Horocyclic surfaces in hyperbolic 3-space

Kazumi Tsukada Department of Mathematics, Ochanomizu University

This talk is based on the joint work [3] with C.Takizawa.

§1 Introduction

Horocyclic surfaces in hyperbolic 3-space are defined analogously to ruled surfaces in the Euclidean space ℝ^{3}:

foliated by

ruled surfaces

in ℝ^{3} ⋯ lines

horocyclic surfaces in H^{3}

⋯

horocycles

The notion of horocyclic surfaces was introduced by S.Izumiya, K.Saji and M.Takahashi ([1]) as one of important objects in their horospherical geometry.

In this talk I explain

• to construct horocyclic surfaces associated with spacelike curves in the lightcone

• to describe geometric properties of horocyclic surfaces in terms of invariants (curvatures and torsions) of corresponding spacelike curves in the lightcone

• in particular to classify singularities of horocyclic surfaces constructed above

§2 Curves and surfaces in hyperbolic 3-space

⟨ The standard Lorentzian model of hyperbolic 3-space ⟩ Let ℝ_{1}^{4} be a 4dimensional vector space ℝ^{4} with the inner product ⟨, ⟩ defined by

⟨x, y⟩ = −x_{0}y_{0} + ∑_{i=1}^{3} x_{i}y_{i} for x = (x_{0}, x_{1}, x_{2}, x_{3}), y = (y_{0}, y_{1}, y_{2}, y_{3}) ∈ ℝ^{4}.

We say that a non-zero vector v ∈ ℝ_{1}^{4} is spacelike, lightlike, or timelike if ⟨v, v⟩ > 0, ⟨v, v⟩ = 0 or ⟨v, v⟩ < 0, respectively.

We define

H_{+}^{3} = {x ∈ ℝ_{1}^{4} | ⟨x, x⟩ = −1, and x_{0} ≥ 0} : the hyperbolic 3-space
LC_{+} = {x ∈ ℝ_{1}^{4} | ⟨x, x⟩ = 0 and x_{0} > 0} : the future light cone
S_{1}^{3} = {x ∈ ℝ_{1}^{4} | ⟨x, x⟩ = 1} : the de Sitter space

Then H_{+}^{3} is a complete Riemannian manifold of constant sectional curvature −1.

⟨ Curves in H_{+}^{3} ⟩ Let γ : I → H_{+}^{3} be a unit speed curve defined on an open interval I of ℝ. We use t(s) for the tangent vector γ′(s) with ∥t(s)∥ = 1. The vector t′(s) − γ(s) is orthogonal to γ(s) and t(s). We assume that t′(s) − γ(s) is not zero. Then we have a unit vector n(s) = (t′(s) − γ(s))/∥t′(s) − γ(s)∥. Moreover we define e(t) such that {γ(s), t(s), n(s), e(s)} is a positive orthonormal basis. Then the following formula (Frene-Seret formula) holds:

( \begin{matrix}γ′(s) \\ t′(s) \\ n′(s) \\ e′(s) \end{matrix} ) = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & κ_{h}(s) & 0 \\ 0 & −κ_{h}(s) & 0 & τ_{h}(s) \\ 0 & 0 & −τ_{h}(s) & 0 \end{matrix} ) ( \begin{matrix}γ(s) \\ t(s) \\ n(s) \\ e(s) \end{matrix} ) .

A curve γ in H_{+}^{3} is called a horocycle if κ_{h}(s) ≡ 1 and τ_{h}(s) ≡ 0. Proposition 2.1 (1) Let γ : I → H_{+}^{3} be a horocycle . Then γ(s) + n(s) is a constant lightlike vector which is denoted by l_{0} and called to be the polar vector of a horocycle γ. Moreover we have

γ(s) = γ(s_{0}) + (s − s_{0})t(s_{0}) + \frac{1}{2}(s − s_{0})^{2}l_{0} for some fixed s_{0} ∈ I.

(2) For vectors l_{0} ∈ LC_{+} and ω_{0} ∈ S_{1}^{3} with ⟨l_{0}, ω_{0}⟩ = 0, {x ∈ H_{+}^{3} | ⟨x, l_{0}⟩ + 1 = 0, ⟨x, ω_{0}⟩ = 0} is a horocycle of H_{+}^{3} whose polar vector is given by l_{0}.

