ISOMETRIC IMMERSIONS OF SPACE FORMS AND SOLITON THEORY

DIRK FERUS AND FRANZ PEDIT

1. Introduction

The study of isometric immersions of space forms into space forms is a classical problem of differential geometry. In its simplest form it arises as the study of surfaces in 3-space of constant (non-zero) Gaussian curvature. In this case the integrability condition reduces to the sin and sinh-Gordon equations. Due to the complicated structure of these equations one focused over the past decades mainly on non-existence results rather then the construction of explicit examples, the most famous one being a result by Hilbert [14] that there are no complete surfaces of negative curvature in Euclidean 3-space. A slightly different spirit was present in the work of Bianchi and Darboux (see also [3]) where one finds many examples, some of which probably have been forgotten over time and only recently been rediscovered in the framework of what might be called soliton geometry. Having its origin in the explicit description of all tori of constant mean curvature [20, 6] the finite gap integration scheme can also be applied to obtain explicit parametrizations of a large class of surfaces of constant Gaussian curvature [16, 15, 5, 4].

In higher dimensions the situation is quite similar: there are a number of interesting non-existence results [21, 9, 17, 18, 19, 13] for isometric immersions f : M^{m}(c) → M̃^{n}(c̃) between space forms M of dimension m, curvature c, and space forms M̃^{n}(c̃) of dimension n, curvature c̃, but no general scheme to construct such immersions. The integrability conditions for such an immersion, the Gauss, Codazzi and Ricci equations, are sometimes referred to as a generalized sin-Gordon system. Certain aspects of its inverse scattering theory [1, 24] as well as solutions obtained by Bäcklund transformations (which generate a finite dimensional solution space starting from a known, trivial, solution) [23] have recently been studied.

On the other hand, exterior differential calculus and the Cartan-Kähler theory [12] leads to a description of the space of local real analytic isometric immersions of space forms (in specific codimensions) in terms of finitely many functions of a single variable. In particular this shows that this space is infinite dimesional.

In this paper we will develop the finite gap integration theory for the integrability conditions for isometric immersions of space forms. By this we mean that we will construct a hierarchy of commuting finite dimensional ODE in Lax form whose solutions will give local real analytic isometric immersions of space forms. We call those isometric immersions of finite type. The flows evolve on a certain loop Lie algebra and can, at least in principle, be integrated by theta functions on the Jacobian of an algebraic curve, the spectral curve of the Lax flows. In this sense the isometric immersion equations for space forms are a completely integrable system and thus can be regarded as a higher dimesional (w.r.t. the flow variables) soliton equation. The explicit algebraic integration so far has only be carried out in detail in the case of surfaces in 3-space [15, 5] and the higher dimensional case would require substantial more work on the description of Riemann surfaces with two (commuting) involutions and certain reality conditions which will be done elsewhere. The initial condition for the Lax flows is a matrix valued function of one variable so that our solution space has at least functionally the correct dimension given by Cartan-Kähler theory and we expect that an appropiate closure of the finite type solutions will in fact account for all the local real analytic isometric immersions of space forms.

Second author was partially supported by NSF grant DMS-9205293, the SFB 288 at Technische Universität Berlin and the Graduierten Kolleg at Humboldt Universität Berlin.

In order to avoid degenerate cases we will make the following assumptions on our isometric immersions f : M^{m}(c) → M̃(c̃) throughout the paper :

i. c ≠ 0 ≠ c̃, c ≠ c̃, and ii. the normal bundle of f : M^{m}(c) → M̃(c̃) is flat.

Note that for c < c̃ and n = 2m − 1 the second assumption is always satisfied [17] and, in this case, there are no (local) isometric immersions for n ≤ 2m − 2 [21]. It is a classical result due to Hilbert [14, 22] that there are no complete isometric immersions M^{2}(c) → M̃^{3}(c̃) if c < c̃, c < 0 but it is yet unknown (though conjectured) whether this result extends to complete isometric immersions M^{m}(c) → M̃^{2m−1}(c̃) for c < c̃, c < 0. Note that for c = 0 one always has the Clifford tori and c > 0 cannot occur due to the fact that such immersions induce global Chebycheff coordinates [17, 19]. In contrast, in the case c > c̃ one always has the totally umbilical hypersurfaces. If n ≤ 2m − 1 and the immersion has no umbilic points [18] then n = 2m − 1 and the normal bundle is again flat. Generally, flatness of the normal bundle is a necessary assumption in higher codimensions to guarantee that those isometric immersions are not too flabby to arise from completely integrable systems [11]. The case c = c̃ is treated exhaustively in [10].

