LECTURE 4: SURFACES IN EUCLIDEAN AND AFFINE SPACES

1. Introduction

In the previous section, we saw that a curve x : I → 𝔼^{3} parametrized by arc length s is completely determined up to rigid motions of 𝔼^{3} by its curvature κ(s) and torsion τ(s). We may express this by saying that the curvature and torsion form a complete set of invariants for curves in 𝔼^{3}. In general, Important Lemma 1 tells us when we have found a complete set of invariants for a “nice” submanifold f : Σ → G/H: assuming that there is a canonical, invariant way of choosing a lifting f̃: Σ → G (this is what “nice” means), a complete set of invariants is contained in f̃^{∗}ω, the pullback via f̃ of the Maurer-Cartan form ω of G.

One question we didn’t address was whether the functions κ and τ could be prescribed arbitrarily. In other words, given arbitrary functions κ(s), τ(s), does there necessarily exist a curve x : I → 𝔼^{3} which is parametrized by arc length and has curvature κ(s) and torsion τ(s)? The answer is yes, but this result is particular to one-dimensional submanifolds of homogenous spaces G/H. It follows from the following lemma:

Important Lemma 2: Let G be a Lie group with Lie algebra 𝔤, and suppose that φ is a 𝔤-valued 1-form on a connected and simply connected manifold Σ. Then there exists a smooth map f̃: Σ → G with f̃^{∗}ω = φ if and only if φ satisfies the equation

dφ = −φ ∧ φ.

When Σ is a curve, this condition is automatically satisfied since any 2-form vanishes on Σ. But when Σ is a surface it will give compatibility conditions that must be satisfied in order for a surface with given invariants to exist.

2. Surfaces in 𝔼^{3}

Consider a smooth, embedded surface x : Σ → 𝔼^{3}, where Σ is an open set in ℝ^{2}. (For ease of notation, we will not distinguish between Σ and its image x(Σ).) We want to choose, for each point x ∈ Σ, an orthonormal frame {e_{1}(x), e_{2}(x), e_{3}(x)} for the tangent space T_{x}𝔼^{3}.

Since x is an embedding, there is a well-defined tangent plane T_{x}Σ for each point x ∈ Σ, so we can make our first adaptation by requiring that e_{3}(x) be orthogonal to T_{x}Σ. This choice is clearly invariant under the action of E(3). e_{1}(x) and e_{2}(x) will then form a basis for T_{x}Σ; for now we allow e_{1}(x), e_{2}(x) to be an arbitrary orthonormal basis of T_{x}Σ.

Now consider the Maurer-Cartan forms of E(3) restricted to this frame. Since the tangent plane of T_{x}Σ is spanned by e_{1}(x), e_{2}(x), the vector-valued 1-form dx must be a linear combination of e_{1} and e_{2}. From the structure equation

dx = ∑_{i=1}^{3} e_{i} ω^{i}

we see that we must have ω^{3} = 0 on this frame, and the linearly independent 1-forms ω^{1}, ω^{2} form a basis for the space of 1-forms on Σ. In fact, the first fundamental form of Σ is given by

I = ⟨dx, dx⟩ = (ω^{1})^{2} + (ω^{2})^{2}.

Now since ω^{3} = 0, we must also have dω^{3} = 0. Therefore

0 = dω^{3} = −ω_{1}^{3} ∧ ω^{1} − ω_{2}^{3} ∧ ω^{2}.

Since ω^{1}, ω^{2} are linearly independent 1-forms, Cartan’s Lemma states that there exist functions h_{11}, h_{12}, h_{22} such that

[\begin{matrix}ω_{1}^{3} \\ ω_{2}^{3} \end{matrix}] = [\begin{matrix}h_{11} & h_{12} \\ h_{12} & h_{22} \end{matrix}] [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] .

The second fundamental form of Σ is given by

II = ω_{1}^{3} ω^{1} + ω_{2}^{3} ω^{2} = h_{11}(ω^{1})^{2} + 2h_{12}ω^{1}ω^{2} + h_{22}(ω^{2})^{2}.