⟨ Surfaces in H_{+}^{3} ⟩ Let M be a 2-dimensional connected oriented manifold and f : M → H_{+}^{3} an immersion. We define two kinds of “ Gauss maps”:

Gauss map  𝔼 : M → S_{1}^{3} a unit normal vector field in H_{+}^{3}
Lightcone Gauss map  𝕃^{±} = f ± 𝔼 : M → LC_{+}

and define associated shape operators at each point p ∈ M:

Shape operator  A_{p} : T_{p}M → T_{p}M defined by d𝔼 = −df(A_{p})
Hyperbolic shape operator  S_{p}^{±} = −1_{T_{p}M} ± A_{p} where d𝕃^{±} = df_{p} ± d𝔼_{p} = −df_{p}(S_{p}^{±})

We denote the eigenvalues of A_{p} by κ_{i}(p) and those of S_{p}^{±} by κ̄_{i}^{±}(p) (i = 1, 2), respectively. Evidently, A_{p} and S_{p}^{±} have same eigenvectors and the relation κ̄_{i}^{±}(p) = −1 ± κ_{i}(p). We call κ_{i}(p) principal curvatures and κ̄_{i}^{±}(p) hyperbolic principal curvatures of (M, f) at p ∈ M, respectively. The Gauss curvature of M at p is given by K(p) = −1 + det A_{p} = −1 + κ_{1}(p)κ_{2}(p) and similarly the hyperbolic Gauss curvature K_{h}^{±} is defined by K_{h}^{±}(p) = det S_{p}^{±} = κ̄_{1}^{±}(p)κ̄_{2}^{±}(p).

Now we recall the geometry of horospheres. For a non-zero vector v ∈ ℝ_{1}^{4} and a real number c ∈ ℝ, we define the hyperplane by HP (v, c) = {x ∈ ℝ_{1}^{4} | ⟨x, v⟩ = c}. Then the intersections M = H_{+}^{3} ∩ HP (v, c) (if not empty) are totally umbilical surfaces of H_{+}^{3} with κ^{2} = \frac{c^{2}}{⟨v,v⟩+c^{2}} , where κ denotes the principal curvature . In particular for v ∈ LC_{+} and c < 0, M = H_{+}^{3} ∩ HP (v, c) has the constant principal curvature 1 or −1 and hence its Gauss curvature vanishes. It is called a horosphere. Let M = H_{+}^{3} ∩ HP (v, −1) with v ∈ LC_{+} be a horosphere. We consider an inclusion map ι : M → H_{+}^{3}. For x ∈ M, 𝔼(x) = v − x is the unit normal vector of M at x ∈ M. Therefore we have the lightcone Gauss map 𝕃^{+}(x) = x + 𝔼(x) = v at x ∈ M. It is a constant map and hence the hyperbolic Gauss curvature K_{h}^{+} vanishes. Definition 2.2 ([1]) An immersed surface (M, f) in H_{+}^{3} is a horospherical flat surface (briefly a horo-flat surface) if K_{h}^{+}(p) = 0 or K_{h}^{−}(p) = 0 at any point p ∈ M .

Horospheres are examples of horo-flat surfaces and they are also totally umbilical. Now we assume that an immersed surface (M, f) is a horo-flat surface without umbilical points. Then the shape operator A of (M, f) has two different eigenvalues 1 and κ(≠ 1) . We denote by F_{1} the line foliation corresponding to the principal curvature 1. Then the lines of F_{1} are horocycles in H_{+}^{3} whose polar vectors coincide with the lightcone Gauss map 𝕃 (Theorem 4.4 in [1]). Definition 2.3 ([1]) A surface (M, f) in H_{+}^{3} is called a horocyclic surface if it is foliated by horocycles in H_{+}^{3}.

§3 Spacelike curves in the lightcone

We introduce the Frenet-Serret formula for spacelike curves in the lightcone LC_{+}. Let l : I → LC_{+} be a unit speed spacelike curve defined on an open interval I of ℝ. We set t(t) = l′(t) and define a function κ on I by κ(t) = ⟨l′′(t), l′′(t)⟩. We put n(t) = −\frac{κ(t)}{2}l(t) − l′′(t) . Then we have ⟨n(t), n(t)⟩ = 0 and ⟨n(t), l(t)⟩ = 1. We denote by e(t) the unit spacelike vector which is orthogonal to l(t), t(t) and n(t) such that the basis {l(t), t(t), e(t), n(t)} has a positive orientation. Then we have an pseudo orthonormal frame field {l(t), t(t), e(t), n(t)} along l. We define a function τ on I by τ(t) = ⟨n′(t), e(t)⟩. Then we have the following formula of Frenet-Serret type :

(3.1)  ( \begin{matrix}l′(t) \\ t′(t) \\ e′(t) \\ n′(t) \end{matrix} ) = ( \begin{matrix}0 & 1 & 0 & 0 \\ −\frac{κ(t)}{2} & 0 & 0 & −1 \\ −τ(t) & 0 & 0 & 0 \\ 0 & \frac{κ(t)}{2} & τ(t) & 0 \end{matrix} ) ( \begin{matrix}l(t) \\ t(t) \\ e(t) \\ n(t) \end{matrix} ) .