First we discuss how the structural equations for an isometric immersion between space forms can be rewritten as a zero curvature condition involving an auxiliary (spectral) parameter. Thus each such isometric immersion is part of a 1-parameter family of isometric immersions and the deformation parameter turns out to be related to the induced curvature. We then express this fact as the flatness condition on a loop algebra valued 1-form with an algebraic constraint. Such a reformulation of the equations at hand is reminiscent in the theory of integrable systems and will be the starting point for their integration. To carry this through we had to slightly modify AKS or R-matrix theory [7], since our equations could not be treated inside the standard setup. This modification was geometrically rooted since the Lax flows so obtained yield framings of the isometric immersions which parallelize the normal bundle and thus are adapted to the geometry of the situation.

A major part of this research was done while the second author was a visiting member at the SFB 288 at Technische Universität Berlin and enjoyed the hospitality, interest and encouragment present at this institution. Finally we would like to thank Josef Dorfmeister for inspiring us to think about a modification of standard AKS and R-matrix formalism.

2. Associated family of isometric immersions

Since our discussion will be local, we may (after scaling) assume M̃^{n}(c̃) to be either the Euclidean sphere S^{n} ⊂ ℝ^{n+1} of curvature c̃ = 1 or hyperbolic space H^{n} ⊂ ℝ_{1}^{n+1} of curvature c̃ = −1 realized as one sheet of the hyperboloid in Lorentz space. We denote by S either of these two standard spaces.

Let f : M^{m}(c) → S be an isometric immersion and let F : M^{m}(c) → SO_{ϵ}(n + 1) be an adapted framing, that is to say F_{0} = f, F_{1}, …, F_{m} are tangential and F_{m+1}, …, F_{n} are normal. We denote by

A = F^{−1}dF

the pull back of the left Maurer-Cartan form of SO_{ϵ}(n+1) by F. (Here and in the sequel ϵ = c̃ = ±1 and indicates whether one deals with the Euclidean or Lorentzian version of the corresponding object). Then A is an so_{ϵ}(n + 1)-valued 1-form and since F is adapted,

A = (\begin{matrix}0 & −ϵθ^{T} & 0 \\ θ & ω & β \\ 0 & −β^{T} & η \end{matrix}) , (2.1)

where θ is an ℝ^{m}-valued 1-form on M^{m}(c) (the dual frame to F_{1}, …, F_{m}), ω is an so(m)-valued 1-form (the Levi-Civita connection on M^{m}(c)), η is an so(n − m)-valued 1-form (the connection in the normal bundle) and β is the 2^{nd} fundamental form for f. The integrability conditions for the existence of such a framing F, the Maurer-Cartan equations

dA + A ∧ A = 0 , (2.2)

unravel to the fundamental equations for the isometric immersion f, the structure equation and the Gauss, Codazzi and Ricci equations

a

dθ + ω ∧ θ = 0 , (2.3a)

dω + ω ∧ ω − ϵθ ∧ θ^{T} − β ∧ β^{T} = 0 , (2.3b) dβ + ω ∧ β + β ∧ η = 0 , (2.3c)

dη + η ∧ η − β^{T} ∧ β = 0 . (2.3d)

In addition one has that the induced metric has curvature c,

dω + ω ∧ ω = c θ ∧ θ^{T} , (2.4)

and that the normal bundle is flat,

dη + η ∧ η = 0 . (2.5)

The starting point in soliton theory for integrating the equations at hand is their reformulation as a zero curvature condition (Maurer-Cartan equation) involving some auxiliary (spectral) parameter. Let A be an so_{ϵ}(n+1)-valued 1-form on M^{m}(c) of the form (2.1). We define a family of so_{ϵ}(n+1, ℂ)- valued 1-forms parametrized by λ ∈ ℂ^{∗} by

Ã^{λ} = (\begin{matrix}0 & \frac{−ϵ\sqrt{ϵc}}{2}(λ + λ^{−1})θ^{T} & 0 \\ \frac{\sqrt{ϵc}}{2}(λ + λ^{−1})θ & ω & \frac{\sqrt{ϵc}}{2\sqrt{1−ϵc}}(λ − λ^{−1})β \\ 0 & \frac{−\sqrt{ϵc}}{2\sqrt{1−ϵc}}(λ − λ^{−1})β^{T} & η \end{matrix}) (2.6)

Lemma 2.1. A solves (2.3), (2.4), and (2.5) if and only if Ã^{λ} solves the Maurer-Cartan equation

dÃ^{λ} + Ã^{λ} ∧ Ã^{λ} = 0

for all λ (in a set with accumulation point) in ℂ^{∗}.