In order to make our next frame adaptation, we will compute how the matrix [h_{ij}] varies if we choose a different frame. So suppose that {ẽ_{1}, ẽ_{2}, ẽ_{3}} is any other orthonormal frame of the form

[ẽ_{1} ẽ_{2} ẽ_{3}] = [e_{1} e_{2} e_{3}] [\begin{matrix}cos θ & − sin θ & 0 \\ sin θ & cos θ & 0 \\ 0 & 0 & 1 \end{matrix}] .

Set B = [\begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix}]. Computing the Maurer-Cartan form of the new

frame shows that

[\begin{matrix} & ω̃^{1} \\ & ω̃^{2} \end{matrix}] = B^{−1} [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] , [\begin{matrix} & ω̃_{1}^{3} \\ & ω̃_{2}^{3} \end{matrix}] = ^{t}B [\begin{matrix}ω_{1}^{3} \\ ω_{2}^{3} \end{matrix}]

and therefore,

[\begin{matrix}h̃_{11} & h̃_{12} \\ h̃_{12} & h̃_{22} \end{matrix}] = ^{t}B [\begin{matrix}h_{11} & h_{12} \\ h_{12} & h_{22} \end{matrix}] B.

Since any symmetric matrix can be transformed to a diagonal matrix by such a change of basis, we can choose e_{1}(x), e_{2}(x) so that

[\begin{matrix}h_{11} & h_{12} \\ h_{12} & h_{22} \end{matrix}] = [\begin{matrix}κ_{1} & 0 \\ 0 & κ_{2} \end{matrix}] .

We will assume that κ_{1} ≠ κ_{2} at every point of Σ. This is equivalent to assuming that Σ has no umbilic points, and in this case the condition that [h_{ij}] be diagonal determines e_{1}(x), e_{2}(x) essentially uniquely. This condition is also invariant under the action of E(3). In this basis the second fundamental form of Σ is

II = κ_{1}(ω^{1})^{2} + κ_{2}(ω^{2})^{2}.

κ_{1} and κ_{2} are called the principal curvatures of Σ at x.

We will now show that any surface without umbilic points is determined up to rigid motions by its first and second fundamental forms. (This is actually true even if Σ has umbilic points, but the proof is slightly more involved.) Suppose that x, x̄ : Σ → 𝔼^{3} have the same first and second fundamental forms. Then it is clear that

ω^{1} = ω̄^{1}, ω^{2} = ω̄^{2},

and ω^{3} = ω̄^{3} = 0. Therefore

dω^{1} = dω̄^{1}

which implies that

(ω_{2}^{1} − ω̄_{2}^{1}) ∧ ω^{2} = 0.

By Cartan’s lemma, ω_{2}^{1} − ω̄_{2}^{1} must be a multiple of ω^{2}. But by the same reasoning, the fact that dω^{2} = dω̄^{2} implies that ω_{2}^{1} − ω̄_{2}^{1} must also be a multiple of ω^{1}. Since ω^{1} and ω^{2} are linearly independent, we must have ω_{2}^{1} = ω̄_{2}^{1}. The fact that II = ĪI implies that ω_{1}^{3} = ω̄_{1}^{3} and ω_{2}^{3} = ω̄_{2}^{3}. Now Important Lemma 1 yields the desired result:

Theorem: Two embedded surfaces x_{1}, x_{2} : Σ → 𝔼^{3} without umbilic points differ by a rigid motion if and only if they have the same first and second fundamental forms.

Now we consider the question discussed in the introduction, namely, can the first and second fundamental forms be prescribed arbitrarily? In other words, given 1-forms ω^{1}, ω^{2}, ω_{1}^{3}, ω_{2}^{3} on a surface Σ, what conditions must these forms satisfy in order that there exist an embedding x : Σ → 𝔼^{3} whose first and second fundamental forms are

I = (ω^{1})^{2} + (ω^{2})^{2}
II = ω_{1}^{3} ω^{1} + ω_{2}^{3} ω^{2}?

