Curved Flats and Isothermic Surfaces

F. Burstall, U. Hertrich-Jeromin, F. Pedit^{∗}, U. Pinkall

September 17, 1997

Abstract

We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso’s fourth order partial differential equation is derived and, using a solution of this equation, a Möbius invariant frame for an isothermic surface is built.

1 Introduction

These notes grew out of a series of discussions on a recent paper by J. Cieśliński, P. Goldstein and A. Sym [3]: these authors give a characterization of isothermic surfaces as ”soliton surfaces” by introducing a spectral parameter. In trying to understand the geometric meaning of this spectral parameter, we observed some analogies with the theory of conformally flat hypersurfaces in a four-dimensional space form: Guichard’s nets may be understood as analogues of isothermic parametrizations of Riemannian surfaces (cf.[6, no.3.4.1]), and so it seems natural to look for relations between the theory of isothermic surfaces in three-dimensional space forms and the theory of conformally flat hypersurfaces in four-dimensional space forms. Here we would like to present some results we found — especially the possibility of constructing isothermic surfaces using

2 Curved Flats

A curved flat is the natural generalization of a developable surface in Euclidean space: it is an (intrinsically) flat submanifold of a symmetric space for which the curvature tensor of the ambient space vanishes on each tangent space and whose dimension is equal to the rank of the symmetric space — this is the maximal dimension possible. Thus, a curved flat may be thought of as the enveloping submanifold of a congruence of flats — totally geodesic submanifolds — of the symmetric space^{1}. Taking a regular parametrization γ : M^{r} → G/K of a curved flat, r being the rank of G/K, and a framing F : M → G of this parametrization, the Maurer-Cartan form Φ = F^{−1}dF of the framing has a natural decomposition Φ = Φ_{k} + Φ_{p} according to the Cartan decomposition g = k ⊕ p of the Lie algebra g. Now the condition for γ to parametrize a curved flat may be formulated as^{2}

∗Partially supported by NSF grant DMS 2905293

[Φ_{p} ∧ Φ_{p}] ≡ 0 . (1)

In the case of a pseudo-Riemannian symmetric space this should be taken as the Definition of a curved flat: An immersion γ : M^{r} → G/K is said to parametrize a curved flat, if the p-part in the Cartan decomposition^{3} of the Maurer-Cartan form F^{−1}dF = Φ = Φ_{k} +Φ_{p} of a framing F : M → G of γ defines a congruence p ↦ Φ_{p}|_{p}(T_{p}M) of maximal abelian subalgebras^{4} of g.

At this point we should remark that curved flats naturally arise in one parameter families [5]: setting Φ_{λ} := Φ_{k} + λΦ_{p} (2)

the Maurer-Cartan equation dΦ_{λ} + \frac{1}{2}[Φ_{λ} ∧ Φ_{λ}] = 0 for the loop λ ↦ Φ_{λ} of forms splits into the three equations

0 = dΦ_{k} + \frac{1}{2}[Φ_{k} ∧ Φ_{k}]
0 = dΦ_{p} + [Φ_{k} ∧ Φ_{p}]
0 = [Φ_{p} ∧ Φ_{p}] ,

(3)

and hence the integrability of the loop λ ↦ Φ_{λ} is equivalent to the forms Φ_{λ} being the Maurer-Cartan forms for some framings F_{λ} : M^{r} → G of curved flats γ_{λ} : M^{r} → G/K. Thus integrable systems theory may be applied to produce examples.

Now we will consider the case leading to the theory of isothermic surfaces: let

G := O_{1}(5) and K := O(3) × O_{1}(2) . (4)

The coset space G_{+}(5, 3) = G/K of space-like 3-planes in the Minkowski space IR_{1}^{5} becomes a six dimensional semi-Riemannian symmetric space of signature (3, 3) when endowed with the pseudo Riemannian metric induced by the Killing form. The rank of this symmetric space being 2, we will consider two-dimensional curved flats

1To get developable surfaces in Euclidean space as a special case of curved flats, we have to omit the dimension assumption.

2The product

[Φ ∧ Ψ](v, w) := [Φ(v), Ψ(w)] − [Φ(w), Ψ(v)] defines a symmetric product on the space of Lie algebra valued 1-forms with values in the space of Lie algebra valued 2-forms.

