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Chapter III: Fundamental Forms of a Surface

1.1. The First Fundamental Form

Let D be an open domain in the (u1, u2)-plane. A parametrized surface is a smooth mapping
(1) X: D --> E3
of D into three-dimensional Euclidean space E3. We will denote the value of X at (u1, u2) by X(u1, u2), the position vector from the origin considered as a function of u1 and u2. For the rest of our study of surfaces, we will be considering sums over the indices i = 1,2. We will therefore make adopt a convention, standard in tensor analysis, that all small Latin indices have the range 1,2. We will also use the convention that subscripts on a vector function indicate partial derivatives, e.g.
(2)i = X/ui and Xij = 2X/uiuj, etc.
The parametrized surface defined by X is said to be regular or immersed if X1(u1, u2) x X2(u1, u2) 0 for every (u1, u2) in the domain D. A point where X1(u1, u2) x X2(u1, u2) = 0 is said to be a singular point. In most cases we will consider regular surfaces, without singular points.
A point X of E3 can be given in terms of its coordinates (x1, x2, x3) and the vector function X(u1,u2) can be expressed in terms of its component functions X(u1,u2) = (x1(u1,u2), x2(u1,u2), x3(u1,u2)). In terms of component functions, the regularity condition (6) means that the matrix with rows (x1(u1,u2)/u1, x2(u1,u2)/u1, x3(u1,u2)/u1) and (x1(u1, u2)/u2, x2(u1,u2)/u2, x3(u1,u2)/u2) has rank two everywhere.
Example 1.The graph of the function f
(3) x3 = f(x1, x2)
gives a parametrized surface
(4) X(u1,u2) = (u1, u2, f(u1, u2))
Since x1(u1,u2) = (1, 0, f(u1, u2)/u1) and x2(u1,u2) = ( 0, 1, f(u1,u2)/u2), the rank is always two so the regularity condition is satisfied at all points of the domain of f.
Example 2. The sphere of radius b is defined by the equation
(5) x12 + x22 + x32 = b2
may be parametrized as
(6) X(u1, u2) = (b sin(u1) cos(u2), b sin(u1) sin(u2) , b cos(u1)). Then
(7) x1(u1,u2) = (bcos(u1) cos(u2), b cos(u1) sin(u2), -b sin(u1)) and
(8) x2(u1,u2) = (-bsin(u1) sin(u2), b sin(u1) cos(u2), 0) so
(9) x1(u1, u2) x x2(u1, u2) = b sin(u1) X(u1,u2).
At the points where sin(u1) = 0, i.e. at the north pole (0, 0, 1) and the south pole (0, 0, -1), the parametrized surface is not regular, although from a geometric point of view, all points of the sphere are like all others. By rotating the sphere one quarter turn around the x1-axis, we may obtain a parametrization which is regular at the north and south poles, although singular at two other points. This example shows the importance of defining a surface by using enough parametrizations so that each point is a regular point of at least one of them. The process of defining a surface as a "combination" of parametrized surfaces is at the foundation of differential geometry, and will be discussed in detail in later chapters.
Example 3. The torus of revolution with the parametrization
(10) X(u1,u2) = ((a + bsin(u1))cos(u2), (a + bsin(u1)sin(u2) , b cos(u1))
for constants a > b > 0 is a regular surface for all (u1, u2). Let P = X(u01, u02) be the point having the parameters (u01, u02). The equations
(11) ui = ui(t), i = 1,2.
satisfying the conditions
(12) u0i = ui(t0), i = 1,2.
define a curve through P on the parametrized surface. By the chain rule, the tangent vector to the curve at P is
(13) dX/dt = X/ui dui/dt = Sxi dui/dt.
where the sum is over the indices i = 1,2. When two indices appear in a formula, it will be assumed that they will be summed unless otherwise mentioned. We may then rewrite (13) as
(13a) dX/dt = xi dui/dt
so the tangent vector to the curve is a linear combination of x1(t0) and x2(t0).
All tangent vectors at P to curves passing through P lie in the plane spanned by the linearly independent vectors X1(u01, u02) and X2(u01, u02), and this is called the tangent plane to the surface at P. The line through P perpendicular to this plane is called the normal line at P. It consists of all vectors of the form X(u01, u02) + v(X1(u01, u02) x X2(u01, u02)). The unit normal for the parametrized surface X is defined by the condition
(14) = (X1 x X2)/(X1 x X2)(X1 x X2).
It is the unique unit vector perpendicular to the tangent plane such that the triple product (X1, X2, ) > 0.
A tangent vector V at P can be written as
(15) aiXi = a1X1 + a2X2,
where a1 and a2 are the components of V (relative to the basis, X1, X2). The dot product of V with itself is then
(16) VV = a1a1X1X1 +a1a2X1X2+ a2a1X2X1+ a2a2X2X2
which we abbreviate, using the summation convention, as
(16a) VV = aiajXiXj.
We introduce the metric coefficients
(17) XiXj = gij
so we have a final abbreviation
(17a) VV = gijaiaj.
More generally, if W is another tangent vector at P, with components bi, we many then write
(18) W = biXi
and the dot product of the two vectors is
(19) VW = aibjXiXj = gijaibj.
We will write these expression using the notation
(20) I(V,W) = VV and I(V) = I(V,V) = VV.
This expression is a bilinear form on the tangent space at P, which we call the first fundamental form. Note that this form has the property that I(V) 0, with I(V) = 0 if and only if V is the zero vector. A form with this property is called positive definite.
The first fundamental form then gives a bilinear form on each of the tangent planes of a parametrized surface, and we can use this form to compute the length of a curve X(t) = X(u1(t), u2(t)) between the limits t1 and t2:
(21) s = I(X'(t))dt = gij (du1/dt)(du2/dt) dt, or
(21a) (ds/dt)2 = gij (du1/dt)(du2/dt)
In differential form, this formula for length can be written
(22) ds2 = gijdu1du2.
The first fundamental form thus gives a quadratic differential form.