Important Lemma 2 gives the answer: the forms must satisfy the structure equations of the Maurer-Cartan forms on E(3). The first three of these equations are

dω^{1} = −ω_{2}^{1} ∧ ω^{2}
dω^{2} = ω_{2}^{1} ∧ ω^{1}
dω^{3} = 0 = −ω_{1}^{3} ∧ ω^{1} − ω_{2}^{3} ∧ ω^{2}.

The first two equations uniquely determine the form ω_{2}^{1}, called the Levi-Civita connection form of the metric given by the first fundamental form I = (ω^{1})^{2} + (ω^{2})^{2}. The third equation says that ω_{1}^{3} and ω_{2}^{3} must be symmetric linear combinations of ω^{1} and ω^{2}.

The remaining structure equations are

dω_{2}^{1} = ω_{1}^{3} ∧ ω_{2}^{3}
dω_{1}^{3} = ω_{2}^{3} ∧ ω_{2}^{1}
dω_{2}^{3} = −ω_{1}^{3} ∧ ω_{2}^{1}.

The first of these equations is called the Gauss equation, and the last two are called the Codazzi equations. By Important Lemma 2, we have the following theorem:

Theorem: Suppose that the forms ω^{1}, ω^{2}, ω_{1}^{3}, ω_{2}^{3} on Σ together with the Levi-Civita connection form ω_{2}^{1} determined by ω^{1} and ω^{2} satisfy the Gauss and Codazzi equations. Then there exists an immersed surface x : Σ → 𝔼^{3}, unique up to rigid motion, whose first and second fundamental forms are

I = (ω^{1})^{2} + (ω^{2})^{2}

II = ω_{1}^{3} ω^{1} + ω_{2}^{3} ω^{2}.

3. Surfaces in 𝔸^{3}

Now consider a smooth, embedded surface x : Σ → 𝔸^{3}. In order to compute invariants for such a surface, we need to use the geometry of the surface to construct a unimodular frame {e_{1}(x), e_{2}(x), e_{3}(x)} for the tangent space T_{x}𝔸^{3} for each x ∈ Σ.

In the Euclidean case we began by choosing e_{3} to be orthogonal to the tangent plane T_{x}Σ; it followed that e_{1}, e_{2} must be a basis for T_{x}Σ. In affine space there is no notion of orthogonality, so we cannot normalize e_{3} immediately. However, we can still make our first adaptation by requiring that e_{1}, e_{2} span the tangent plane T_{x}Σ. This condition is clearly invariant under the action of A(3). When we restrict the Maurer-Cartan form to such a frame, the same reasoning as in the Euclidean case tells us that the linearly independent 1-forms ω^{1}, ω^{2} form a basis for the space of 1-forms on Σ and that ω^{3} = dω^{3} = 0. Therefore there exist functions h_{11}, h_{12}, h_{22} such that

[\begin{matrix}ω_{1}^{3} \\ ω_{2}^{3} \end{matrix}] = [\begin{matrix}h_{11} & h_{12} \\ h_{12} & h_{22} \end{matrix}] [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] .

In order to make our next frame adaptation, we will compute how the matrix [h_{ij}] varies if we choose a different frame. So suppose that {ẽ_{1}, ẽ_{2}, ẽ_{3}} is any other affine frame with the property that ẽ_{1}, ẽ_{2} span the tangent space T_{x}Σ. Then {ẽ_{1}, ẽ_{2}, ẽ_{3}} has the form

[ẽ_{1} ẽ_{2} ẽ_{3}] = [e_{1} e_{2} e_{3}] [\begin{matrix}B & \begin{matrix}r_{1} \\ r_{2} \end{matrix} \\ 0 0 & (det B)^{−1} \end{matrix}]

where B ∈ GL(2). Computing the Maurer-Cartan form of the new frame shows that

[\begin{matrix} & ω̃^{1} \\ & ω̃^{2} \end{matrix}] = B^{−1} [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] , [\begin{matrix} & ω̃_{1}^{3} \\ & ω̃_{2}^{3} \end{matrix}] = (det B)^{t}B [\begin{matrix}ω_{1}^{3} \\ ω_{2}^{3} \end{matrix}]

and therefore,

[\begin{matrix}h̃_{11} & h̃_{12} \\ h̃_{12} & h̃_{22} \end{matrix}] = (det B)^{t}B [\begin{matrix}h_{11} & h_{12} \\ h_{12} & h_{22} \end{matrix}] B.