3The subgroup K is not necessarily compact — nevertheless the decomposition g = k ⊕ p is called “Cartan decomposition”, if the characteristic conditions

[k, k] ⊂ k, [k, p] ⊂ p, [p, p] ⊂ k are satisfied.

4As a consequence, γ is curvature isotropic in the sense [5] and hence intrinsically and extrinsically flat. It is not clear, however, whether curvature isotropic surfaces are necessarily curved flats.

γ : M^{2} → G_{+}(5, 3) . (5)

Fixing a pseudo orthonormal basis (e_{1}, …, e_{5}) of the Minkowski space IR_{1}^{5} with

(⟨e_{i}, e_{j}⟩)_{ij} = E_{5} := ( \begin{matrix}I_{3} & 0 \\ 0 & 0 1 1 0 \end{matrix} ) , (6)

we get the matrix representations

\begin{matrix}O_{1}(5) = \{A ∈ Gl(5, IR)|A^{t}E_{5}A = E_{5}\} \\ o_{1}(5) = \{X ∈ gl(5, IR)|(E_{5}X) + (E_{5}X)^{t} = 0\} . \end{matrix} (7)

The subalgebra k and its complementary linear subspace p in the Cartan decomposition of o_{1}(5) are given by the +1- resp. −1-eigenspaces of the involutive automorphism Ad(Q) : o_{1}(5) → o_{1}(5) with Q = ( \begin{matrix}−I_{3} & 0 \\ 0 & I_{2} \end{matrix} ). Writing down the Maurer-Cartan form of a framing F : M^{2} → O_{1}(5) of our curved flat γ : M^{2} → G_{+}(5, 3) with this notation we obtain

F^{−1}dF = Φ = Φ_{k} + Φ_{p} with
Φ_{k} = ( 0 ν^{Ω 0} ) :  T M → o(3) × o_{1}(2)
Φ_{p} = ( 0 η_{−E_{2}η 0} ) : T M → p .

(8)

Since Φ_{p} is abelian, we can put η into the standard form

η = ( \begin{matrix}ω_{1} & −aω_{1} \\ ω_{2} & aω_{2} \\ 0 & 0 \end{matrix} ) (a ∈ C^{∞}(M)) (9)

by taking an O(3)-gauge ( \begin{matrix}H & 0 \\ 0 & I_{2} \end{matrix} ) : M → O(3) × O_{1}(2). Henceforth, we make the regularity assumption that a is never zero^{5}. We may then take a ≡ 1 by applying an O_{1}(2)-gauge.

Calculating the Maurer-Cartan equation using the ansatz

Ω = ( \begin{matrix}0 & ω & −ψ_{1} \\ −ω & 0 & −ψ_{2} \\ ψ_{1} & ψ_{2} & 0 \end{matrix} ) and ν = ( \begin{matrix}ν & 0 \\ 0 & −ν \end{matrix} ) (10)

together with η given by (9), we see that

dω_{1} = dω_{2} = 0 . (11)

5This is equivalent to the assumption that the metric induced by the immersion γ (by pulling back the Killing form) is nondegenerate.

So we are given canonical coordinates (x, y) : M → IR^{2} by integrating^{6} the forms ω_{1} and ω_{2}. Moreover, since we also have dν = 0, we may set ν = −du for a suitable function u ∈ C^{∞}(M) — this gives us ω = u_{y}dx − u_{x}dy, where u_{x} and u_{y} denote the partial derivatives of u in x- resp. y-directions. Finally, the equations ψ_{1} ∧ ω_{1} = 0 and ψ_{2} ∧ ω_{2} = 0 show that ψ_{1} = e^{u}k_{1}dx and ψ_{2} = e^{u}k_{2}dy for two functions k_{i} ∈ C^{∞}(M).