Proof. Since (2.3a) to (2.3d) are equivalent to the Maurer-Cartan equation (2.2) we have to verify that Ã^{λ} solves (2.3a) to (2.3d). Since (2.3a) and (2.3c) are homogeneous in θ and β (and neither ω nor η involve λ) they are trivially satisfied by Ã^{λ}. Also (2.3d) holds because of (2.5). Now (2.3b) and (2.4) imply

(c − ϵ)θ ∧ θ^{T} = β ∧ β^{T} = 0

so that, using (2.4), we have

cθ ∧ θ^{T} − \frac{c}{2}(λ + λ^{−1})^{2}θ ∧ θ^{T} − \frac{ϵc}{4(1 − ϵc)}(λ − λ^{−1})^{2}β ∧ β^{T} = 0 ,

which shows that also (2.3b) holds for Ã^{λ}. To prove the converse we compute the entries in dÃ^{λ} + Ã^{λ} ∧ Ã^{λ} = 0 and compare coefficients of equal powers of λ:

d(\frac{\sqrt{ϵc}}{2}(λ + λ^{−1})θ) + ω ∧ (\frac{\sqrt{ϵc}}{2}(λ + λ^{−1})θ) = 0

dω + ω ∧ ω − \frac{c}{4}(λ + λ^{−1})^{2}θ ∧ θ^{T} − \frac{ϵc}{4(1 − ϵc)}(λ − λ^{−1})^{2}β ∧ β^{T} = 0

d(\frac{\sqrt{ϵc}}{2\sqrt{1 − ϵc}}(λ − λ^{−1})β) + ω ∧ (\frac{\sqrt{ϵc}}{2\sqrt{1 − ϵc}}(λ − λ^{−1})β) + (\frac{\sqrt{ϵc}}{2\sqrt{1 − ϵc}}(λ − λ^{−1})β) ∧ η = 0

dη + η ∧ η − \frac{c}{4(1 − ϵc)}(λ − λ^{−1})^{2}β^{T} ∧ β = 0 .

The first and third identity yield (2.3a) and (2.3c) and the last identity gives (2.5) and (2.3d). Expanding the second identity we obtain

dω + ω ∧ ω − \frac{c}{2}θ ∧ θ^{T} + \frac{ϵc}{2(1 − ϵc)}β ∧ β^{T} − (λ^{2} + λ^{−2})(\frac{c}{4}θ ∧ θ^{T} + \frac{ϵc}{4(1 − ϵc)}β ∧ β^{T}) = 0

and thus

(1 − ϵc)θ ∧ θ^{T} + ϵβ ∧ β^{T} = 0
dω + ω ∧ ω = cθ ∧ θ^{T}

which implies (2.4) and (2.3b). The converse follows by a similar, but easier, calculation.

In the situation of Lemma 2.1 we can integrate (F^{λ})^{−1}dF^{λ} = Ã^{λ} to a (complex) framing

F^{λ} : M^{m} → SO_{ϵ}(n + 1, ℂ)

for each λ. To obtain a real valued framing Ã^{λ} has to take values in so_{ϵ}(n + 1). This is the case if and only if

S = S^{n} S = H^{n}
λ  c^{λ}  c^{λ}
real  (0, 1)  (−1, 0)
imaginary (−∞, 0) (0, ∞)
unitary (1, ∞) (−∞, −1)

(2.7)

In either of these cases F^{λ} : M^{m} → SO_{ϵ}(n + 1) is real and thus we obtain a family of immersions with flat normal bundles (since (2.5) holds for each Ã^{λ})

f^{λ} = F_{0}^{λ} : M^{m} → S (2.8)

from the first column of F^{λ}. The curvature c^{λ} of the induced metric is given by (2.4) expressed in the coframe for f^{λ}:

dω + ω ∧ ω = ϵ\frac{4}{(λ + λ^{−1})^{2}} \frac{\sqrt{c}}{2}(λ + λ^{−1})θ ∧ \frac{\sqrt{c}}{2}(λ + λ^{−1})θ^{T} ,

and thus

c^{λ} = ϵ\frac{4}{(λ + λ^{−1})^{2}} . (2.9)

Depending on the domain of λ the induced curvature c^{λ} ranges over the intervals given in (2.7). Finally, for

λ_{0} = \frac{1}{\sqrt{ϵc}}(1 + \sqrt{1 − ϵc}) (2.10)

we have Ã^{λ_{0}} = A, f^{λ_{0}} = f and c^{λ_{0}} = c, and thus recover the original immersion. We summarize the discussion so far in the following

Lemma 2.2. Let f : M^{m}(c) → S be an isometric immersion with flat normal bundle, F : M^{m}(c) → SO_{ϵ}(n + 1) an adapted framing with induced Maurer-Cartan form A = F^{−1}dF and let Ã^{λ} be given by (2.6). Then dÃ^{λ} + Ã^{λ} ∧ Ã^{λ} = 0 for all λ.

