Josai Mathematical Monographs vol 3 (2001), pp. 3-10

CONSTRUCTION OF FLAT TORI IN THE 3-SPHERE AND ITS APPLICATIONS

YOSHIHISA KITAGAWA

Introduction

In 1975, Yau [16, p.87] posed the problem of the classification of flat tori in the unit 3-sphere S^{3}. Concerning this problem, there is a method for constructing flat surfaces in S^{3} which was obseved by Lawson ([12]). Let p : S^{3} → S^{2} be the Hopf fibration, and let γ be a curve in S^{2}. Then the inverse image p^{−1}(γ) is a flat surface in S^{3}. If the curve γ is closed, the inverse image p^{−1}(γ) is a flat torus in S^{3} and it is called the Hopf torus corresponding to γ ([10]). On the other hand, there is another method for constructing flat surfaces in S^{3} which was obtained by Bianchi[1] and Sasaki[11]. Let α and β be curves in S^{3} whose torsions satisfy τ_{α} = 1 and τ_{β} = −1. Using the group structure on S^{3}, define a surface F : ℝ^{2} → S^{3} by

F (s_{1}, s_{2}) = α(s_{1})β(s_{2}).

Then F is a flat surface such that the s_{i}-curves are the asymptotic curves of the surface ([13, p.139 - 163]). Infinitely many complete flat surfaces in S^{3} are constructed by this method.

Recently, using the Hopf fibration, the author obtained a method for constructing closed curves in S^{3} with τ = ±1. Combining this result with the method of Bianchi and Sasaki, the author established a method for constructing all the flat tori isometrically immersed in S^{3} (Theorem 1). Applying this method, we obtain some interesting results on flat tori in S^{3}. For example, by using the Arf invariant for knots, we see that every embedded flat torus in S^{3} is invariant under the antipodal map of S^{3} (Theorem 4). This implies a rigidity theorem for the Clifford tori in S^{3} (Theorem 5). Furthermore, we obtain the classification of undeformable flat tori in S^{3} (Theorem 8).

The outline of this article is as follows. In Section 1 we explain the method for constructing all the flat tori isometrically immersed in S^{3}. In Section 2 we deal with embedded flat tori in S^{3}, and in Section 3 we explain the result of the classification of undeformable flat tori in S^{3}.

1. Construction of flat tori in S^{3}

We first explain a method for constructing closed curves in S^{3} with τ = ±1. Let ℍ denote the set of all quaternions, and let ℝ^{4} be the 4-dimensional Euclidean space identified with ℍ as follows:

(x_{1}, x_{2}, x_{3}, x_{4}) ⟷ x_{1} + x_{2}i + x_{3}j + x_{4}k.

Then the unit spheres S^{2} and S^{3} are given by

S^{2} = {x ∈ Im ℍ : |x| = 1}, S^{3} = {x ∈ ℍ : |x| = 1}.

Note that the unit sphere S^{3} has a group structure induced by the multiplicative structure of ℍ. Let US^{2} denote the unit tangent bundle of S^{2} identified with a subset of S^{2} × S^{2} as follows:

US^{2} = {(x, v) ∈ S^{2} × S^{2} : ⟨x, v⟩ = 0},

where the canonical projection p_{1} : US^{2} → S^{2} is given by p_{1}(x, v) = x. Let p_{2} : S^{3} → US^{2} be a double covering map defined by p_{2}(a) = (aia^{−1}, aja^{−1}). Then the Hopf fibration p : S^{3} → S^{2} is given by p = p_{1} ∘ p_{2}. We now consider a regular curve γ : ℝ → S^{2} with a period l > 0, and define a curve γ̂ : ℝ → US^{2} by

γ̂(s) = (γ(s), γ′(s)/|γ′(s)|).