We now perform a final O_{1}(2)-gauge ( \begin{matrix}I_{3} & 0 \\ 0 & \begin{matrix}e^{u} & 0 \\ 0 & e^{−u} \end{matrix} \end{matrix} ) : M → O(3) × O_{1}(2) and

insert the spectral parameter λ to obtain the Maurer-Cartan form discussed in (cf.[3]):

Φ_{λ} = ( \begin{matrix}0 & u_{y}dx − u_{x}dy & −e^{u}k_{1}dx & λe^{u}dx & −λe^{−u}dx \\ −u_{y}dx + u_{x}dy & 0 & −e^{u}k_{2}dy & λe^{u}dy & λe^{−u}dy \\ e^{u}k_{1}dx & e^{u}k_{2}dy & 0 & 0 & 0 \\ λe^{−u}dx & −λe^{−u}dy & 0 & 0 & 0 \\ −λe^{u}dx & −λe^{u}dy & 0 & 0 & 0 \end{matrix} ) . (12)

We are now lead directly to the theory of

3 Isothermic Surfaces

In the context of Möbius geometry the three sphere S^{3} is viewed as the projective lightcone IP L^{4} in IR_{1}^{5} while the Lorentzian sphere {v ∈ IR_{1}^{5}|⟨v, v⟩ = 1} should be interpreted as the space of (oriented) spheres in the three sphere^{7} (cf.[1]). Now, denoting by

\begin{matrix}n := F e_{3} : M → S_{1}^{5} = \{v ∈ IR_{1}^{5}|⟨v, v⟩ = 1\} \\ f := F e_{4} : M → L^{4} = \{v ∈ IR_{1}^{5}|⟨v, v⟩ = 0\} \\ f̂:= F e_{5} : M → L^{4} \end{matrix} (13)

one of the sphere congruences resp. the two immersions given by our frame F, we see that Theorem: The sphere congruence n given by our curved flat is a Ribeaucour sphere congruence^{8}, which is enveloped by two isothermic immersions f and f̂ (cf.[1, S.362]): Since

\begin{matrix}⟨f, n⟩ = 0 and ⟨df, n⟩ ≡ 0, \\ ⟨f̂, n⟩ = 0 and ⟨df̂, n⟩ ≡ 0 , \end{matrix} (14)

the immersions f and f̂ do envelop the sphere congruence n and, since the bilinear forms

\begin{matrix}⟨df, dn⟩ = λe^{2u}(k_{1}dx^{2} + k_{2}dy^{2}), \\ ⟨df̂, dn⟩ = λ(−k_{1}dx^{2} + k_{2}dy^{2}) \end{matrix} (15)

6Since our theory is local, all closed forms may be assumed to be exact.

7Or, equivalently, it may be interpreted as the space of (oriented) spheres and planes in Euclidean three space IR^{3}: the polar hyperplane to a vector v of the Lorentz sphere intersects the three sphere — thought of as the absolute quadric in projective four space — in a two sphere. Stereographic projection yields a sphere in IR^{3} or, if the projection center lies on the sphere, a plane.

8The curvature lines on the two enveloping immersions correspond.

are diagonal with respect to the induced metrics

\begin{matrix}⟨df, df⟩ = λ^{2}e^{2u}(dx^{2} + dy^{2}), \\ ⟨df̂, df̂⟩ = λ^{2}e^{−2u}(dx^{2} + dy^{2}) , \end{matrix} (16)

the two immersions f and f̂ are isothermic^{9}.

It is quite difficult to calculate the first and second fundamental forms of these isothermic immersions, when they are projected to S^{3} resp. IR^{3}, but applying a (constant) conformal change (constant O_{1}(2)-gauge)

f ⇝ \frac{1}{λ}f and f̂ ⇝ λf̂ or

f ⇝ λf and f̂ ⇝ \frac{1}{λ} f̂ (17)

and sending λ → 0, f̂ resp. f becomes a constant vector — Φ_{λ=0}e_{5} resp. Φ_{λ=0}e_{4} vanishes. This constant light-like vector may be interpreted as the point at infinity and we therefore obtain an isothermic immersion f : M → IR^{3} with first and second fundamental forms

\begin{matrix}I = e^{2u}(dx^{2} + dy^{2}) \\ II = e^{2u}(k_{1}dx^{2} + k_{2}dy^{2}) \end{matrix} (18)

resp. its Euclidean dual surface f̂: M → IR^{3} with first and second fundamental forms

\begin{matrix}I = e^{−2u}(dx^{2} + dy^{2}) \\ II = −k_{1}dx^{2} + k_{2}dy^{2} . \end{matrix} (19)