If Ã^{λ} satisfies the reality conditions (2.7) and F^{λ} integrates Ã^{λ} then F^{λ} : M^{m} → SO_{ϵ}(n + 1) is an adapted framing for the isometric immersion f^{λ} = F_{0}^{λ} : M^{m}(c^{λ}) → S with flat normal bundle and induced curvature c^{λ} = ϵ\frac{4}{(λ+λ^{−1})^{2}} . The original immersion f is recovered at λ = \frac{1}{\sqrt{ϵc}}(1 + \sqrt{1 − ϵc}).

Thus isometric immersions of space forms with flat normal bundle come naturally in 1-parameter familes which we call the associated family.

For our sequel developments it will be crucial that Lemma 2.2 has a converse:

Lemma 2.3. Let

Ã^{λ} = (\begin{matrix}0 & −ϵ(λ + λ^{−1})θ^{T} & 0 \\ (λ + λ^{−1})θ & ω & (λ − λ^{−1})β \\ 0 & −(λ − λ^{−1})β^{T} & η \end{matrix}) (2.11)

be a family of so_{ϵ}(n + 1)-valued 1-forms on M^{m}, where the forms θ and β may be imaginary (to fulfill the reality conditions (2.7) ), satisfying the Maurer-Cartan equation. If θ^{T} = (θ^{1}, …, θ^{m}) are linearly independent then F^{λ} : M^{m} → SO_{ϵ}(n + 1) integrating (F^{λ})^{−1}dF^{λ} = Ã^{λ} is an adapted framing for the isometric immersion f^{λ} = F_{0}^{λ} : M^{m}(c^{λ}) → S with induced curvature c^{λ} = ϵ\frac{4}{(λ+λ^{−1})^{2}} and flat normal bundle.

Proof. Since Ã^{λ} solves the Maurer-Cartan equation (2.2) for all λ it follows on the one hand that we can integrate to an adapted frame F^{λ} for an immersion f^{λ} = F_{0}^{λ}. On the other hand, comparing coefficients at powers of λ, we obtain

dη + η ∧ η = 0 ,
dω + ω ∧ ω = c^{λ}(λ + λ^{−1})θ ∧ (λ + λ^{−1})θ^{T}

which shows that f^{λ} has flat normal bundle and induced curvature c^{λ}.

The gist of this reformulation is that the construction of isometric immersions of space forms with flat normal bundle is equivalent to the construction of a certain family of so_{ϵ}(n + 1) valued 1-forms (2.11) satisfying the Maurer-Cartan equation. Notice that such a reformulation is well known in the theory of harmonic maps of Riemann surfaces into Lie groups and symmetric spaces [8].

3. Loop algebra formulation

In the previous section we discussed how the equations for an isometric immersion f : M^{m} → S with flat normal bundle can be written as the zero curvature condition for a family (“loop”) of Lie algebra valued 1-forms (2.11). The proper setting for this are loop algebras. Let 𝔤 = so_{ϵ}(n + 1) with complexification 𝔤^{ℂ} = so_{ϵ}(n + 1, ℂ) and define the loop algebra

Λ𝔤^{ℂ} = {ξ : ℂ^{∗} → 𝔤^{ℂ}; ξ polynomial in λ and λ^{−1}}

which is a (complex) Lie algebra under pointwise bracket. If C denotes either of ℝ^{∗}, iℝ^{∗} or S^{1} we have the real subalgebras (c.f. also (2.7))

Λ𝔤 = {ξ : C → 𝔤} ⊂ Λ𝔤^{ℂ}

corresponding to the conjugations

ξ̄(λ) = \overline{ξ(λ̄)} ,  ξ̄(λ) = \overline{ξ(−λ̄)} ,  ξ̄(λ) = \overline{ξ(1/λ̄)} . (3.1)

Note that

Ã^{λ} = (\begin{matrix}0 & −ϵ(λ + λ^{−1})θ^{T} & 0 \\ (λ + λ^{−1})θ & ω & (λ − λ^{−1})β \\ 0 & −(λ − λ^{−1})β^{T} & η \end{matrix})

for λ ∈ C can be regarded as a Λ𝔤-valued 1-form with the following symmetries:

Ã^{−λ} = Ad P Ã^{λ}
Ã^{1/λ} = Ad QÃ^{λ}

where

P = ( \begin{matrix}-1 \\ & 1 \\ & & -1 \end{matrix} ) and Q = ( \begin{matrix}1 \\ & 1 \\ & & -1 \end{matrix} ).

This motivates to consider the following involutions on Λ𝔤:

(σξ)(λ) = Ad P ξ(−λ)
(τξ)(λ) = Ad Qξ(1/λ) .

We let

Λ𝔤_{σ,τ} = {ξ ∈ Λ𝔤; σξ = ξ, τξ = ξ}

be the subalgebra fixed under σ and τ.

