An extension of the classical Ribaucour transformation

Marcos Dajczer & Ruy Tojeiro

In this paper we extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with flat normal bundle admitting a global system of principal coordinates. Named and extensively used by Bianchi ([5]), this transformation is a correspondence preserving lines of curvature between the focal surfaces of a two-parameter congruence of spheres.

In the process of proving the converse of Guichard’s theorem on the deformations of quadrics of revolution, Bianchi gave a positive answer to the question whether among the Ribaucour transforms of a surface with constant mean or Gaussian curvature there exist other surfaces with the same property. One of our main achievements is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature. As Bianchi in the surface case, we are able to write down a parameterization, in terms of solutions of a completely integrable linear first order system of PDEs, of all Ribaucour transforms of a given holonomic submanifold with constant sectional curvature that also have the same constant curvature.

It is a classical fact that surfaces in ℝ^{3} of constant negative curvature and umbilic–free surfaces of constant positive curvature are in correspondence with solutions of the sine–Gordon and elliptic sinh–Gordon equations, respectively. Similar correspondences between n–dimensional submanifolds of constant sectional curvature c ≠ 0 in ℝ^{2n−1} (free of weak–umbilics when c > 0) and solutions of completely integrable systems of partial differential equations, named the Generalized sine–Gordon equation if c < 0 and the Generalized elliptic sinh–Gordon equation if c > 0, were obtained in [2],

[11] and [20], and extended to broader classes of submanifolds in [3], [4], [11], [19] and [20]. Here, we further generalize these results by allowing the codimension of the submanifolds to be arbitrary.

The analytical counterpart of Bianchi’s result is a method for generating a family of new solutions of the sine–Gordon and elliptic sinh–Gordon equations from a given one. Similarly, our results yield a process for constructing a family of new solutions of the aforementioned systems of PDEs starting from a given one. Moreover, the new solutions are explicitly computed in terms of the former and the solutions of a linear first order system of PDEs.

Basically, the same method for a similar class of systems of PDEs was developed by Bianchi as a result of his transformation theory for n–orthogonal systems in manifolds of constant sectional curvature (see [5]). The close relation between his analytical results and ours is due to the fact that some of these n–orthogonal systems can be realized as principal coordinate systems of holonomic constant curvature submanifolds.

We apply our method to produce new parametrized examples of (noncomplete!) n–dimensional submanifolds with constant sectional curvature c ≠ 0 in euclidean space ℝ^{2n−1}. First, starting with the trivial solution of the Generalized sine–Gordon and elliptic sinh–Gordon equations, we construct the higher dimensional analogues of the constant curvature surfaces in ℝ^{3} of Joachimsthal-type discovered by Bianchi, Kuen and Sievert (see [14]). As in the surface case, they have the remarkable property that the curvature lines of one family are contained in 2–dimensional planes through a fixed axis, whereas those of all other families lie in 2–dimensional spheres with centers on the axis. It is in general a quite difficult task to produce explicit examples if one starts with a nontrivial solution. Nevertheless, we are able to carry this out for the n–dimensional pseudosphere with constant negative sectional curvature in ℝ^{2n−1} discovered by Schur (see [18]).

As far as we know, our examples of negative curvature submanifolds are the first explicit local isometric immersions of hyperbolic space ℍ^{n} into ℝ^{2n−1}, n ≥ 4, besides Schur’s. The ones of positive curvature are the first explicit local isometric immersions of the standard sphere 𝕊^{n} into ℝ^{2n−1}, n ≥ 4, that cannot be obtained as compositions of the inclusion 𝕊^{n} ↪ ℝ^{n+1} with local isometric immersions of ℝ^{n+1} into ℝ^{2n−1}. It follows from the main result in [10] that only compositions can occur in lower codimension.

We point out that other examples for n = 3 have been constructed in [7] and [17] as multi-rotational submanifolds with curves as profiles. A more explicit description of these examples, as well as a characterization of all multirotational submanifolds with constant sectional curvature and flat normal bundle was obtained in [11].

We recall that a different geometric transformation for n–dimensional submanifolds of constant sectional curvature in Riemannian (2n−1)–dimensional space forms with higher sectional curvature was developed in [19], [20] and [21] by extending the classical Bäcklund transformation for focal surfaces of a pseudo–spherical line congruence. In particular, this provided the so called Bäcklund transformation for generating new solutions of the Generalized sine–Gordon and wave equations from a given one by solving a completely integrable system of partial differential equations of Ricatti type.

Another main result in the paper is a permutability theorem for the Ribaucour transformation, similar to Bianchi’s for the surface case. A pair of Ribaucour transforms of a submanifold determines a one-parameter family of Ribaucour transforms, which admits a conjugate family all of whose elements are Ribaucour transforms of any submanifold of the former. Furthermore, starting with a submanifold of constant curvature and the family determined by two of its Ribaucour transforms with the same curvature, we show that the conjugate family contains precisely one other submanifold with the same curvature. Moreover, the latter can be parameterized just by algebraic computations involving the data associated to the starting submanifolds. Analytically, the permutability theorem provides a tool for constructing algebraically a new solution of the corresponding PDEs from a given one and two of its transforms.

The Ribaucour transformation has proved to be a valuable tool in the study of certain classes of submanifolds. In [12] and [22], it is applied to Lagrangian submanifolds with constant sectional curvature of complex space forms. In [9], it is used to give a parametric description of all holonomic Dupin submanifolds.

Acknowledgment. We would like to thank the referee for several useful comments.

Holonomic submanifolds and associated PDEs

The main goal of this section is to establish a correspondence between holonomic submanifolds and solutions of certain completely integrable systems of partial differential equations.

For the sake of simplicity, throughout this paper we assume that any holonomic submanifold f: M^{n} → ℚ_{s}^{n+p}(c) of a Riemannian manifold into a pseudo–Riemannian space form of constant sectional curvature c and index 0 ≤ s ≤ p is simply connected. Moreover, (u_{1}, …, u_{n}) always denotes its global principal coordinate system and {X_{1}, …, X_{n}} the orthonormal principal frame given by X_{j} = v_{j}^{−1}∂/∂u_{j}, where v_{j} > 0 and ds^{2} = ∑_{j=1}^{n} v_{j}^{2}du_{j}^{2}. Finally, we agree that a global orthonormal normal frame {ξ_{1}, …, ξ_{p}} that is parallel in the normal connection has been fixed.