We denote by I(γ) the element of the homology group H_{1}(US^{2}) represented by the closed curve γ̂ : [0, l] → US^{2}. Let c : ℝ → S^{3} be a lift of the curve γ̂ with respect to the covering p_{2}. Since H_{1}(US^{2}) ≅ ℤ_{2} and the double covering p_{2} satisfies the relation p_{2}(−a) = p_{2}(a), we obtain

c(s + l) = \begin{cases}c(s) & I(γ) = 0, \\ −c(s) & I(γ) = 1. \end{cases} (1)

On the other hand we obtain

c(s)^{−1}c′(s) = \frac{1}{2}|γ′(s)|(k + k(s)i),

where k(s) denotes the geodesic curvature of γ(s). This implies that the torsions of the curves c and c^{−1} satisfy

τ_{c} = 1, τ_{c^{−1}} = −1. (2)

Combining the observations above with the method of Bianchi and Sasaki, we obtain a method for constructing all the flat tori isometrically immersed in S^{3}.

Definition . A periodic admissible pair (p.a.p.) is a pair of periodic regular curves γ_{i} : ℝ → S^{2}, i = 1, 2, such that

(a) k_{1}(s_{1}) > k_{2}(s_{2}) for all s_{1}, s_{2} ∈ ℝ,

(b) |γi^{′}(s)|\sqrt{1 + k_{i}(s)^{2}} = 2,

where k_{i}(s) denotes the geodesic curvature of γ_{i}(s).

CONSTRUCTION OF FLAT TORI IN THE 3-SPHERE AND ITS APPLICATIONS5

Let Γ = (γ_{1}, γ_{2}) be a p.a.p., and let c_{i} be a lift of γ̂_{i} with respect to the covering p_{2}. Using the group structure of S^{3}, we define the map F_{Γ} : ℝ^{2} → S^{3} by

F_{Γ} (s_{1}, s_{2}) = c_{1}(s_{1})c_{2}(s_{2})^{−1}. (3)

Then it follows that the map F_{Γ} is an immersion and induces a flat Riemannian metric g_{Γ} on ℝ^{2}. We now consider the group

G(Γ) = {φ ∈ Diff(ℝ^{2}) : F_{Γ} ∘ φ = F_{Γ} }.

Since each element of G(Γ) is a parallel translation of ℝ^{2}, we identify the group G(Γ) with a subgroup of ℝ^{2}. Then we obtain a flat torus M_{Γ} = (ℝ^{2}, g_{Γ})/G(Γ), and an isometric immersion

f_{Γ} : M_{Γ} → S^{3}

satisfying the relation f_{Γ} ∘ π_{Γ} = F_{Γ}, where π_{Γ} : ℝ^{2} → M_{Γ} denotes the canonical projection.

Theorem 1 ([5, 7]). Let f : M → S^{3} be an isometric immersion of a flat torus M into the unit sphere S^{3}. Then there exist a p.a.p. Γ and a covering map ρ : M → M_{Γ} such that A ∘ f = f_{Γ} ∘ ρ for some isometry A : S^{3} → S^{3}.

Remark 1. If the image of f_{Γ} contains a great circle of S^{3}, either γ_{1} or γ_{2} is a circle in S^{2}. So it follows that there exists a p.a.p. Γ such that the image of f_{Γ} is not congruent to any Hopf torus in S^{3}.

Remark 2. Let l_{i} be the period of γ_{i}. Then

generators of G(Γ) = \begin{cases}(l_{1}, 0), (0, l_{2}) & if I(γ_{1}) = 0, I(γ_{2}) = 0, \\ (2l_{1}, 0), (0, l_{2}) & if I(γ_{1}) = 1, I(γ_{2}) = 0, \\ (l_{1}, 0), (0, 2l_{2}) & if I(γ_{1}) = 0, I(γ_{2}) = 1, \\ (l_{1}, l_{2}), (l_{1}, −l_{2}) & if I(γ_{1}) = 1, I(γ_{2}) = 1. \end{cases} (4)

Remark 3. Consider two curves a_{1} and a_{2} in M_{Γ} given by

a_{1}(s) = π_{Γ} (s, 0), a_{2}(s) = π_{Γ} (0, s). (5)

Then it follows from (4) that they are simple closed curves in M_{Γ}.