Lemma 3.1. Let ξ = ∑_{k∈ℤ} λ^{k}ξ_{k} ∈ Λ𝔤. Then ξ ∈ Λ𝔤_{σ,τ} if and only if ξ_{0} ∈ 𝔤^{P} ∩ 𝔤^{Q}, ξ_{even} ∈ 𝔤^{P} , ξ_{odd} ∈ 𝔤^{−P} and ξ_{−k} = Ad Qξ_{k}. (Here 𝔤^{±P} , 𝔤^{±Q} denote the ±1-eigenspaces of Ad P resp. Ad Q on 𝔤).

Proof. This follows at once by comparing coefficients in σξ = τξ = ξ.

For d ∈ ℕ let

Λ_{d} = {ξ ∈ Λ𝔤_{σ,τ}; ξ = ∑_{|k|≤d} λ^{k}ξ_{k}}

be the subspace of Laurent polynomial loops of degree at most d.

Corollary 3.1. ξ ∈ Λ_{1} if and only if

ξ = ( \begin{matrix}0 & −ϵ(λ + λ^{−1})a^{T} & 0 \\ (λ + λ^{−1})a & A & (λ − λ^{−1})C \\ 0 & −(λ − λ^{−1})C^{T} & B \end{matrix} )

Proof. This follows from Lemma 3.1 together with the fact that

𝔤^{P} = \{( \begin{matrix}* & & * \\ & * \\ * & & * \end{matrix} )\}, 𝔤^{−P} = \{( \begin{matrix}& * \\ * & & * \\ & * \end{matrix} )\} (3.2)

𝔤^{Q} = \{( \begin{matrix}* & * \\ * & * \\ & & * \end{matrix} )\}, 𝔤^{−Q} = \{( \begin{matrix}& & * \\ & & * \\ * & * \end{matrix} )\} (3.3)

Corollary 3.1 together with the reformulation of the isometric immersion equations in Lemma 2.2 and Lemma 2.3 now yield

Corollary 3.2. There is a natural correspondence between isometric immersions f : M^{m} → S of space forms with flat normal bundle and Λ_{1}-valued 1-forms Ã : T M → Λ_{1} satisfying the Maurer-Cartan equation (and whose first row Ã_{0,−} = (0, ϵ(λ + λ^{−1})θ^{T} , 0) has θ^{T} = (θ^{1}, …, θ^{m}) linearly independent).

The construction of flat loop algebra valued 1-forms satisfying an algebraic constraint (i.e. taking values in Λ_{1}) is reminiscent in the theory of soliton equations: they appear as solutions to certain completely integrable Lax-type equations [7].

4. Integration of the isometric immersion equations for space forms

We are now going to discuss a recipe for the construction of Λ_{1}-valued flat 1-forms on ℝ^{m} from finite dimensional commuting Lax flows. Following existing nomenclature we will call the isometric immersions so obtained finite type isometric immersions. Such immersions will be real analytic by construction (in fact given by theta functions) and thus cannot account for all isometric immersions of space forms. Our solutions will be parametrized by a finite number of functions in one variable which relates to well known results obtained by Cartan-Kähler theory [12]. We begin by recalling some facts about the integration of certain Lax equations on Lie algebras. Let G be a Lie algebra (in our case a loop algebra) which has a vector space direct sum decomposition

G = P ⊕ A ⊕ M

with K = A ⊕ P, B = A ⊕ M and A Lie subalgebras with commutation relations [A, P] ⊆ P and [A, M] ⊆ M. Denote the corresponding projections by π_{P} , π_{A} and π_{M}. An ad-equivariant vector field on G is a map V : G → G satisfying

d_{ξ}V [ξ, η] = [V (ξ), η] (4.1) for ξ, η ∈ G. Natural examples of such maps are the gradients (with respect to an invariant inner product) of ad-invariant functions on G. Notice that (4.1) implies that two ad-equivariant vector fields V, Ṽ : G → G commute pointwise so that

[V (ξ), Ṽ(ξ)] = 0 (4.2)

for ξ ∈ G. Given an ad-equivariant vector field V on G we define a vector field X^{V} on K by

X^{V} (ξ) = [ξ, π_{P}V (ξ)], ξ ∈ K .

In order to make sense of the derivative of a map on G without specifying a topology we assume that the images under the map of finite dimensional (vector) subspaces are contained in finite dimensional (vector) subspaces. Note that if V has this property so automatically has X^{V}.

Lemma 4.1. Let V, Ṽ : G → G be ad-equivariant vector fields. Then, for ξ ∈ K,

[X^{V} , XṼ]_{C^{∞}}(ξ) = [ξ, π_{A}[π_{P}V (ξ), π_{P} Ṽ(ξ)]] .