We now recognise the remaining three equations from the Maurer-Cartan equation for Φ_{λ}

\begin{matrix}0 = Δu + e^{2u}k_{1}k_{2} \\ 0 = k_{1y} + (k_{1} − k_{2})u_{y} \\ 0 = k_{2x} − (k_{1} − k_{2})u_{x} \end{matrix} (20)

as the Gauß and Codazzi equations of the Euclidean immersion f resp. its dual f̂^{10} [2], [3]. As a consequence, we can invert our construction and build a curved flat from an isothermic surface: Theorem. Given an isothermic surface f : M^{2} → IR^{3} and its Euclidean dual surface f̂: M → IR^{3} we get a curved flat γ : M → G_{+}(5, 3) integrating the Maurer-Cartan form (12), which we are able to write down knowing the first and second fundamental forms of the immersions f and f̂^{11}.

9The bundle defined by span{n, f, f̂} over M is flat (cf.(12)) and so the map p ↦ d_{p}f(T_{p}M) defines a normal congruence of circles [4]: for each p ∈ M

t ↦ f_{t}(p) := \frac{1}{\sqrt{2}} sin t · n(p) + \frac{1}{2}(1 + cos t) · f(p) − \frac{1}{2}(1 − cos t) · f̂(p)

parametrizes the circle (d_{p}f(T_{p}M))^{⊥} meeting the sphere n(p) in f(p) and f̂(p) orthogonal. Since n, f and f̂ are parallel sections in this bundle, the maps p ↦ f_{t}(p) (which generically are not degenerate)

parametrize the surfaces orthogonal to this congruence of circles.

In general the immersions f and f̂ = f_{π} will be the only isothermic surfaces among the surfaces.

10When the normal congruence of circles mentioned in footnote 9 (p.5) is projected to Euclidean three space IR^{3}, we see that, in the limit λ → 0, the circles become straight lines — circles meeting the collapsed surface f̂ resp. f in the point at infinity — while the Ribeaucour sphere congruence enveloped by the two surfaces f and f̂ becomes the congruence of tangent planes of f resp. f̂.

Another way to obtain these two Euclidean immersions is presented in [3]. Applying Sym’s formula to the associated family of frames F = F (λ), we obtain a map

(\frac{∂}{∂λ}F)F^{−1}|_{λ=0} : M → p ; (21)

interpreting p as two copies of Euclidean three space^{12} IR^{3} this map gives us the immersion f, and in the other copy of IR^{3}, its dual f̂: this can be seen by looking at the differential

\begin{matrix}d(\frac{∂}{∂λ}F)F^{−1}|_{λ=0} = F_{0}Φ_{p}F_{0}^{−1} \\ ≅ H_{3} ( \begin{matrix}e^{u}dx & −e^{−u}dx \\ e^{u}dy & e^{−u}dy \\ 0 & 0 \end{matrix} ) . \end{matrix} (22)

Here F_{0} = ( \begin{matrix}H_{3} & 0 \\ 0 & I_{2} \end{matrix} ) solves the equation F_{0}^{−1}dF_{0} = Φ_{k} and thus we may view H_{3} :

M → O(3) as an Euclidean framing of f resp. f̂.

There is another possibility for producing isothermal surfaces in Euclidean space IR^{3} (or S^{3}): that is, by using a solution of

4 Calapso’s equation

To understand this, we write down the Maurer-Cartan form of a frame F : M → O_{1}(5), which is Möbius-invariantly connected to a given immersion: taking f = F e_{4} the (unique) isometric lift of the isothermic immersion and n = F e_{3} the central sphere congruence (conformal Gauß map) of the immersion, the frame is determined by the assumption of being an adapted frame (i.e. F e_{1} = f_{x} and F e_{2} = f_{y}). The associated Maurer-Cartan form will be

Φ = ( \begin{matrix}0 & 0 & kdx & dx & χ_{1} \\ 0 & 0 & −kdy & dy & χ_{2} \\ −kdx & kdy & 0 & 0 & τ \\ −χ_{1} & −χ_{2} & −τ & 0 & 0 \\ −dx & −dy & 0 & 0 & 0 \end{matrix} ) , (23)

k^{2} being the conformal factor relating the metric induced by the central sphere congruence to the isometric one, and the 1-forms χ_{1}, χ_{2} and τ to be determined. From the Maurer-Cartan equation for this form we learn that

11Since this construction depends on the conformal rather than the Euclidean geometry of the ambient space, we generally get a whole three parameter family of “loops” of curved flats from one isothermic surface: when viewing our given isothermic surface as a surface in the three sphere S^{3}, we may choose the point at infinity arbitrarily.

