
=1
= −A(e1 cos θ + e2 sin θ) · (e1 cos θ + e2 sin θ) =
= (k1 cos θ · e1 + k2 sin θ · e2) · (e1 cos θ + e2 sin θ) = k1 cos 2 θ + k2 sin 2 θ.
Theorem 4.16.1. Let S be an oriented surface. Then the normal curvature at a point of
the surface, in the direction of a vector e, is given by the Euler’s formula:
kn(e) = k1 cos 2 θ + k2 sin 2 θ, (4.16.1)
where k1 and k2 are the principal curvature of the surface, while θ = ∠(e, e1).
An immediate consequence of the Euler’s formula is:
Theorem 4.16.2. The principal curvatures of a surface at a point are extreme values
of the normal curvature of the surface in the direction of a vector e, when the vector e
rotates around the origin of the tangent space of the surface at that point.
Proof. From the Euler’s formula, we have
kn(e) = k1 cos 2 θ + k2(1 − cos 2 θ) = k2 + (k1 − k2) cos 2 θ.
It is clear now that the maximum value of the normal curvature is reached for θ = 0 (we
assumed that k1 ≥ k2!) and, in this case, we get kn = k1, while the minimal value – for
θ = π
2 , obtaining kn = k2. �
Definition 4.16.5. The quantities Kt = k1·k2 and Km = 1
2 (k1+k2) are called, respectively,
the total (Gaussian) and mean curvature of the surface.
The total and mean curvatures of a surface can be computed easily if the matrix of
the shape operator in an arbitrary basis is known. Indeed, we have:
Proposition 4.16.2.
Km = − 1
2 Tr A (4.16.2)
Kt = det A. (4.16.3)