The proof consists of a straightforward calculation using (4.1) and (4.2). Observe that in the case A = 0, i.e., when G = K ⊕ B is the (vector space) direct sum of two Lie subalgebras, the vector fields X^{V} and XṼ commute, and one is in the realm of standard AKS or R-matrix theory [2, 7].

Lemma 4.2. Let V_{1}, …, V_{m} be ad-equivariant vector fields on G and assume that for all ξ ∈ K

π_{A}[π_{P}V_{i}(ξ), π_{P}V_{j}(ξ)] = 0 . (4.3)

Then the system of ODE

dξ = ∑_{i=1}^{m} X^{V_{i}}(ξ)dx^{i} = ∑_{i=1}^{m}[ξ, π_{P}V_{i}(ξ)]dx^{i} , ξ(0) ∈ K

has a unique (local) solution ξ : U ⊂ ℝ^{m} → K, and the P-valued 1-form on U,

Ã = ∑_{i=1}^{m} π_{P}V_{i}(ξ)dx^{i} ,

satisfies the Maurer-Cartan equation

dÃ + \frac{1}{2}[Ã ∧ Ã] = 0 .

Proof. By Lemma 4.1 the vector fields X^{V_{i}} on K commute. Thus the system

dξ = ∑_{i=1}^{m} X^{V_{i}}(ξ)dx^{i}

is integrable and so has a unique solution to any initial condition ξ(0) ∈ K. The final statement follows from an analogous calculation as in the proof of Lemma 4.1 which gives

dÃ + \frac{1}{2}[Ã ∧ Ã] = ∑_{i,j=1}^{m} π_{A}[π_{P}V_{i}(ξ), π_{P}V_{j}(ξ)]dx^{i} ∧ dx^{j} = 0 .

Remark . The previous two Lemmas are simple modifications of standard results in AKS and Rmatrix theory. Even though one cannot expect the commutativity conditions (4.3) to hold for general ad-equivariant vector fields there are interesting geometric applications where they do in fact hold. One of them are the isometric immersion equations (Corollary 3.2), which do not fit into the standard scheme, but can be treated with this more general setup. A more detailed study of the abstract general situation will perhaps be done elsewhere.

To apply the above considerations to the integration of the isometric immersion equations we first work in the complex setup and put

G = Λ𝔤_{σ}^{ℂ} = {ξ ∈ Λ𝔤^{ℂ}; σξ = ξ} ,
K = Λ𝔤_{σ,τ}^{ℂ} = {ξ ∈ Λ𝔤^{ℂ}; σξ = τξ = ξ} .

A natural vector space complement M to K in Λ𝔤_{σ}^{ℂ} is given as follows: note that

G = G_{−} ⊕ G_{0} ⊕ G_{+} (4.4)

with

G_{±} = {ξ ∈ Λ𝔤_{σ}^{ℂ}; ξ = ∑_{k><0} λ^{k}ξ_{k}} , G_{0} = (𝔤^{ℂ})^{P} .

Given ξ ∈ Λ𝔤_{σ}^{ℂ} we write ξ = ξ_{−} + ξ_{0} + ξ_{+} according to (4.4) and decompose

ξ = (ξ_{−} + τξ_{−} + \frac{ξ_{0} + τξ_{0}}{2} ) + (ξ_{+} − τξ_{−} + \frac{ξ_{0} − τξ_{0}}{2} ) (4.5)

to obtain the factorization G = K ⊕ M where M = (𝔤^{P} ∩ 𝔤^{−Q})^{ℂ} ⊕ G_{+} . Unfortunately M is not a Lie subalgebra since

0 ≠ [𝔤^{P} ∩ 𝔤^{−Q}, 𝔤^{P} ∩ 𝔤^{−Q}] ⊆ 𝔤^{P} ∩ 𝔤^{Q} .

In fact,

𝔤^{P} ∩ 𝔤^{Q} = so_{ϵ}(1 + m) ⊕ so(n − m) =: 𝔞_{1} ⊕ 𝔞_{2} = \{( \begin{matrix}* \\ & * \end{matrix} )\}

and

[𝔤^{P} ∩ 𝔤^{−Q}, 𝔤^{P} ∩ 𝔤^{−Q}] ⊆ 𝔞_{2} .

Thus, setting

A = 𝔞_{2}^{ℂ} = {(  * )} ,
P = {ξ ∈ K; ξ_{0} ∈ 𝔞_{1}^{ℂ}} ,

we arrive at

G = P ⊕ A ⊕ M

with K = P⊕A,B = A⊕M and A Lie subalgebras satisfying the commutation relations [A, P] ⊆ P and [A, M] ⊆ M.

Finally we introduce the ad-equivariant vector fields relevant to our situation.