12Here the Euclidean metric is induced by the quadratic form \frac{1}{2}trΦ_{p}^{t}Φ_{p} instead of the Killing form.

\begin{matrix}τ = k_{x}dx − k_{y}dy \\ χ_{1} = (\frac{1}{2}k^{2} − u)dx − \frac{k_{xy}}{k} dy \\ χ_{2} = −\frac{k_{xy}}{k} dx + (\frac{1}{2}k^{2} + u)dy , \end{matrix} (24)

where u ∈ C^{∞}(M) is a function satisfying the differential equation

du = −((\frac{k_{xy}}{k})_{y} + (k^{2})_{x})dx + ((\frac{k_{xy}}{k})_{x} + (k^{2})_{y})dy (25)

— the integrability condition of this equation is a fourth order partial differential equation closely related to Calapso’s original equation [2]:

0 = Δ(\frac{k_{xy}}{k}) + 2(k^{2})_{xy} (26)

This shows, that Theorem: Any isothermic surface gives rise to a solution of Calapso’s equation.

Conversely, from a solution k ∈ C^{∞}(M) of Calapso’s equation we can construct a Möbius invariant frame of an isothermic surface by integrating the Maurer-Cartan form (23), where the function u is a solution of (25).

Now, applying a conformal change f ⇝ \frac{1}{k}f while fixing the central sphere congruence n ⇝ n, the Maurer-Cartan form of the associated frame becomes

Φ = ( \begin{matrix}0 & ω & kdx & \frac{1}{k} dx & χ_{1} \\ −ω & 0 & −kdy & \frac{1}{k} dy & χ_{2} \\ −kdx & kdy & 0 & 0 & 0 \\ −χ_{1} & −χ_{2} & 0 & 0 & 0 \\ − \frac{1}{k} dx & − \frac{1}{k} dy & 0 & 0 & 0 \end{matrix} ), (27)

where

\begin{matrix}ω = −\frac{k_{y}}{k} dx + \frac{k_{x}}{k} dy \\ χ_{1} = k(\frac{k_{xx}}{k} − \frac{k_{x}^{2}+k_{y}^{2}}{2k^{2}} + \frac{1}{2}k^{2} − u)dx \\ χ_{2} = k(\frac{k_{yy}}{k} − \frac{k_{x}^{2}+k_{y}^{2}}{2k^{2}} + \frac{1}{2}k^{2} + u)dy \end{matrix} . (28)

Here we see that the central sphere congruence of an isothermic surface is a Ribeaucour sphere congruence, which actually is a characterisation of isothermic surfaces (cf.[1, S.374]), and hence it has flat normal bundle as a codimension two surface in the Lorentz sphere S_{1}^{4}.

In general, the second enveloping surface of the central sphere congruence of an isothermic surface will not be an isothermic surface and it seems to be difficult to built a curved flat starting with it. But in a quite simple case this is possible:

5 Example

Starting with a surface of revolution

f(x, y) = (r(x) cos y, r(x) sin y, z(x)) , (29) the functions r and z satisfying the differential equation

r^{2} = r^{′2} + z^{′2} , (30)

i.e. the curve (r, z) being parametrized by arc length (thought of as a curve in the Poincaré half plane), we may write down the loop of Maurer-Cartan forms

Φ_{λ} = ( \begin{matrix}0 & − \frac{r′}{r} dy & − \frac{r′z′′−r′′z′}{r^{2}} dx & λrdx & − \frac{λ}{r} dx \\ \frac{r′}{r} dy & 0 & − \frac{z′}{r} dy & λrdy & \frac{λ}{r} dy \\ \frac{r′z′′−r′′z′}{r^{2}} dx & \frac{z′}{r} dy & 0 & 0 & 0 \\ \frac{λ}{r} dx & − \frac{λ}{r} dy & 0 & 0 & 0 \\ −λrdx & −λrdy & 0 & 0 & 0 \end{matrix} ) , (31)

which gives us the immersion f and its dual f̂ in the limit λ → 0.