First we prove that isometric immersions f: M^{n} → ℚ_{s}^{n+p}(c) with flat normal bundle and first normal spaces of high dimension are always holonomic. Recall that the first normal space of f at x ∈ M^{n} is the subspace of the normal space spanned by the image of the second fundamental form α_{f}: T M × T M → T_{f}^{⊥}M at x ∈ M^{n}. The following result also appears in [1] in a somewhat different form.

Proposition 1. Let f: M^{n} → ℚ_{s}^{n+p}(c) be an isometric immersion with flat normal bundle. Assume that either the first normal spaces have everywhere dimension n or that they have everywhere dimension n−1 and f has nowhere an umbilical vector field. Then f is holonomic.

Proof: Given c̃ < c, we denote by i: ℚ_{s}^{n+p}(c) → ℚ_{s}^{n+p+1}(c̃) the umbilical inclusion and by e a unit normal vector field. Since the second fundamental forms of f and g = i ∘ f are related by

α_{g}(X, Y) = α_{f}(X, Y) + \sqrt{c − c̃}⟨X, Y ⟩e,  (1)

either of the assumptions on the first normal spaces of f implies that the first normal spaces of g have dimension n everywhere. Let {X_{1}, …, X_{n}} be an orthonormal principal frame for both immersions. Choose c̃ so that |α_{g}(X_{i}, X_{i})| ≠ 0, 1 ≤ i ≤ n, and write α_{g}(X_{i}, X_{i}) = λ_{i}ξ_{i}, where λ_{i} > 0 and ξ_{1}, …, ξ_{n} are unit vector fields. Then we easily obtain from the Codazzi equation that [λ_{j}^{−1}X_{j}, λ_{i}^{−1}X_{i}] = 0.

Before proceeding, we recall an elementary classical fact on orthogonal coordinate systems ds^{2} = ∑_{i=1}^{n} v_{i}^{2}du_{i}^{2}; see [13]. Namely, set

h_{ij} = v_{i}^{−1}∂v_{j}/∂u_{i}.  (2)

and X_{j} = v_{j}^{−1}∂/∂u_{j}. Then

∇_{∂/∂u_{i}}X_{j} = h_{ji}X_{i}, i ≠ j.  (3)

In classical notation, the functions v_{i} and h_{ij} are usually written H_{i} and β_{ij} and are known as the Lamé and rotation coefficients.

For a holonomic submanifold f: M^{n} → ℚ_{s}^{n+p}(c̃), define V_{ir} ∈ C^{∞}(M) by

A_{ξ_{r}}X_{i} = v_{i}^{−1}V_{ir}X_{i},  (4)

where A_{ξ_{r}} stands for the shape operator of f in the direction of ξ_{r}. Set v = (v_{1}, …, v_{n}), h = (h_{ij}) for h_{ij} as in (2), and V = (V_{ir}) ∈ M_{p×n}(ℝ). We call (v, h, V) the triple associated to f.

Proposition 2. The triple (v, h, V) associated to f satisfies the completely integrable system of partial differential equations

(I) \begin{cases}(i) \frac{∂v_{i}}{∂u_{j}} = h_{ji}v_{j}, (ii) \frac{∂h_{ij}}{∂u_{i}} + \frac{∂h_{ji}}{∂u_{j}} +∑_{k} h_{ki}h_{kj} +∑_{r} ϵ_{r}V_{ir}V_{jr} + c̃v_{i}v_{j} = 0, \\ (iii) \frac{∂h_{ik}}{∂u_{j}} = h_{ij}h_{jk}, (iv) \frac{∂V_{ir}}{∂u_{j}} = h_{ji}V_{jr}, i ≠ j ≠ k ≠ i, e_{r} = ⟨ξ_{r}, ξ_{r}⟩. \end{cases}

Conversely, let (v, h, V) be a solution of (I) on an open simply connected subset U ⊂ ℝ^{n} such that v_{i} ≠ 0 everywhere. Then there exists a holonomic immersion f: U → ℚ_{s}^{n+p}(c̃) that has (v, h, V) as associated triple.

Proof: Equation (i) in (I) is merely the definition of h_{ij}, and the remaining ones follow using (3) by computing the Gauss and Codazzi equations. For the converse, endow U with the metric ds^{2} = ∑_{i} v_{i}^{2} du_{i}^{2} and set M^{n} = {U, ds^{2}}. Consider the trivial vector bundle E = M^{n}×ℝ^{p}, where ℝ^{p} = span{e_{1}, …, e_{p}} is endowed with the inner product ⟨e_{r}, e_{s}⟩ = ϵ_{r}δ_{rs}. Introduce on E the flat compatible vector bundle connection ∇′ that makes {e_{1}, …, e_{p}} into a parallel frame of E and define α: T M × T M → E by

α(∂/∂u_{i}, ∂/∂u_{j}) = ∑_{i=1}^{p} ϵ_{r}δ_{ij}v_{i}V_{ir}e_{r}.

Then the Gauss and Codazzi equations follow from (I) and the Ricci equation is satisfied because ∇′ is flat and α is orthogonally diagonalizable. The conclusion now follows from the fundamental theorem of submanifolds.

Next we consider Riemannian submanifolds f: M^{n}(c) → ℚ_{s}^{n+p}(c̃) of constant sectional curvature c. We denote by 𝕆_{s}^{t}(p×n) the subspace of M_{p×n}(ℝ) of all matrices V satisfying V^{t}JV = J̃, where J_{ij} = ϵ_{i}δ_{ij} and J̃_{ij} = ϵ̃_{i}δ_{ij}, the ϵ_{i} being −1 for s of the indices 1, …, p and 1 for the others, and the ϵ̃_{i} being −1 for t of the indices 1, …, n and 1 for the remaining ones.