Remark 4. Weiner [14, 15] studied the Gauss map of a flat torus in S^{3}, and obtained a method for constructing all the flat tori in S^{3} which is different from ours. For the Gauss map of the immersion f_{Γ} : M_{Γ} → S^{3}, see Section 7 of [7]

In the rest of this section we deal with the intrinsic structure of M_{Γ}. For each p.a.p. Γ = (γ_{1}, γ_{2}), we set

L(γ_{i}) = ∫_{0}^{l_{i}} |γi^{′}(s)|ds, K(γ_{i}) = ∫_{0}^{l_{i}} k_{i}(s)|γi^{′}(s)|ds,

where l_{i} denotes the period of γ_{i}. Then it follows that the intrinsic structure of the flat torus M_{Γ} = (ℝ^{2}, g_{Γ})/G(Γ) is determined by the following data:

{I(γ_{i}), K(γ_{i}), L(γ_{i})}_{i=1,2}.

More precisely, we obtain

Theorem 2 ([8]). Let W (Γ) be a lattice of ℝ^{2} generated by the following vectors.

v⃗_{1}, v⃗_{2}  if I(γ_{1}) = 0, I(γ_{2}) = 0,
2v⃗_{1}, v⃗_{2}  if I(γ_{1}) = 1, I(γ_{2}) = 0,
v⃗_{1}, 2v⃗_{2}  if I(γ_{1}) = 0, I(γ_{2}) = 1,
v⃗_{1} ± v⃗_{2}  if I(γ_{1}) = 1, I(γ_{2}) = 1,

where v⃗_{i} = \frac{1}{2}(K(γ_{i}), L(γ_{i})). Then the flat torus M_{Γ} is isometric to ℝ^{2}/W (Γ).

2. Embedded flat tori in S^{3}

In this section, using the method explained in Section 1, we study the embedded flat tori in S^{3}. Let Γ = (γ_{1}, γ_{2}) be a p.a.p., and let I(Γ) = (I(γ_{1}), I(γ_{2})).

Theorem 3 ([7]). If f_{Γ} : M_{Γ} → S^{3} is an embedding, then I(Γ) = (1, 1).

Proof (Outline). Assume that I(Γ) ≠ (1, 1). Using the embedding f_{Γ}, we identify M_{Γ} with a subset of S^{3}. Let a_{1} and a_{2} be the simple closed curves in M_{Γ} given by (5), and let a_{i}^{+} be a simple closed curve in S^{3} − M_{Γ} obtained by pushing the curve a_{i} a very small amount along a unit normal vector field ξ of M_{Γ}. Then we obtain the links {a_{1}, a_{1}^{+}} and {a_{2}, a_{2}^{+}} in S^{3}, and it follows that the linking numbers of these links satisfy

lk(a_{1}, a_{1}^{+}) ≡ lk(a_{2}, a_{2}^{+}) ≡ 1, mod 2. (6)

We now consider a disk D ⊂ M_{Γ} which does not intersect the union a_{1}∪a_{2}, and let K be a knot in S^{3} given by K = ∂D. Since I(Γ) ≠ (1, 1), it follows from (4) that {a_{1}, a_{2}} is a canonical basis of the homology group H_{1}(V), where V is a Seifert surface of the knot K given by V = M_{Γ} −D. So, by using [4, Chapter10], we see that the Arf invariant of K is given by

Arf(K) ≡ lk(a_{1}, a_{1}^{+}) lk(a_{2}, a_{2}^{+}), mod 2.

Hence, (6) implies Arf(K) = 1. On the other hand, since K = ∂D, we obtain Arf(K) = 0. This is a contradiction.