On the other hand, denoting by H = \frac{1}{2}(\frac{z′}{r^{2}} + \frac{r′z′′−r′′z′}{r^{3}} ) the mean curvature of our surface of rotation, the central sphere congruence of f is n + Hf. The metric it induces has conformal factor k^{2} (relative to the metric induced by f) given by

k = \frac{1}{2r^{2}} (rz′ − r′z′′ + r′′z′) . (32)

Since k_{y} ≡ 0, this is obviously a solution of Calapso’s equation and a function u solving (25) is u = λ^{2} − k^{2}. So the Maurer-Cartan form (23) becomes

Φ_{λ} = ( \begin{matrix}0 & 0 & kdx & dx & ( \frac{3}{2} k^{2} − λ^{2})dx \\ 0 & 0 & −kdy & dy & (− \frac{1}{2} k^{2} + λ^{2})dy \\ −kdx & kdy & 0 & 0 & k_{x}dx \\ −( \frac{3}{2} k^{2} − λ^{2})dx & ( \frac{1}{2} k^{2} − λ^{2})dy & −k_{x}dx & 0 & 0 \\ −dx & −dy & 0 & 0 & 0 \end{matrix} ) . (33)

A change n ⇝ n+kf of the sphere congruence, enveloped by f, followed by an O_{1}(2)-gauge f ⇝ λf and f̂ ⇝ λ^{−1}f̂ gives us the Maurer-Cartan form

Φ_{λ} = ( \begin{matrix}0 & 0 & 2kdx & λdx & −λdx \\ 0 & 0 & 0 & λdy & λdy \\ −2kdx & 0 & 0 & 0 & 0 \\ λdx & −λdy & 0 & 0 & 0 \\ −λdx & −λdy & 0 & 0 & 0 \end{matrix} ) . (34)

of a curved flat, quite different from that coming from (31).

To understand the geometry of the two enveloping immersions f = F e_{4} and f̂ = F e_{5}, we remark that the sphere congruence n = F e_{3} depends only on one variable and hence the two immersions parametrize a channel surface; moreover all spheres of the congruence are perpendicular to the fixed circle^{13} given by span{F e_{2}, F (e_{4} + e_{5})}, which may be thought as an axis of rotation: the immersions f and f̂ parametrize pieces of a surface of revolution^{14}, f and f̂ being axisymmetric^{15}. Taking now the limit λ → 0, we obtain a cylinder resp. its dual, which is an (axial) reflection of the original cylinder.

13We have Φe_{2} = −(e_{4} + e_{5})dy and Φ(e_{4} + e_{5}) = 2e_{2}dy.

14The meridian curve is given by \frac{1}{\sqrt{2}}(f − f̂) — which only depends on one variable — thought as a curve in the Poincaré half plane; its tangent field is given by F e_{1} and its unit normal field by n = F e_{3}.

15The circles {F (p)e_{1}, F (p)e_{2}}^{⊥} intersecting the sphere n(p) orthogonally in f(p) and f̂(p) all meet the axis (cf. footnote 9, page 5).

References

[1] W. Blaschke, Vorlesungen über Differentialgeometrie III, Berlin (1929)

[2] P. Calapso, Sulla superficie a linee di curvatura isoterme, Rend. Circ. Mat. Palermo 17 (1903) 275-286

[3] J. Cieśliński, P. Goldstein, A. Sym, Isothermic Surfaces in E^{3} as Soliton Surfaces, Short Report (1994)

[4] J. Coolidge, Congruences and Complexes of Circles, Trans. AMS 15 (1914) 107-134 [5] D. Ferus, F. Pedit, Curved Flats in Symmetric Spaces, Manuscript (1993)

[6] U. Hertrich-Jeromin, Über konform flache Hyperflächen in vierdimensionalen Raumformen, Thesis (1994)