Lemma 4.3. (i) The map V : G → G defined by

V (ξ) = λ^{2k}ξ^{2ℓ−1}, k ∈ ℤ, ℓ ∈ ℕ

is an ad-equivariant vector field on G.

(ii) Let d ∈ ℕ be odd and define V_{ℓ}(ξ) = λ^{d(2ℓ−1)−1}ξ^{2ℓ−1}, ℓ ∈ ℕ. Then the corresponding vector fields on K

X^{V_{ℓ}}(ξ) = [ξ, π_{P}V_{ℓ}(ξ)] ,

are tangential to the finite dimensionsal subspace Λ_{d}^{ℂ} = {ξ ∈ K; ξ = ∑_{|k|≤d} λ^{k}ξ_{k} } ⊂ K and

π_{P}V_{ℓ}(ξ) ∈ Λ_{1}^{ℂ} .

(iii) [X^{V_{i}}, X^{V_{j}}]_{C^{∞}} = 0, i.e., the vector fields X^{V_{ℓ}} commute.

(iv) (Reality conditions) If ξ ∈ Λ_{d} is real (w.r.t. any of the three reality conditions (3.1) λ ∈ ℝ^{∗},iℝ^{∗} or S^{1}) then also π_{P}V_{ℓ}(ξ) ∈ Λ_{1} is real (and thus the vector fields X^{V_{ℓ}} are real, i.e., remain tangent to Λ_{d}).

Proof. (i) Since ξ ∈ G = Λ𝔤_{σ}^{ℂ} we have

\begin{aligned}σV (ξ)(λ) & = (−λ)^{2k}AdP ξ(−λ)^{2ℓ−1} = λ^{2k}(AdP ξ(−λ))^{2ℓ−1} \\ & = λ^{2k}ξ(λ)^{2ℓ−1} = V (ξ)(λ) \end{aligned}

so that V : G → G is well-defined. To verify (4.1) we simply compute

d_{ξ}V [ξ, η] = λ^{2k} ∑_{i+j=2ℓ−2} ξ^{i}[ξ, η]ξ^{j} = λ^{2k}(ξ^{2ℓ−1}η − ηξ^{2ℓ−1}) = [V (ξ), η] .

(ii) Let ξ = ∑_{|j|≤d} λ^{j}ξ_{j} ∈ Λ_{d}^{ℂ} then

V_{ℓ}(ξ) = λ^{d(2ℓ−1)−1}ξ^{2ℓ−1} = λ^{−1}ξ_{−d}^{2ℓ−1} + ∑_{j=0}^{2d(2ℓ−1)−1} λ^{j}η̃_{j} ,

where

η̃_{0} = ∑_{k=0}^{2l−2}(ξ_{−d})^{k}ξ_{−d+1}(ξ_{−d})^{2l−2−k} . (4.6)

Thus, by (4.5),

π_{P}V_{ℓ}(ξ) = λ^{−1}ξ_{−d}^{2ℓ−1} + η_{0} + λAdQξ_{−d}^{2ℓ−1} ∈ Λ_{1}^{ℂ} (4.7)

with η_{0} = π_{𝔞_{1}^{ℂ}} \frac{η̃_{0}+AdQη̃_{0}}{2} ∈ 𝔞_{1}^{ℂ}. From this it is clear that X^{V_{ℓ}}(ξ) = [ξ, π_{P}V_{ℓ}(ξ)] ∈ Λ_{d}^{ℂ}, i.e., X^{V_{ℓ}} is tangential to Λ_{d}^{ℂ}.

(iii) Due to Lemma 4.1 we have to verify that for ξ ∈ Λ_{d}^{ℂ}

π_{A}[π_{P}V_{i}(ξ), π_{P}V_{j}(ξ)] = 0 .

From (4.7) and the fact that η_{0} ∈ 𝔞_{1}^{ℂ} it suffices to show that π_{A}[ξ_{−d}^{2i−1}, AdQξ_{−d}^{2j−1}] = 0 . But this last is seen from the more general fact that π_{A}[X, Y] = 0 for X, Y ∈ (𝔤^{−P})^{ℂ} implies π_{A}[X, AdQ Y] = 0, which follows at once from (3.2) and the specific form of Q.

(iv) If λ is real then the statement is obvious. In case λ is purely imaginary ξ = ∑_{|k|≤d} λ^{k}ξ_{k} ∈ Λ_{d} is equivalent to \overline{ξ_{k}} = (−1)^{k}ξ_{k}. From this and (4.7), (4.6) it is clear that π_{P}V_{ℓ}(ξ) is real. Finally, if λ is unitary the reality condition (3.1) for ξ ∈ Λ_{d} gives ξ_{−k} = \overline{ξ_{k}} = AdQξ_{k} so that by (4.7), (4.6) we again conclude that π_{P}V_{ℓ}(ξ) is real.