This transformation has the property that det[h̃_{ij}] = (det B)^{4} det[h_{ij}], so the sign of the determinant is fixed. We will assume that det[h_{ij}] > 0; in this case the surface is said to be elliptic. Then we can choose the matrix B so that [h_{ij}] is the identity matrix. This determines the frame up to a transformation of the form

[ẽ_{1} ẽ_{2} ẽ_{3}] = [e_{1} e_{2} e_{3}] [\begin{matrix}B & \begin{matrix}r_{1} \\ r_{2} \end{matrix} \\ 0 0 & 1 \end{matrix}]

with B ∈ SO(2).

The quadratic form

I = ω_{1}^{3} ω^{1} + ω_{2}^{3} ω^{2} = (ω^{1})^{2} + (ω^{2})^{2}

is now well-defined. It is called the affine first fundamental form, and it defines a metric on Σ which is invariant under the action of A(3).

We still don’t have a well-defined normal vector for Σ because we still allow changes of frame with

ẽ_{3} = e_{3} + r_{1}e_{1} + r_{2}e_{2}.

Now consider the connection form ω_{3}^{3}. Under a change of frame as above, we can compute that

ω̃_{3}^{3} = ω_{3}^{3} + r_{1}ω^{1} + r_{2}ω^{2}.

Since ω^{1}, ω^{2} are linearly independent, there is a unique choice of r_{1}, r_{2} for which ω_{3}^{3} = 0. In particular, imposing this condition yields a unique, affineinvariant choice of e_{3}, called the affine normal of Σ.

Since we now have ω_{3}^{3} = 0, we must have dω_{3}^{3} = 0. Therefore

0 = dω_{3}^{3}
= −ω_{1}^{3} ∧ ω_{3}^{1} − ω_{2}^{3} ∧ ω_{3}^{2}
= ω_{3}^{1} ∧ ω^{1} + ω_{3}^{2} ∧ ω^{2}.

By Cartan’s lemma, there exist functions ℓ_{11}, ℓ_{12}, ℓ_{22} such that

[\begin{matrix}ω_{3}^{1} \\ ω_{3}^{2} \end{matrix}] = [\begin{matrix}ℓ_{11} & ℓ_{12} \\ ℓ_{12} & ℓ_{22} \end{matrix}] [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] .

It is not difficult to show that the quadratic form

II = ω_{3}^{1} ω^{1} + ω_{3}^{2} ω^{2} = ℓ_{11}(ω^{1})^{2} + 2ℓ_{12} ω^{1} ω^{2} + ℓ_{22}(ω^{2})^{2}

is well-defined; this is the affine second fundamental form. The quantity L = \frac{1}{2}(ℓ_{11} + ℓ_{22}) is called the affine mean curvature of Σ. It is identically zero if and only if Σ is a critical point of the affine area functional, just as in the Euclidean case.

There are still more invariants; differentiating the equations

ω_{1}^{3} = ω^{1}, ω_{2}^{3} = ω^{2}

and using Cartan’s Lemma shows that there exist functions h_{111}, h_{112}, h_{122}, h_{222} such that

[\begin{matrix}2ω_{1}^{1} \\ ω_{2}^{1} + ω_{1}^{2} \\ 2ω_{2}^{2} \end{matrix}] = [\begin{matrix}h_{111} & h_{112} \\ h_{112} & h_{122} \\ h_{122} & h_{222} \end{matrix}] [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] .

Furthermore, since we now have ω_{1}^{1} + ω_{2}^{2} = 0 (Exercise: why?), it follows that

h_{122} = −h_{111}, h_{112} = −h_{222}.

The cubic form

P = ∑_{i,j,k=1}^{2} h_{ijk}ω^{i}ω^{j}ω^{k}

is called the Fubini-Pick form of Σ. Together with the first and second affine fundamental forms, it forms a complete set of invariants for elliptic affine surfaces.