Corollary 3. Let f: M^{n}(c) → ℚ_{s}^{n+p}(c) be an isometric immersion with flat normal bundle and nondegenerate first normal bundle of index t of a simply connected Riemannian manifold of constant sectional curvature c. Then f is holonomic, the triple (v, h, V) associated to f satisfies the completely integrable system of partial differential equations

(II) \{ \begin{matrix}(i) \frac{∂v_{i}}{∂u_{j}} = h_{ji}v_{j}, (ii) \frac{∂h_{ij}}{∂u_{i}} + \frac{∂h_{ji}}{∂u_{j}} + ∑_{k} h_{ki}h_{kj} + cv_{i}v_{j} = 0, \\ (iii) \frac{∂h_{ik}}{∂u_{j}} = h_{ij}h_{jk}, (iv) \frac{∂V_{ir}}{∂u_{j}} = h_{ji}V_{jr}, i ≠ j ≠ k ≠ i, \end{matrix}

and V ∈ 𝕆_{s}^{t}(p × n).

Conversely, let (v, h, V) be a solution of (II) on an open simply connected subset U ⊂ ℝ^{n} such that V ∈ 𝕆_{s}^{t}(p × n) and v_{i} ≠ 0 everywhere. Then there exists a holonomic immersion f: U → ℚ_{s}^{n+p}(c) with nondegenerate first normal bundle of rank n and index t that has (v, h, V) as associated triple, and whose induced metric has constant sectional curvature c.

Proof: For an orthonormal principal frame {X_{1}, …, X_{n}} for f, the Gauss equation and the assumption on the first normal bundle imply that the vectors η_{i} = α_{f}(X_{i}, X_{i}), 1 ≤ i ≤ n, are pairwise orthogonal, ⟨η_{i}, η_{i}⟩ ≠ 0 and ⟨η_{i}, η_{i}⟩ < 0 for t of the indices 1, …, n. Set η_{i} = v_{i}^{−1}ξ_{i}, where v_{i} > 0, 1 ≤ i ≤ n, and ⟨ξ_{i}, ξ_{j}⟩ = ϵ̃_{i}δ_{ij}, with ϵ̃_{i} = ±1. It follows from Proposition 1 that f is holonomic and that ds^{2} = ∑_{j=1}^{n} v_{j}^{2}du_{j}^{2}. It is now easy to see that V = (V_{ir}) given by (4) with respect to any parallel orthonormal frame satisfies V ∈ 𝕆_{s}^{t}(p × n), and hence equation (I) − (ii) for the triple (v, h, V) associated to f reduces to (II) − (ii).

Conversely, by Proposition 2 there exists a holonomic isometric immersion f: U → ℚ_{s}^{n+p}(c) that has (v, h, V) as associated triple. It follows from V ∈ 𝕆_{s}^{t}(p × n) and the Gauss equations that the first normal bundle is nondegenerate, has rank n and index t, and that the metric induced by f has constant sectional curvature c.

For a holonomic submanifold f: M^{n}(c) → ℚ_{s}^{n+p}(c̃) with c̃ ≠ c, the augmented matrix V̂ ∈ M_{(p+1)×n}(ℝ) is defined as

V̂_{ir} = V_{ir}, 1 ≤ r ≤ p, and V̂_{i(p+1)} = \sqrt{|c − c̃|} v_{i}.

Corollary 4. Let f: M^{n}(c) → ℚ_{s}^{n+p}(c̃), c̃ ≠ c, be an isometric immersion with flat normal bundle such that

θ_{i} = ⟨α_{f}(X_{i}, X_{i}), α_{f}(X_{i}, X_{i})⟩ + c̃ − c ≠ 0, 1 ≤ i ≤ n,  (5)

for an orthonormal principal frame {X_{1}, …, X_{n}}. Then f is holonomic and its associated triple (v, h, V) satisfies (II). Moreover, the augmented matrix V̂ belongs to 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n), where ϵ_{o} = 0 or 1 according to c̃ > c or c̃ < c, respectively, and t is the number of indices such that θ_{i} < 0.

Conversely, assume that c̃ ≠ c and that (v, h, V) is a solution of (II) on an open simply connected subset U ⊂ ℝ^{n} such that v_{i} ≠ 0 everywhere and V̂ ∈ 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n). Then there exists a holonomic isometric immersion f: U → ℚ_{s}^{n+p}(c̃) with θ_{i} < 0 for t of the indices that has (v, h, V) as associated triple and whose induced metric has constant sectional curvature c.

Proof: If j: ℚ_{s}^{n+p}(c̃) → ℚ_{s+ϵ_{o}}^{n+p+1}(c) is the umbilical inclusion, then the second fundamental forms of f and g = j ∘ f are related by

α_{g}(X, Y) = α_{f}(X, Y) + \sqrt{|c̃ − c|}⟨X, Y ⟩e,  (6)

where e is normal to j and ⟨e, e⟩ = (c̃ − c)/|c̃ − c|. It follows that

⟨α_{g}(X_{i}, X_{j}), α_{g}(X_{i}, X_{j})⟩ = δ_{ij}θ_{i}, i ≠ j,

and hence (5) is equivalent to the first normal bundle of g being nondegenerate everywhere. By Corollary 3, there exists a global principal coordinate system on M^{n}(c) for g, which by (6) is also principal for f. If (v, h, V) is the triple associated to f with respect to a parallel orthonormal normal frame {ξ_{1}, …, ξ_{p}} then (v, h, V̂) is the triple associated to g with respect to the frame {j_{∗}ξ_{1}, …, j_{∗}ξ_{p}, e}, and the direct statement follows from Corollary 3.

For the converse, let g: U → ℚ_{s+ϵ_{o}}^{n+p+1}(c) be the isometric immersion associated to (v, h, V̂) by Corollary 3 with respect to the standard coordinates and some parallel orthonormal normal frame {ξ_{1}, …, ξ_{p+1}}. Then ξ_{p+1} is a parallel normal vector field satisfying ⟨ξ_{p+1}, ξ_{p+1}⟩ = (c̃ − c)/|c̃ − c| and A_{ξ_{p+1}}^{g} = \sqrt{|c̃ − c|} I. Hence g(U) is contained in an umbilical hypersurface ℚ_{s}^{n+p}(c̃) and, as an isometric immersion into ℚ_{s}^{n+p}(c̃), g has (v, h, V) as associated triple with respect to the same coordinates and the frame {ξ_{1}, …, ξ_{p}}.

