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A Brief Survey of Differential Geometry
Adrian Down
August 29, 2006
1 Introduction
Differential geometry is a branch of mathematics that uses the tools of cal- culus and linear algebra to study geometry. In this course, we will write equations describing geometric objects such as curves and surfaces. We will analyze these equations using the tools of math 53 and 54 gain geometric insight from these equations.
Note. The notation in this field is very precise and should be followed exactly. The formulas will quickly become complicated and it will be useful to follow the standardized notation.
2 Curvature
2.1 Motivation
Imagine a curve in a plane. Let x^1 and x^2 refer to the coordinates in this plane (superscripts refer to coordinates). We would like to define the curvature κ of the curve at a particular point on the plane. This definition of curvature should meet at least two basic requirements:
- The curvature of a straight line should be identically equal to 0 every- where along the curve.
- The curvature of a circle should be constant and nonzero at every point on the circle.
Following the general method above, we should write a formula describing the curve and manipulate it using calculus. The simplest way to write an equation parameterizing a plane curve is by y = f (x) for some function f (x). All of the information about the curve must be contained in the function f.
2.2 Possible (incorrect) definitions
It is instructive to try a few possible definitions for the curvature and see how they fail to meet the requirements above. One possibility is κ = f ′(x). This definition fails to give κ ≡ 0 for a straight line with nonzero slope. We could also try κ = f ′′(x). This definition does ensure that κ ≡ 0 for any straight line. We can verify if the second condition is also satisfied. The unit circle can be parameterized,
S =
x, y ∈ R
∣ (^) x^2 + y^2 = 1
⇒ y = f (x) = (1 − x^2 )
(^12)
Taking the derivative,
f ′(x) =
−x (1 − x^2 ) (^12)
f ′′(x) = −
x^2 (1 − x^2 )
3 2
(1 − x^2 )
5 2
f ′′(x) is not constant on the unit circle, and so κ = f ′′(x) cannot be the correct definition of curvature.
Note. Another argument against the choice κ = f ′′(x) is that this definition is not invariant under rotation of the coordinate system.
2.3 Correct definition of curvature
The correct definition of curvature is,
κ =
|f ′′(x)| (1 + f ′(x))
(^32)
In can be shown that this formula is invariant to coordinate transforma- tion. The denominator can be thought of as the necessary factor to make the definition invariant.
4 Surfaces in space
4.1 Parameterization
Now we consider two-dimensional surfaces lying in a three-dimensional space. By analogy with the mapping from the real line to R^3 considered in the case of curves, surfaces are parameterized by a function x(u^1 , u^2 ) that maps from a two dimensional space U with coordinates (u^1 , u^2 ) to R^3. The function x can be written in component form,
x(u^1 , u^2 ) = (x^1 (u^1 , u^2 ), x^2 (u^1 , u^2 ), x^3 (u^1 , u^2 ))
The surface is completely described by these three functions xi.
4.2 Geometric interpretation
We can now attempt to extract geometric information from this parameter- ization of the surface. In analogy with the orthonormal basis constructed before, we will use the function x to construct the most convenient comoving basis in which to examine the surface. We will construct a comoving frame at any point from the two vectors describing the plane tangent at that point, denoted by x 1 and x 2 , and the normal to this tangent plane, denoted by n. We will see that the tangent vectors are formed using partial differentiation,
x 1 =
∂x ∂u^1
∂x^1 ∂u^1
∂x^2 ∂u^1
∂x^3 ∂u^1
x 2 =
∂x ∂u^2
∂x^1 ∂u^2
∂x^2 ∂u^2
∂x^3 ∂u^2
Note. The comoving basis constructed to analyze surfaces is not in gen- eral orthonormal. This is different than the basis used to analyze curves, { T, N, B }, which was orthonormal.
We would like an analog of the Frenet-Serret formula. Since the basis chosen to analyze the surface is not orthonormal, the form of this formula is
more complicated,
xij = Lij n +
∑^3
k=
Γkij xk 1 ≤ i, j, ≤ 2
nj = −
∑^2
k=
Lkj xk 1 ≤ j ≤ 2
where Lij is called the coefficient of the second fundamental form and γijk is called the Christoffel symbol of the second kind. The first of these equations is called Gauss’s formula. The second is Weingarten’s formula and represents how the surface is changing as the co- ordinates xi^ change.
4.3 Intrinsic verses extrinsic properties
We would like to have a geometric interpretation of the quantities that we have just discovered. In particular, we would like to identify which of these quantities is intrinsic, meaning they can be measured at any particular point on the surface without knowledge of the rest of the surface. We will see that the Christoffel symbols are intrinsic, whereas the coefficients of the second fundamental form are not. One way to think of intrinsic quantities is as those that could be measured by a bunch of mathematically-inclined bugs living on the surface that are unaware of the larger three-dimensional space in which the curve lies.
