DIFFERENTIAL GEOMETRY COURSE

12. *THE FIRST FUNDAMENTAL FORM*

*Explanation 103 :* We have to consider differentials first.

If f = f(x) then df = f '(x) dx ; for example, if f = cos(x) then df = -sin(x) dx ; df is the first order approximation of Δf := f(x+dx) - f(x) (calculate df and Δf in the example, with
x = π/2, dx = 0.001).

If f = f(x,y) then df = f_{x}dx + f_{y}dy ('total differential') ; for example, if f = x/y then df = (1/y)dx -(x/y^{2})dy ;
df is the first order approximation of Δf := f(x+dx,y+dy) - f(x,y).

*Explanation 104 :* Now we have to consider curves on the surface.

The image curve has parametrisation * x*(u

The arc length s of

*Definition 105 :* We define the first fundamental form of the surface by (ds)^{2} = a_{1 1} (du_{1})^{2} + 2 a_{1 2} du_{1} du_{2} +
a_{2 2} (du_{2})^{2}, where a_{i j} = __x___{ui} . __x___{uj}.

*Examples 106 :*

Plane: * x*(u,v) = (u,v,0). Then (ds)

Sphere:

Graph of a function:

For each surface we have: a_{1 2} = 0 in the points where the parameter lines are perpendicular to each other.

*Explanation 107 :* We will now see that the first fundamental form determines the metrics of the surface:

i) The arc length of a curve on the surface, say * x*(u

ii) The angle between two curves on the surface, say

cos(φ) = (

= ( a

iii) The area of a domain on the surface, say * x*(u

∫ ∫

So arc length, angle and area are variables we calculate with the help of the coefficients of the first fundamental form and of variables we calculate in the u_{1},u_{2}-plane ;
therefore we call them 'intrinsic variables'.

From the following definition and explanation it will become clear why we call them 'invariant under bending' as well.

*Definition 108 :* Consider the following picture (__x__^{1} and __x__^{2} are differentiable):

If the surfaces __x__^{1}(u_{1},u_{2}) and __x__^{2}(u_{1},u_{2}) have the same first fundamental form, we call them
bents of each other, and we call the mapping __x__^{2} o (__x__^{1}) ^{-1} an isometry.

*Explanation 109 :* Check by calculation that the right circular cylinder * x*(u,v) = (cos u, sin u, v) is a bent of the plane (u,v,0);
if we shade on a piece of paper a domain and draw two curves on it, and roll up the piece of paper so that it becomes a right circular cylinder, then the area of the shaded domain, the lengths of the curves,
and the angle between the curves in the intersection point don't change.

*Problem 110 :* Calculate the angle between the curves (t^{2} cos(t^{3}), t^{2} sin(t^{3}), 2t^{2}) and
(t^{3} cos(t^{2}), t^{3} sin(t^{2}), 2t^{3}) on the surface of revolution (u cos v, u sin v, 2u) in the point with parameter value t=1.

Do this in the direct way first, and then with the help of the first fundamental form.

*Problem 111 :* Consider the surface V with parametrisation (u+v, u^{2}+v^{2}, uv). Prove that the points on V where the parameter lines are perpendicular to each other are lying on a
parabola.

*Problem 112 :* Calculate the area of the part of the sphere with center (0,0,0) and radius 1 that lies between the parallels of latitude θ = θ_{1} and θ = θ_{2}
by making use of the first fundamental form.

*Problem 113 :* Show that the right circular cone u(cos v, sin v, 1) (a kind of conical cap) is a bent of a plane (first use for the parametrisation of the plane
polar coordinates u and v, and then make some little adaptations).