Remarks 5. (i) Clearly, (5) is always satisfied for s = 0 when c < c̃. In this case, flatness of the normal bundle is automatic in the least possible codimension p = n − 1, as shown in [8]. For c > c̃, it was proved in [16] that (5) can only hold if p ≥ n − 1. Flatness of the normal bundle was shown in [15] to be also automatic whenever (5) holds and p = n − 1. Moreover, it was shown in [10] that, along an open dense subset, isometric immersions f: M^{n}(c) → ℚ_{0}^{n+p}(c̃), c > c̃, such that (5) fails and either p ≤ n − 1 or f has flat normal bundle, are locally compositions f = i ∘ g of local isometric immersions g: ℚ_{0}^{n+1}(c̃) → ℚ_{0}^{n+p}(c̃) with the umbilical inclusion of M^{n}(c) into ℚ_{0}^{n+1}(c̃).

(ii) An equivalent version of Corollary 3, but stated only for s = 0 or 1, was given in [11]. Corollary 4 for c < c̃, s = 0 and p = n − 1 was obtained in [2] and independently in [20] and [21] for c̃ = 0, and later in [19] for c̃ ≠ 0. A proof of Corollary 4 for p = n − 1 using moving frames was given in [4].

(iii) If s = 0 and p = n − 1 in Corollary 4, then V̂ ∈ 𝕆(n) or 𝕆(1, n) according to c < c̃ or c > c̃, respectively. In the former case, (II) is called the Generalized sine–Gordon or Wave equation ([19], [20], [21]) and, in the latter, the Generalized sinh–Gordon or Laplace equations ([11]) according to c ≠ 0 or c = 0, respectively. For n = 2 and setting

V̂ = ( \begin{matrix}cos ϕ & sin ϕ \\ − sin ϕ & cos ϕ \end{matrix} ) or V̂ = ( \begin{matrix}cosh ϕ & sinh ϕ \\ − sinh ϕ & cosh ϕ \end{matrix} ) ,

it reduces to the standard version of either one of those differential equations.

The Ribaucour transformation

Classically, two surfaces S, S′ in ℝ^{3} are said to be related by a Ribaucour transformation when there exists a diffeomorphism ψ: S → S′ preserving lines of curvature such that, for each point x ∈ S the normals to S and S′ at x and ψ(x), respectively, intersect at a point that is equidistant from both points. The intersection points describe, generically, a surface in ℝ^{3}, and the 2-parameter congruence of spheres centered at points of this surface, and having the common distances to corresponding points as radii, has S and S′ as focal surfaces. The intersection surface degenerates into a curve when the common distance function is a principal curvature of both surfaces that is constant along its curvature lines.

This very same definition applies well for hypersurfaces of arbitrary dimension. In fact, the study of the Ribaucour transformation for hypersurfaces goes back to Bianchi ([6]). Extending the definition to submanifolds of higher codimension, however, requires some extra care. First, in the presence of normal spaces of higher dimension it is not clear a priori how to extend the condition on the intersections of corresponding normals. Another difficulty arises when trying to extend the condition on preservation of curvature lines, a fundamental property of the transformation for surfaces. The latter is overcome when working with the class of holonomic submanifolds, for which one can also take advantage of the underlying systems of PDEs.

Definition 6. Two holonomic isometric immersions f: M^{n} → ℝ_{s}^{n+p} and f̃: M̃^{n} → ℝ_{s}^{n+p} are said to be related by a Ribaucour transformation (or f̃ is said to be a Ribaucour transform of f) when there exist a curvature-lines-preserving diffeomorphism ψ: M^{n} → M̃^{n} with |f − f̃ ∘ ψ| ≠ 0 everywhere, a vector bundle isometry P̄: T_{f}^{⊥}M → T ⊥ ; f̃M̃ covering ψ, and a vector field ζ ∈ T_{f}^{⊥}M that is nowhere a principal curvature normal of f, such that:

(a) P̄(ξ) − ξ = ⟨ξ, ζ⟩(f − f̃∘ ψ) for all ξ ∈ T_{f}^{⊥}M;

(b) P̄ is parallel.

Geometrically (a) means that, if ⟨ξ, ζ⟩ ≠ 0, then the lines in ℝ_{s}^{n+p} through f(x) and f̃(ψ(x)) tangent to ξ and P̄(ξ), respectively, intersect at a point that is at a common distance |ξ|/⟨ξ, ζ⟩ to f(x) and f̃(ψ(x)). When ⟨ξ, ζ⟩ = 0 the lines are parallel. Condition (b) says that ∇_{ψ_{∗}X}^{⊥} P̄ξ = P̄∇_{X}^{⊥}ξ for all X ∈ T M.

Recall that a vector field η ∈ T_{f}^{⊥}M is called a principal curvature normal of f at a point x ∈ M^{n} if the subspace

E(η) = {X ∈ T_{x}M : A_{ξ}^{f}X = ⟨ξ, η⟩X for all ξ ∈ T_{x}^{⊥}M}

is at least one–dimensional. To shed some light on the assumption that ζ be nowhere a principal curvature normal of f, let g_{ξ} = f + ⟨ξ, ζ⟩^{−1}ξ be the set described by the intersection points correspondent to the normal vector field ξ. Then it is easily seen that (g_{ξ})_{∗}X = 0 for some X ∈ T M and every parallel normal vector field ξ if and only if X ∈ E(ζ) and ∇_{X}^{⊥}ζ = 0. Hence, our assumption can be regarded as the analogue of the condition for the surface case assuring that the intersection set be also a surface.

If f̃: M̃^{n} → ℝ_{s}^{n+p} is a Ribaucour transform of f: M^{n} → ℝ_{s}^{n+p} with data (ψ, P̄, η), then g = f̃ ∘ ψ: M̄^{n} → ℝ_{s}^{n+p} is a Ribaucour transform of f with data (id, P̄ ∘ ψ^{−1}, η ∘ ψ^{−1}), where M̄^{n} stands for M^{n} with the metric induced by g. Thus, f̃ may be given as f̃ = g ∘ ψ̃, where g: M^{n} → ℝ_{s}^{n+p} is an immersion defined on the same manifold M^{n} as f such that g: M̄^{n} → ℝ_{s}^{n+p} is a Ribaucour transform of f and ψ̃ := ψ^{−1}: M̄^{n} → M^{n} is an isometry. Therefore, from now on we consider with no loss of generality those immersions g: M^{n} → ℝ_{s}^{n+p} (for which we still use the notation f̃: M^{n} → ℝ_{s}^{n+p}) as Ribaucour transforms of a given isometric immersion f: M^{n} → ℝ_{s}^{n+p}.