It follows from Theorem 3 and (1) that if f_{Γ} is an embedding, then the image of f_{Γ} is invariant under the antipodal map of S^{3}. Hence, Theorem 1 implies the following

Theorem 4 ([7]). If f : M → S^{3} is an isometric embedding of a flat torus M, then the image f(M) is invariant under the antipodal map of S^{3}.

As an application of this theorem, we obtain a rigidity theorem for the Clifford tori in S^{3}. For positive numbers R_{1} and R_{2} satisfying R_{1}^{2} + R_{2}^{2} = 1, let S^{1}(R_{1}) × S^{1}(R_{2}) denote the Clifford torus in S^{3} given by

S^{1}(R_{1}) × S^{1}(R_{2}) = {x ∈ ℝ^{4} : x_{1}^{2} + x_{2}^{2} = R_{1}^{2}, x_{3}^{2} + x_{4}^{2} = R_{2}^{2}},

and let i : S^{1}(R_{1}) × S^{1}(R_{2}) → S^{3} denote the inclusion map.

Theorem 5 ([3]). If f : S^{1}(R_{1}) × S^{1}(R_{2}) → S^{3} is an isometric embedding, then there exists an isometry A of S^{3} such that f = A ∘ i.

Proof (Outline). We can show that if f : S^{1}(R_{1}) × S^{1}(R_{2}) → S^{3} is an isometric immersion whose extrinsic diameter is equal to π, then there exists an isometry A of S^{3} such that f = A ∘ i. So the assertion follows from Theorem 4.

Remark 5. RecentlyCDadok-Sha [2] obtained the same result as we have proved in Theorem 4. Their proof is different from ours.

Remark 6. In Theorem 4 the word “embedding” cannot be replaced by “immersion”. In fact, there exists an isometric immersion of a flat torus into S^{3} whose image is not invariant under the antipodal map of S^{3} ([7, Theorem 4.4]). However the author does not know whether the extrinsic diameter of any flat torus isometrically immersed in S^{3} is equal to π. If this is true, the conclusion of Theorem 5 is valid for every isometric immersion f : S^{1}(R_{1}) × S^{1}(R_{2}) → S^{3}.

3. The classification of undeformable flat tori in S^{3}

An isometric immersion f : M → S^{3} is said to be deformable if there exists a nontrivial isometric deformation of f. As a corollary of Theorem 5, it follows that the inclusion map i : S^{1}(R_{1}) × S^{1}(R_{2}) → S^{3} is not deformable ([6]). In this section we give the classification of undeformable flat tori isometrically immersed in S^{3}.

Theorem 6 ([8]). Let f : M → S^{3} be an isometric immersion of a flat torus M into S^{3}. If the mean curvature of the immersion f is not constant, then f is deformable.

Proof (Outline). It follows from Theorem 1 that there exist a p.a.p. Γ = (γ_{1}, γ_{2}) and a covering map ρ : M → M_{Γ} such that

f = A ∘ f_{Γ} ∘ ρ,

where A denotes an isometry of S^{3}. So it is sufficient to show that f_{Γ} is deformable. On the other hand, we see that the mean curvature of f_{Γ} is constant if and only if both γ_{1} and γ_{2} are circles in the unit sphere S^{2}. So, by the assumption, either γ_{1} or γ_{2} is not a circle. This ensures the existence of a nontrivial deformation of Γ which preserves the data {I(γ_{i}), K(γ_{i}), L(γ_{i})}_{i=1,2}. Hence, Theorem 2 implies a nontrivial isometric deformation of f_{Γ}.