Putting together the above discussion with the results in the previous sections we obtain a recipe for the construction of (local) isometric immersions of space forms into S^{n} and H^{n} from a hierarchy of finite dimensional ODE.

Theorem 4.1. Let d ∈ ℕ be odd, m := [\frac{n+1}{2}] and recall the ad-equivariant vector fields V_{ℓ}(ξ) = λ^{d(2ℓ−1)−1}ξ^{2ℓ−1}, ℓ = 1, …, m.

(i) The system of ODE

dξ = [ξ, ∑_{ℓ=1}^{m} π_{P}V_{ℓ}(ξ)dx^{ℓ}], ξ(0) = ξ^{∘} ∈ Λ_{d} (4.8)

has a unique (local) solution ξ : U ⊂ ℝ^{m} → Λ_{d} for any initial condition ξ^{∘} ∈ Λ_{d}.

(ii) If ξ : U → Λ_{d} is a solution to (4.8) then

Ã = ∑_{ℓ=1}^{m} π_{P}V_{ℓ}(ξ)dx^{ℓ}

is a Λ_{1}-valued 1-form on U ⊂ ℝ^{m} solving the (matrix) Maurer-Cartan equation

dÃ + Ã ∧ Ã = 0 .

Thus, integrating (F^{λ})^{−1}dF^{λ} = Ã^{λ}, F^{λ}(0) = 1, to a framing F^{λ} : U → SO_{ϵ}(n + 1) gives a family of isometric immersion

f^{λ} = F_{0}^{λ} : U → S = S^{n} or H^{n} (4.9)

with induced curvature c^{λ} = ϵ\frac{4}{(λ+λ^{−1})^{2}} .

Proof. (i) follows from Lemma 4.2 together with Lemma 4.3. (ii) is a consequence of the description of isometric immersions as special loops of flat so_{ϵ}(1 + n)-valued 1-forms, c.f. Lemma 2.3 and Corollary 3.2, together with Lemmas 4.3 and 4.2.

Remark . (i) Due to the fact that

Ã = ∑_{ℓ=1}^{m} π_{P}V_{ℓ}(ξ)dx^{ℓ} ∈ \{( \begin{matrix}& * \\ * & * & * \\ & * \end{matrix} )\}

the framing constructed from the Lax flows (4.8) parallelizes the normal bundle of the isometric immersion (4.9) which makes the flows geometrically adapted.

(ii) Since the ad-equivariant vector fields V_{ℓ} arise from (shifts of) gradients of ad-invariant functions on the finite dimensional Lie algebra so_{ϵ}(1+n) only rank = [\frac{n+1}{2}] many vector fields are independent, i.e., one cannot construct in this way isometric immersions of space forms of dimension exceeding m = [\frac{n+1}{2}]. This fact is also geometrically rooted: it is well-known that there are no isometric immersions f : M^{m}(c) → M̃^{n}(c̃) in case c < c̃ and n ≤ 2m − 2.

(iii) Using Cartan-Kähler theory one can show that local real analytic isometric immersions f : M^{m}(c) → M̃^{2m−1}(c̃) with c < c̃ depend on finitely many functions in one variable. Our scheme

produces such real analytic isometric immersions from an arbitrary initial condition ξ^{∘} ∈ Λ_{d}, d ∈ ℕ odd, which is a certain so_{ϵ}(1+n)-valued Laurent polynomial (of arbitrary odd degree) in the variable λ. This indicates that an appropriate closure of the solutions so constructed should account for all real analytic solutions, but at present we have little idea how to address this issue.

(iv) If the target space form is the sphere S^{n} and hence the corresponding Lie algebra 𝔤 = so_{ϵ}(1 + n) compact, then there exists an ad-invariant positive definite (appropriately weighted L^{2}) inner product on Λ𝔤 which makes the flows (4.8) evolve on Euclidean spheres in Λ_{d} and thus complete.

But in general we cannot show that the flows (4.8) have global, i.e., defined on all of ℝ^{m}, solutions, in which case there still remains the question of whether the isometric immersion (4.9) is globally defined or whether its differential drops rank. This is of course related to the conjecture that there are no complete isometric immersions f : M^{m}(c) → M̃^{2m−1}(c̃) for c < c̃, c < 0 which is a classical Theorem first proved by Hilbert in the case of surfaces in Euclidean 3-space.

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Fachbereich Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 136, 10622 Berlin, Germany

E-mail address, Dirk Ferus: ferus@sfb288.math.tu-berlin.de

Department of Mathematics, University of Massachusetts, Amherst, MA 01003 E-mail address, Franz Pedit: franz@gang.umass.edu