Proposition 7. Let f: M^{n} → ℝ_{s}^{n+p} be a holonomic isometric immersion, and let f̃: M^{n} → ℝ_{s}^{n+p} be a Ribaucour transform of f with data (P̄, ζ). Then there exist a vector bundle isometry P: f^{∗}T ℝ_{s}^{n+p} → f̃^{∗}T ℝ_{s}^{n+p} extending P̄ and a smooth vector field δ ∈ f^{∗}T ℝ_{s}^{n+p} whose normal component is ζ such that

PZ − Z = ⟨δ, Z⟩(f − f̃) for all Z ∈ f^{∗}T ℝ_{s}^{n+p}.  (7)

Proof: Differentiating (a) in Definition 6 and using that P̄ is parallel yields

f̃_{∗}B̃_{ξ}X − P̄∇_{X}^{⊥}ξ + X(⟨ξ, ζ⟩)(f − f̃) = f_{∗}B_{ξ}X − ∇_{X}^{⊥}ξ,  (8)

where B_{ξ} = A_{ξ} − ⟨ξ, ζ⟩I and B̃_{ξ} = Ã_{P̄(ξ)} − ⟨ξ, ζ⟩I. Therefore,

f̃_{∗}B̃_{ξ}X + P̄η = f_{∗}B_{ξ}X + η,  (9)

where η = −∇_{X}^{⊥}ξ if ⟨ξ, ζ⟩ = 0 and η = X(⟨ξ, ζ⟩)⟨ξ, ζ⟩^{−1}ξ − ∇_{X}^{⊥}ξ otherwise. Since ζ is nowhere a principal curvature normal of f, there exists ξ̄ ∈ T_{f}^{⊥}M such that Bξ̄ is invertible everywhere. Define P′: f_{∗}T M → f̃_{∗}T M by

P′ ∘ f_{∗} ∘ Bξ̄ = f̃_{∗} ∘ B̃ξ̄.

Then P′ is an isometry by (9), and from (a) in Definition 6 and (8) we obtain

P′f_{∗}Bξ̄X − f_{∗}Bξ̄X = −⟨∇_{X}^{⊥}ζ, ξ̄⟩(f − f̃).

Finally, consider the one-form defined by

ω(X) = −⟨∇_{B−1 ; ξ̄X}^{⊥}ζ, ξ̄⟩.

To conclude the proof define P = P′ ⊕ P̄ and the tangent component of δ as the vector field dual to ω.

Given an isometric immersion f: M^{n} → ℚ_{s}^{n+p}(c̃), from now on we set F = i ∘ f: M^{n} → ℝ_{s+ϵ_{o}}^{n+p+1}, where i: ℚ_{s}^{n+p}(c̃) → ℝ_{s+ϵ_{o}}^{n+p+1} is the umbilical inclusion and ϵ_{o} = 0 or 1 according to c̃ > 0 or c̃ < 0, respectively.

Definition 8. We say that f̃: M^{n} → ℚ_{s}^{n+p}(c̃) is a Ribaucour transform of f: M^{n} → ℚ_{s}^{n+p}(c̃) with data (P̄, ζ) if F̃: M^{n} → ℝ_{s+ϵ_{o}}^{n+p+1} is a Ribaucour transform of F: M^{n} → ℝ_{s+ϵ_{o}}^{n+p+1} with data (P̂, ζ̂), where ζ̂ = ζ − c̃F and P̂: T_{F}^{⊥}M → T ⊥ ; F̃M is the extension of P̄ defined by setting P̂(F) = F̃.

Geometrically, the geodesics in ℚ_{s}^{n+p}(c̃) through f(x) and f̃(x) tangent to ξ and P̄(ξ), respectively, intersect at a point that is at a common distance to both points.

We now state and prove the main result in this section.

Theorem 9. Let f: M^{n} → ℚ_{s}^{n+p}(c̃) be a holonomic submanifold with associated triple (v, h, V). Then any Ribaucour transform f̃: M^{n} → ℚ_{s}^{n+p}(c̃) of f is given by

F̃ = F − 2νφ(∑_{i} γ_{i}F_{∗}X_{i} + ∑_{r} β_{r}ξ_{r} + c̃φF),  (10)

where (φ, γ, β) := (φ, γ_{1}, …, γ_{n}, β_{1}, …, β_{p}) is a solution of

R_{1} = \{ (i) \frac{∂φ}{∂u_{i}} = v_{i}γ_{i}, (ii) \frac{∂γ_{j}}{∂u_{i}} = h_{ji}γ_{i}, i ≠ j, (iii) \frac{∂β_{r}}{∂u_{i}} = −ϵ_{r}V_{ir}γ_{i},

and ν^{−1} := ϑ = ∑_{i} γ_{i}^{2} + ∑_{r} ϵ_{r}β_{r}^{2} + c̃φ^{2}.

Conversely, for a given solution (φ, γ, β) of R_{1} define

B_{i} = ∂γ_{i}/∂u_{i} + ∑_{j≠i} h_{ji}γ_{j} − ∑_{r} β_{r}V_{ir} + c̃v_{i}φ,  (11)

and let U ⊂ M^{n} be an open subset where φϑ ≠ 0, ṽ_{i} := v_{i} − 2νφB_{i} ≠ 0 for 1 ≤ i ≤ n, and φV_{ir} + ϵ_{r}β_{r}v_{i} ≠ 0 for each 1 ≤ i ≤ n and some 1 ≤ r ≤ p. Then F̃ defined on U by (10) satisfies F̃ = i ∘ f̃, where f̃|_{U} is a Ribaucour transform of f|_{U}.

Proof: By Proposition 7, there is an isometry P: F^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} → F̃^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} and a smooth vector field δ ∈ F^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} such that

P(Z) − Z = ⟨δ, Z⟩(F − F̃) and P(F) = F̃.  (12)

Set e_{j} = F_{∗}X_{j}, 1 ≤ j ≤ n, e_{j} = ξ_{j−n}, n + 1 ≤ j ≤ n + p, e_{n+p+1} = ϵ_{0}\sqrt{|c̃|}F and ẽ_{j} = P(e_{j}), 1 ≤ j ≤ n + p + 1. Let μ ∈ C^{∞}(M) and η ∈ F^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} with μ ≠ 0 and ⟨η, η⟩ = ϵ = ±1 everywhere be defined by

F̃ = F + μη,

and set η = ∑_{j=1}^{n+p+1} α_{j}e_{j}. Then 2α_{n+p+1} = −|c̃|μ and ẽ_{j} = e_{j} − ⟨δ, e_{j}⟩μη by (12). Taking norms of both sides of the last equation yields

ẽ_{j} = e_{j} − 2ϵϵ_{j}α_{j}δ, 1 ≤ j ≤ n + p + 1.