§4 Construction of horocyclic surfaces

We construct horocyclic surfaces in H_{+}^{3} by a slightly different method to [1] and study their properties. Let l : I → LC_{+} and ω : I → S_{1}^{3} be smooth maps from an open interval I of ℝ with ⟨l(t), ω(t)⟩ = 0 for any t ∈ I. Now we set subsets M̃ of H_{+}^{3} × I and M of H_{+}^{3} as follows:

M̃ = {(x, t) ∈ H_{+}^{3} × I | ⟨x, l(t)⟩ + 1 = 0, ⟨x, ω(t)⟩ = 0},
M = {x ∈ H_{+}^{3} | there exists t ∈ I with (x, t) ∈ M̃}.

Then the set M̃ is a 2-dimensional submanifold of H_{+}^{3} × I. We restrict the projections of H_{+}^{3} ×I onto H_{+}^{3} and of H_{+}^{3} ×I onto I to M̃, which are denoted by π_{1} and π_{2}, respectively. Then M = π_{1}(M̃) and M is a horocyclic surface in H_{+}^{3} possibly with singularites. In fact, for each t ∈ I we put

C_{t} = π_{1} ∘ π_{2}^{−1}(t) = {x ∈ H_{+}^{3} | ⟨x, l(t)⟩ + 1 = 0, ⟨x, ω(t)⟩ = 0}.

By Proposition 2.1 (2), it follows that C_{t} is a horocycle in H_{+}^{3} whose polar vector is l(t).

Now we will construct a parametrization f : ℝ×I → H_{+}^{3} with f(ℝ×I) = M. We choose a smooth curve γ̃(t) = (γ(t), t) in M̃. Then γ is a smooth curve in H_{+}^{3} which is contained in M. For each t ∈ I , we put a_{2}(t) = l(t) − γ(t) and a_{3}(t) = ω(t). Then γ(t), a_{2}(t), and a_{3}(t) are orthonormal. We define a unit spacelike vector a_{1}(t) such that a_{1}(t) is orthogonal to γ(t), a_{2}(t), a_{3}(t) and the basis {γ(t), a_{1}(t), a_{2}(t), a_{3}(t)} has a positive orientation. Then we have a_{1}(t) ∈ T_{γ(t)}C_{t} and the horocycle C_{t} is parametrized as in Proposition 2.1 (1). We now define a map f of ℝ × I to H_{+}^{3} by

f(s, t) = γ(t) + sa_{1}(t) + \frac{1}{2}s^{2}l(t).

Then f(ℝ × I) = M.

In order to study the detailed geometric properties of f , we consider a special class of horocyclic surfaces which satisfy the following condition: (HC) we assume that l : I → LC_{+} has a unit speed and that l′(t) = ω(t). Under the assumption above, the structure equation for the orthonormal frame fields {γ(t), a_{1}(t), a_{2}(t), a_{3}(t)} is of the following form:

( \begin{matrix}γ′(t) \\ a′_{1}(t) \\ a′_{2}(t) \\ a′_{3}(t) \end{matrix} ) = ( \begin{matrix}0 & 0 & 0 & c_{3}(t) \\ 0 & 0 & 0 & c_{5}(t) \\ 0 & 0 & 0 & c_{6}(t) \\ c_{3}(t) & −c_{5}(t) & −c_{6}(t) & 0 \end{matrix} ) ( \begin{matrix}γ(t) \\ a_{1}(t) \\ a_{2}(t) \\ a_{3}(t) \end{matrix} ) , c_{3}(t) + c_{6}(t) = 1.

The induced metric g on ℝ × I by f is given by

g = ds^{2} + \frac{1}{4}(s^{2} + 2c_{5}(t)s + 2c_{3}(t))^{2}dt^{2}.

This implies the following: Lemma 4.1 Under the assumption (HC), a point (s_{0}, t_{0}) ∈ ℝ × I is a singular point of f if and only if

s_{0}^{2} + 2c_{5}(t_{0})s_{0} + 2c_{3}(t_{0}) = 0.

At a non-singular point (s, t) ∈ ℝ × I, the shape operator A is given by

A (\frac{∂}{∂s}) = \frac{∂}{∂s}, A (\frac{∂}{∂t}) = (1 − \frac{2}{s^{2} + 2c_{5}(t)s + 2c_{3}(t)}) \frac{∂}{∂t}.

Consequently we obtain the following. Proposition 4.2 A horocyclic surface M constructed above is horo-flat.

Now we describe the geometric properties of a horocyclic surface M in terms of invariants κ, τ defined in (3.1) of the corresponding spacelike curve l : I → LC_{+} . First we show the criteria for the existence of singularities on the horocycle C_{t} for t ∈ I. Proposition 4.3 (1) If κ(t) > 1, there exist two singular points of f on the horocycle C_{t}.

(2) If κ(t) = 1, there exists only one singular point of f on the horocycle C_{t}.

(3) If κ(t) < 1, there exists no singularity of f on the horocycle C_{t}.