Exercises

1. Any invariant of a surface Σ ⊂ 𝔼^{3} that can be expressed purely in terms of the first fundamental form

I = (ω^{1})^{2} + (ω^{2})^{2}

is called an intrinsic invariant of the surface. For instance, arc length and area are intrinsic quantities on Σ. The principal curvatures κ_{1}, κ_{2}, however, are not; they depend not only on the metric, but also on how the surface is embedded.

The function K = κ_{1}κ_{2} on Σ is called the Gauss curvature of Σ. Even though κ_{1}, κ_{2} are not intrinsic quantities, Gauss’ Theorem Egregium states that their product K is in fact intrinsic. We will prove this in several steps.

The 1-forms ω^{1}, ω^{2} are determined by the first fundamental form up to a transformation of the form

[\begin{matrix} & ω̃^{1} \\ & ω̃^{2} \end{matrix}] = [\begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix}] [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] .

a) Show that the area form

dA = ω^{1} ∧ ω^{2}

is an intrinsic quantity.

b) Show that if ω_{2}^{1} is the Levi-Civita connection form corresponding to {ω^{1}, ω^{2}}, then

ω̃_{2}^{1} = ω_{2}^{1} + dθ.

Conclude that dω_{2}^{1} is an intrinsic quantity.

c) Show that dω_{2}^{1} = ω_{1}^{3} ∧ ω_{2}^{3} = K ω^{1} ∧ ω^{2}. Conclude that K must be an intrinsic quantity.

2. In terms of local coordinates (u, v) on a surface x : Σ → 𝔼^{3}, the first and second fundamental forms are usually written

I = E du^{2} + 2F du dv + G dv^{2}
II = e du^{2} + 2f du dv + g dv^{2}

It is always possible to parametrize a surface locally so that the coordinate curves are principal curves on the surface, and for any such parametrization F = f = 0.

a) For a parametrization with F = f = 0, show that the frame

e_{1} = \frac{1}{\sqrt{E}} x_{u},  e_{2} = \frac{1}{\sqrt{G}} x_{v},  e_{3} = e_{1} × e_{2}

is orthonormal, and that its dual forms are

ω^{1} = \sqrt{E} du,  ω^{2} = \sqrt{G} dv,  ω^{3} = 0.

b) Use the structure equations for dω^{1}, dω^{2} to show that

ω_{2}^{1} = \frac{1}{2\sqrt{EG}} (E_{v} du − G_{u} dv).

c) Use the second fundamental form to show that

ω_{1}^{3} = \frac{e}{E} ω^{1} = \frac{e}{\sqrt{E}} du

ω_{2}^{3} = \frac{g}{G} ω^{1} = \frac{g}{\sqrt{G}} dv

d) Show that the Gauss equation is equivalent to

\frac{eg}{EG} = −\frac{1}{2\sqrt{EG}} [(\frac{E_{v}}{\sqrt{EG}})_{v} + (\frac{G_{u}}{\sqrt{EG}})_{u}] .

The left-hand side is, by definition, the Gauss curvature K, and this equation shows that K is in fact completely determined by the coefficients of the first fundamental form.

e) Show that the Codazzi equations are equivalent to

e_{v} = \frac{1}{2}E_{v} (\frac{e}{E} + \frac{g}{G})

g_{u} = \frac{1}{2}G_{u} (\frac{e}{E} + \frac{g}{G}) .

3. Let x : Σ → 𝔸^{3} be an elliptic affine surface, and suppose that the affine second fundamental form is a multiple of the affine first fundamental form, so that

ω_{3}^{1} = λ ω^{1}, ω_{3}^{2} = λ ω^{2}

for some function λ.

a) Prove that λ is constant. (Hint: use the structure equations to differentiate the equations above and use Cartan’s lemma.)

b) Show that if λ = 0, then de_{3} = 0, and therefore the affine normals of Σ are all parallel. Such surfaces are called improper affine spheres.

c) Show that if λ ≠ 0, then d(x − \frac{1}{λ}e_{3}) = 0. Therefore, all the affine normals intersect at the point x_{0} = x − \frac{1}{λ}e_{3}. Such surfaces are called proper affine spheres.