From ⟨(F + μη)_{∗}e_{i}, ẽ_{j}⟩ = 0 if 1 ≤ i ≤ n and 1 ≤ j ≠ i ≤ n + p + 1, we easily obtain that

μ(e_{i}(α_{j}) − ϵ_{j}⟨ ˜∇_{e_{i}}e_{j}, δ⟩) = α_{j}(e_{i}(μ) + 2ϵα_{i}),

where ˜∇ stands for the derivative in ℝ_{s+ϵ_{0}}^{n+p+1}. Thus μ and the α_{j} satisfy the system of differential equations

\{ \begin{matrix}(i) e_{i}(μ) = ω_{i}μ − 2ϵα_{i}, 1 ≤ i ≤ n, \\ (ii) e_{i}(α_{j}) = ω_{i}α_{j} + ϵ_{j}⟨ ˜∇_{e_{i}}e_{j}, δ⟩, 1 ≤ i ≤ n, 1 ≤ j ≠ i ≤ n + p + 1, \end{matrix}

where (i) just defines ω_{i}. Equations (ii) for j = n+p+1 reduce to (i), and an easy computation shows that the integrability conditions for (i) are equivalent to the one–form ω = ∑_{i=1}^{n} ω_{i}e_{i}^{∗} being closed, where e_{i}^{∗} stands for the dual one–form to e_{i}. Thus there exists θ ∈ C^{∞}(M) such that ω = −θ^{−1} dθ, and the system takes the form

\{ \begin{matrix}e_{i}(θμ) = −2ϵθα_{i}, 1 ≤ i ≤ n \\ e_{i}(θα_{j}) = ϵ_{j}θ⟨ ˜∇_{e_{i}}e_{j}, δ⟩, 1 ≤ i ≤ n, 1 ≤ j ≠ i ≤ n + p. \end{matrix}  (13)

Define 2φ = −ϵθμ, γ_{i} = θα_{i} and β_{r} = θα_{n+r}. Then (13) reduces to R_{1}. Moreover,

F := ∑_{i=1}^{n} γ_{i}F_{∗}X_{i} + ∑_{r=1}^{p} β_{r}ξ_{r} + c̃φF = θη,

thus ν^{−1} = ⟨F, F⟩ = ϵ θ^{2}, and hence F̃ − F = μη = −2φνF.

Conversely, let F̃: U → ℝ_{s+ϵ_{o}}^{n+p+1} be given by (10) on an open subset U as in the statement. First notice that ⟨F̃, F̃⟩ = 1/c̃, and hence there exists f̃: U → ℚ_{s}^{n+p}(c̃) such that F̃ = i∘f̃. Define P: F^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} → F̃^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} by

P(Z) = Z − 2ν⟨F, Z⟩F.  (14)

It is easily seen that P is a vector bundle isometry and that P(F) = F̃. Moreover, a long but straightforward computation shows that

F̃_{∗}∂/∂u_{i} = ṽ_{i}PF_{∗}X_{i}.

Therefore F̃ is an immersion and P restricts to a vector bundle isometry

P̂: T_{F}^{⊥}M → T ⊥ ; F̃M. Now define ζ = −φ^{−1} ∑_{r} β_{r}ξ_{r} and ζ̂ = ζ − c̃F. Then,

P̂(ξ_{r}) − ξ_{r} = −2ϵ_{r}νβ_{r}F = ⟨ζ̂, ξ_{r}⟩(F − F̃).

Furthermore, another long computation yields

˜∇_{∂/∂u_{i}} P̂(ξ_{r}) = −Ṽ_{ir}P̂(F_{∗}X_{i}), where Ṽ_{ir} = V_{ir} + 2ϵ_{r}νβ_{r}B_{i}.

Therefore, P̂ is parallel and the coordinates are also principal for F̃. Finally, the last assumption assures that ζ̂ is nowhere a principal curvature normal of F. We conclude that F̃ is a Ribaucour transform of F |_{U} with data (P̂, ζ̂), and hence f̃|_{U} is a Ribaucour transform of f|_{U} with data (P̄, ζ̄), where P̄: T_{f}^{⊥}M → T ⊥ ; f̃M is the restriction of P̂ to the normal bundle of f.

Given a holonomic submanifold f: M^{n} → ℚ_{s}^{n+p}(c̃) with associated triple (v, h, V), let L(f) denote the space of solutions of R_{1}. By Theorem 9, the Ribaucour transforms of f are determined by elements (φ, γ, β) of L(f) such that φϑ ≠ 0 and φV_{ir} + ϵ_{r}β_{r}v_{i} ≠ 0 for each 1 ≤ i ≤ n and some 1 ≤ r ≤ p. Moreover, it is easily seen that (φ, γ, β) and (φ′, γ′, β′) ∈ L(f) yield the same Ribaucour transform of f if and only if (φ′, γ′, β′) = λ(φ, γ, β) for some λ ≠ 0. The set of equivalence classes w = [(φ, γ, β)] of such elements of L(f) under the equivalence relation defined by the preceding equation will be denoted by L̄(f), and the Ribaucour transform of f determined by w by R_{w}(f).

The Ribaucour transforms of a holonomic submanifold f: M^{n} → ℝ_{s}^{n+p} determined by the “trivial” elements w ∈ L̄(f) are listed below.

Examples 10. (i) Given P_{0}, v ∈ ℝ_{s}^{n+p} with ⟨v, v⟩ = ϵ = ±1, define φ = ⟨f − P_{0}, v⟩, γ_{i} = ⟨f_{∗}X_{i}, v⟩ and β_{r} = ⟨ξ_{r}, v⟩. Then (φ, γ, β) ∈ L(f) and

R_{w}(f) = f − 2ϵ⟨f − P_{0}, v⟩v

whenever w = [(φ, γ, β)] ∈ L̄(f), that is, R_{w}(f) is obtained from f by a reflection with respect to the hyperplane through P_{0} orthogonal to v.