Exercises for Section 1.1

  1. Find the metric coefficients gij for the ellipsoid with equation
    (23) X(u1, u2) = (a sin(u1) cos(u2), b sin(u1) sin(u2) , c cos(u1)),
    where a, b, and c are constants. At which points does the surface fail to be regular? Find the unit normal vector at regular points.
  1. Find the metric coefficients gij for the surface of revolution
    2. (24) X(u1,u2) = (u1 cos(u2), u1 sin(u2) , f(u1)).
    At what points will the surface fail to be regular? What is the length of the curve u1(t) = c, u2(t) = t? Find an expression for the unit normal vector.
  1. Find the metric coefficients gij for the right helicoid
    2. (25) X(u1,u2) = (u1 cos(u2), u1 sin(u2) , g(u2)).
    What is the length of the curve u1(t) = t, u2(t) = c? Find an expression for the unit normal vector.
  1. Find the metric coefficients for the catenoid, with equation
    2. (26) X(u1,u2) = ( cosh(u1) cos(u2), cosh(u1) sin(u2), u1).
    What is the length of the curve u1(t) = t, u2(t) = c? What is the length of the curve u1(t) = c, u2(t) = t? Find an expression for the unit normal vector.
  2. Find the metric coefficients for the helicoid, with equation
    3. (27) X(u1,u2) = ( sinh(u1) sin(u2), -sinh(u1) cos(u2), u2).
    What is the length of the curve u1(t) = t, u2(t) = c? What is the length of the curve u1(t) = c, u2(t) = t? Find an expression for the unit normal vector.
  1. Find the metric coefficients and the unit normal vector for the function graph X(u1,u2) = (u1, u2, f(u1, u2)).
  1. Find the metric coefficients and the unit normal vector of the torus
    2. X(u1,u2) = ((a + bsin(u1))cos(u2), (a + bsin(u1)sin(u2) , b cos(u1)).
Next: Geometry in the Tangent Plane