We now consider an isometric immersion f : M → S^{3} of a flat torus M into S^{3} with constant mean curvature. In this case, it is easy to see that f is congruent to the immersion

F/G : ℝ^{2}/G → S^{3}, (7)

where F is a covering map of ℝ^{2} onto a Clifford torus S^{1}(R_{1}) × S^{1}(R_{2}) defined by

F (x_{1}, x_{2}) = (R_{1} cos \frac{x_{1}}{R_{1}} , R_{1} sin \frac{x_{1}}{R_{1}} , R_{2} cos \frac{x_{2}}{R_{2}} , R_{2} sin \frac{x_{2}}{R_{2}} ) ,

and G is a subgroup of the covering transformation group of F such that ℝ^{2}/G is compact. The covering transformation group of F, which consists of parallel translations of ℝ^{2}, is generated by the vectors e⃗_{1} = (2πR_{1}, 0) and e⃗_{2} = (0, 2πR_{2}). So the group G is generated by

a⃗ = a_{1}e⃗_{1} + a_{2}e⃗_{2}, b⃗ = b_{1}e⃗_{1} + b_{2}e⃗_{2}, (8)

where a_{i}, b_{i} ∈ ℤ and a_{1}b_{2} − a_{2}b_{1} ≠ 0.

Theorem 7 ([9]). The following statements (a) and (b) are equivalent.

(a) F/G : ℝ^{2}/G → S^{3} is not deformable,

(b) g.c.d.(a_{1} + a_{2}, b_{1} + b_{2}) = g.c.d.(a_{1} − a_{2}, b_{1} − b_{2}) = 1.

Proof (Outline). (a) ⇒ (b). Assume that g.c.d.(a_{1} + a_{2}, b_{1} + b_{2}) = n ≥ 2. Let G_{0} denote the covering transformation group of F, which is generated by e⃗_{1} and e⃗_{2}, and let W (n) be a subgroup of G_{0} given by

W (n) = {n_{1}e⃗_{1} + n_{2}e⃗_{2} : n_{1} + n_{2} ∈ nℤ}.

Then G ⊂ W (n) and the immersion F/W (n) is congruent to the Hopf torus p^{−1}(γ), where γ denotes a n-fold circle in S^{2}. Since the Hopf torus p^{−1}(γ) is deformable for n ≥ 2, we see that the immersion F/W (n) is deformable. So it follows from G ⊂ W (n) that the immersion F/G is deformable. Similarly, the immersion F/G is deformable if g.c.d.(a_{1} − a_{2}, b_{1} − b_{2}) ≠ 1.

(b) ⇒ (a). Let f_{t} : ℝ^{2}/G → S^{3} be an isometric deformation of F/G, and let F_{t} be the isometric deformation of F : ℝ^{2} → S^{3} induced by f_{t}. Then each F_{t} is invariant under the group G. Furthermore we can show that each F_{t} is σ(G)-invariant, where σ denotes an automorphism of G_{0} satisfying

σ(e⃗_{1}) = e⃗_{2}, σ(e⃗_{2}) = e⃗_{1}.

On the other hand, the assumption (b) implies G + σ(G) = G_{0}, and so F_{t} is G_{0}-invariant. Hence we obtain an isometric deformation F_{t}/G_{0} of the isometric embedding F/G_{0} : ℝ^{2}/G_{0} → S^{3}. Since the embedding is congruent to the inclusion map i : S^{1}(R_{1}) × S^{1}(R_{2}) → S^{3}, it follows from Theorem 5 that for each t there exists an isometry A_{t} of S^{3} such that F_{t}/G_{0} = A_{t} ∘ (F/G_{0}). Hence f_{t} = F_{t}/G = A_{t} ∘ (F/G), and so the immersion F/G is not deformable.

By Theorems 6 and 7, we obtain the following classification of undeformable flat tori in S^{3}.

Theorem 8. Let f : M → S^{3} be an isometric immersion of a flat torus M into S^{3}. Then the immersion f is not deformable if and only if it is congruent to the immersion F/G defined by (7) such that the group G is generated by a⃗ and b⃗ satisfying (8) and g.c.d.(a_{1} + a_{2}, b_{1} + b_{2}) = g.c.d.(a_{1} − a_{2}, b_{1} − b_{2}) = 1.

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Department of Mathematics, Utsunomiya University, Mine-machi, Utsunomiya 321-8505, Japan

  E-mail address: kitagawa@@cc.utsunomiya-u.ac.jp