(ii) Given P_{0} ∈ ℝ_{s}^{n+p} and r > 0, define 2φ = ⟨f − P_{0}, f − P_{0}⟩ − ϵr^{2}, γ_{i} = ⟨f − P_{0}, f_{∗}X_{i}⟩ and β_{r} = ⟨f − P_{0}, ξ_{r}⟩. Then (φ, γ, β) ∈ L(f) and

R_{w}(f) = P_{0} + ϵr^{2}⟨f − P_{0}, f − P_{0}⟩^{−1}(f − P_{0})

whenever w = [(φ, γ, β)] ∈ L̄(f), that is, R_{w}(f) is obtained from f by an inversion with pole P_{0}.

(iii) Let ξ ∈ T_{f}^{⊥}M be a nonzero parallel normal section. Define φ = c ∈ ℝ, γ_{i} = 0 and β_{r} = −⟨ξ, ξ_{r}⟩. Then (φ, γ, β) ∈ L(f) and

R_{w}(f) = f + aξ, where a = 2c⟨ξ, ξ⟩^{−1},

whenever w = [(φ, γ, β)] ∈ L̄(f), that is, R_{w}(f) is a parallel immersion to f in the direction of ξ.

Analytically, Theorem 9 yields the following result.

Corollary 11. If (v, h, V) and (φ, γ, β) are solutions of (I) and R_{1}, then a new solution (ṽ, h̃, Ṽ) of (I) is given by

ṽ_{i} = v_{i} − 2νφB_{i}, h̃_{ij} = h_{ij} − 2νγ_{i}B_{j} and Ṽ_{ir} = V_{ir} + 2ϵ_{r}νβ_{r}B_{i}. (15)

Proof: Assume first that both solutions are defined on a simply connected open subset U where the v_{i} are nowhere vanishing, and let f: U → ℚ_{s}^{n+p}(c̃) be an immersion with (v, h, V) as associated triple. Suppose further that w = [(φ, γ, β)] ∈ L̄(f). The proof of Theorem 9 shows that the triple associated to R_{w}(f) is (ṽ, h̃, Ṽ), and hence (ṽ, h̃, Ṽ) is a new solution of (I) by Proposition 2. One can now check by a direct computation that the statement holds with no extra assumptions.

For later use, in the next result we obtain the data that determine the inverse of a Ribaucour transformation.

Proposition 12. Let f: M^{n} → ℚ_{s}^{n+p}(c̃) be a holonomic submanifold, and let f̃ = R_{w}(f) be the Ribaucour transform of f determined by w = [(φ, γ, β)] ∈ L̄(f). Then f = Rw̃(f̃), where w̃ = [(φ̃, γ̃, β̃)] ∈ L̄(f̃) is given by

φ̃ = −νφ, γ̃_{i} = −νγ_{i} and β̃_{r} = −φβ_{r}.  (16)

Proof: Let (ṽ, h̃, Ṽ) be the triple associated to f̃ given by (15). An easy computation using that ∂ν/∂u_{i} = −2ν^{2}γ_{i}B_{i} shows that (φ̃, γ̃, β̃) ∈ L(f̃). Moreover, we have that ϑ̃ = ν, and hence φ̃ϑ̃ = −φν^{2} ≠ 0. Furthermore,

φ̃Ṽ_{ir} +ϵ_{r}β̃_{r}ṽ_{i} = −νφ(V_{ir} +2ϵ_{r}νβ_{r}B_{i})−ϵ_{r}νβ_{r}(v_{i} −2νφB_{i}) = −ν(φV_{ir} +ϵ_{r}β_{r}v_{i}).

Therefore w̃ = [(φ̃, γ̃, β̃)] ∈ L̄(f̃). Now let P: F^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} → F̃^{∗}T ℝ_{s+ϵ_{0}}^{n+p+1} be the vector bundle isometry given by (14). Since P(F) = −F, we obtain that

F̃ = ∑_{i=1}^{n} γ̃_{i}F̃_{∗}X̃_{i} + ∑_{r=1}^{p} β̃_{r}P(ξ_{r}) + c̃φ̃F̃ = −νP(F) = νF,

and hence Rw̃(F̃) = F̃ − 2φ̃ν̃F̃ = F − 2φνF + 2φνF = F.

The transformation for constant curvature submanifolds

In this section we derive the main application of our theory in this paper, namely, a transformation for submanifolds with constant sectional curvature.

Theorem 13. Let f: M^{n}(c) → ℚ_{s}^{n+p}(c̃) satisfy the assumptions of either Corollary 3 or Corollary 4 according to c̃ = c or c̃ ≠ c, respectively, and let f̃ = R_{w}(f) be the Ribaucour transform of f determined by w = [(φ, γ, β)] ∈ L̄(f). If the metric induced by f̃ has also constant curvature c, n ≥ 3 and the functions B_{i} given by (11) are nowhere vanishing, then (φ, γ, β) satisfies

∂γ_{i}/∂u_{i} = − ∑_{j≠i} h_{ji}γ_{j} + (1 − C) ∑_{r} β_{r}V_{ir} − (C(c − c̃) + c̃)v_{i}φ  (17)

and

ϑ − C(∑_{r} ϵ_{r}β_{r}^{2} − (c − c̃)φ^{2}) = 0.  (18)

Conversely, the left hand side of (18) is a constant K ∈ ℝ for any solution (φ, γ, β) of the system R_{2} obtained by adjoining (17) to R_{1}. Moreover, if K = 0 and U ⊂ M^{n} is an open subset as in Theorem 9, then the metric induced by the Ribaucour transform of f|_{U} determined by w = [(φ, γ, β)] has also constant sectional curvature c.