Applying the criteria above, we can construct complete non-singular horo-flat horocyclic surfaces.Let l : ℝ → LC_{+} be a unit speed spacelike curve defined on the whole ℝ. Suppose that there exists a positive number ε > 0 such that κ(t) ≤ 1 − ε for any t ∈ ℝ.

Then by Proposition 4.3 (3), f is an immersion of ℝ × ℝ into H_{+}^{3} . Moreover we see that the induced Riemannian metric g on ℝ × ℝ by f is geodesically complete.

§5 Classification of singularities

Now we study the cases of Proposition 4.3 (1) and (2) in the previous section. For a fixed t_{0} ∈ I with κ(t_{0}) ≥ 1, we may assume that a point (0, t_{0}) ∈ ℝ × I is a singular point of f. By Lemma 4.1, it follows that c_{3}(t_{0}) = 0 and the other singular point on C_{t_{0}} is given by (−2c_{5}(t_{0}), t_{0}). Moreover we assume that c_{5}(t_{0}) ≥ 0. Then we have the following theorem:

Theorem 5.1 (A) Suppose that κ(t_{0}) > 1. Then we have the following:

(1) The point (0, t_{0}) is the cuspidal edge if \frac{1}{2}κ′(t_{0}) − \sqrt{κ(t_{0}) − 1}τ(t_{0}) ≠ 0.

(2) The point (0, t_{0}) is the swallowtail if \frac{1}{2}κ′(t_{0}) − \sqrt{κ(t_{0}) − 1}τ(t_{0}) = 0 and \frac{1}{2}κ′′(t_{0}) − \sqrt{κ(t_{0}) − 1}τ′(t_{0}) − τ(t_{0})^{2} ≠ 0.

(3) The point (−2c_{5}(t_{0}), t_{0}) is the cuspidal edge if \frac{1}{2}κ′(t_{0}) + \sqrt{κ(t_{0}) − 1}τ(t_{0}) ≠ 0.

(4) The point (−2c_{5}(t_{0}), t_{0}) is the swallowtail if \frac{1}{2}κ′(t_{0}) + \sqrt{κ(t_{0}) − 1}τ(t_{0}) = 0 and \frac{1}{2}κ′′(t_{0}) + \sqrt{κ(t_{0}) − 1}τ′(t_{0}) − τ(t_{0})^{2} ≠ 0. (B) Suppose that κ(t_{0}) = 1. Then we have c_{5}(t_{0}) = 0 and the point (0, t_{0}) is only one singular point on C_{t_{0}}. Moreover we have the following:

(1) The point (0, t_{0}) is the cuspidal edge if κ′(t_{0}) ≠ 0.

(2) The point (0, t_{0}) is the cuspidal beaks if κ′(t_{0}) = 0, κ′′(t_{0}) > 0, and κ′′(t_{0}) ≠ 2τ(t_{0})^{2}.

(3) The point (0, t_{0}) is the cuspidal lips if if κ′(t_{0}) = 0, κ′′(t_{0}) < 0.

Here the cuspidal edge is a germ of surface diffeomorphic to {(x_{1}, x_{2}, x_{3})|x_{1} = u^{2}, x_{2} = u^{3}, x_{3} = v}, the swallowtail is a germ of surface diffeomorphic to {(x_{1}, x_{2}, x_{3})|x_{1} = 3u^{4} + u^{2}v, x_{2} = 4u^{3} + 2uv, x_{3} = v}, the cuspidal beaks is a germ of surface diffeomorphic to {(x_{1}, x_{2}, x_{3})|x_{1} = 3u^{4} − 2u^{2}v^{2}, x_{2} = u^{3} − uv^{2}, x_{3} = v} and the cuspidal lips is a germ of surface diffeomorphic to {(x_{1}, x_{2}, x_{3})|x_{1} = 3u^{4} + 2u^{2}v^{2}, x_{2} = u^{3} + uv^{2}, x_{3} = v}. Remark This theorem is an another formulation of the classification theorem (Theorem 6.2 in [1]) proved by S.Izumiya, K.Saji and M.Takahashi. Compared with their theorem, our classification is to use invariants (the curvatures κ and the torsions τ ) of spacelike curves in the lightcone.

Recently useful criteria in order to recognize the singularities of surfaces have been developed (cf. [2], [1]). We apply their recognition lemmas and prove the Theorem above.

References

[1] S.Izumiya, K.Saji and M.Takahashi: Horospherical flat surfaces in hyperbolic 3space,preprint.

[2] M.Kokubu, W.Rossman, K.Saji, M.Umehara and K.Yamada: Singularities of flat fronts in hyperbolic space , Pacific J.Math.,221(2005),303-351

[3] C.Takizawa and K.Tsukada : Horocyclic surfaces in hyperbolic 3-space , preprint