Proof: We consider the case c̃ ≠ c; the proof for the case c̃ = c is similar. Since f satisfies the assumptions of Corollary 4, it follows that the triple (v, h, V) associated to f satisfies (II) and that the augmented matrix V̂ belongs to 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n), where ϵ_{o} = 0 or 1 according to c̃ > c or c̃ < c, respectively. Let (ṽ, h̃, Ṽ) be the triple associated to f̃ given by (15). Since also M̃^{n}(c) has constant sectional curvature c, for any i ≠ j we have that

0 = ∑_{r} ϵ_{r}Ṽ_{ir}Ṽ_{jr} + (c̃ − c)ṽ_{i}ṽ_{j} = 2νB_{i}A_{j} + 2νB_{j}A_{i} + 4ν^{2}B_{i}B_{j}ρ,  (19)

where

A_{i} = ∑_{r} V_{ir}β_{r} + (c − c̃)v_{i}φ and ρ = ∑_{r} ϵ_{r}β_{r}^{2} − (c − c̃)φ^{2}. (20)

Thus, (A_{i} + νρB_{i})B_{j} + (A_{j} + νρB_{j})B_{i} = 0 if i ≠ j. Since n ≥ 3 and B_{i} is nowhere vanishing for 1 ≤ i ≤ n, it follows that

A_{i} + νρB_{i} = 0, 1 ≤ i ≤ n.  (21)

An easy computation shows that

∂A_{i}/∂u_{j} = h_{ji}A_{j}, and ∂B_{i}/∂u_{j} = h_{ji}B_{j}, i ≠ j.

Using this, it follows by differentiating (21) that ∂(νρ)/∂u_{i} = 0. Therefore, there exists C ≠ 0 such that ν^{−1} = Cρ, and hence B_{i} = −CA_{i} for 1 ≤ i ≤ n. This yields (17) and (18).

Conversely, it is easily seen that all derivatives of the left-hand-side of (18) vanish for a solution (φ, γ, β) of R_{2}. If (18) holds, then going backwards in the proof of the direct statement shows that (19) is satisfied, and hence the metric induced by f̃ has also constant sectional curvature c.

The analytical counterpart of Theorem 13 is the following.

Corollary 14. Let (v, h, V) be a solution of (II) such that the augmented matrix V̂ ∈ 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n). Then R_{2} is completely integrable and has the first integral

ν^{−1} − C(∑_{r} ϵ_{r}β_{r}^{2} − (c − c̃)φ^{2}) = K ∈ ℝ.  (22)

If K = 0 in (22), then (ṽ, h̃, Ṽ) defined for ρ and A_{i} as in (20) by

ṽ_{i} = v_{i} + 2φρ^{−1}A_{i}, h̃_{ij} = h_{ij} + 2γ_{i}ρ^{−1}A_{j} and Ṽ_{ir} = V_{ir} − 2ϵ_{r}β_{r}ρ^{−1}A_{i} (23)

is a new solution of (II) with ˆ Ṽ ∈ 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n).

The permutability theorems

We now turn our attention to the permutability results referred to in the introduction. We start with the following basic result.

Proposition 15. Let f: M^{n} → ℚ_{s}^{n+p}(c̃) be a holonomic submanifold, and let f̃ = R_{w}(f) be the Ribaucour transform of f determined by w = [(φ, γ, β)] ∈ L̄(f). Then any (φ̄, γ̄, β̄) ∈ L(f̃) is given by

φ̄ = ντ′φ + φ′, γ̄_{i} = ντ′γ_{i} + γ′_{i} and β̄_{r} = ντ′β_{r} + β′_{r},  (24)

where (φ′, γ′, β′) ∈ L(f) and dτ′ = −2 ∑_{i=1}^{n} B_{i}γ′_{i}du_{i}.

Proof: If (φ̄, γ̄, β̄) ∈ L(f̃), we claim that ω = ν ∑_{i=1}^{n} B_{i}γ̄_{i}du_{i} is a closed one–form. In fact, using that

∂B_{i}/∂u_{j} = h_{ji}B_{j}, ∂ν/∂u_{j} = −2ν^{2}γ_{j}B_{j}

and

∂γ̄_{i}/∂u_{j} = h̃_{ij}γ̄_{j} = (h_{ij} − 2νγ_{i}B_{j})γ̄_{j},

we obtain that

∂(νB_{i}γ̄_{i})/∂u_{j} = −2ν^{2}B_{i}B_{j}(γ_{j}γ̄_{i} + γ_{i}γ̄_{j}) + ν(h_{ji}B_{j}γ̄_{i} + h_{ij}B_{i}γ̄_{j}),

which is symmetric in the indices. For ρ ∈ C^{∞}(M) with dρ = −2ω, define

φ′ = φ̄ − ρφ, γ′_{i} = γ̄_{i} − ργ_{i} and β′_{r} = β̄_{r} − ρβ_{r}.

An easy computation shows that (φ′, γ′, β′) ∈ L(f). Moreover, setting τ′ = ν^{−1}ρ we obtain that dτ′ = −2 ∑_{i=1}^{n} B_{i}γ′_{i}du_{i}.

Given w = [(φ, γ, β)], w′ = [(φ′, γ′, β′)] ∈ L̄(f), let ℓ := ℓ_{w,w′} denote the projective line through w and w′, that is,

ℓ = {[c(φ, γ, β) + c′(φ′, γ′, β′)] : c, c′ ∈ ℝ},

and call the one-parameter family R_{ℓ}(f) of Ribaucour transforms of f determined by ℓ the associated family to ℓ. For f̃ = R_{w}(f) as in Proposition 15, let w̃ = [(φ̃, γ̃, β̃)] be the element in L̄(f̃) given by (16) such that Rw̃(f̃) = f. Notice that τ′ in (24) is determined up to a constant, and that different choices of this constant give rise to elements in L(f̃) that differ by a constant multiple of (φ̃, γ̃, β̃). Therefore, (24) defines a projective line ℓ̃ through w̃. Moreover, it is easily verified that any element in ℓ gives rise to the same projective line ℓ̃, which we denote by R_{w}(ℓ).

Theorem 16. Let f: M^{n} → ℚ_{s}^{n+p}(c̃) be a holonomic submanifold, and let ℓ be a projective line through w = [(φ, γ, β)] ∈ L̄(f). Then the permutability formula

R_{R_{w}(ℓ)}(R_{w}(f)) = R_{R_{w′}(ℓ)}(R_{w′}(f))  (25)

holds for any w′ = [(φ′, γ′, β′)] ∈ ℓ. More precisely, any element of either family is given by

F̄ = F − Γ(ϑ′φ + τφ′)F − Γ(ϑφ′ + τ′φ)F′,  (26)

where τ + τ′ = −2⟨F, F′⟩ and Γ = 2/(ϑϑ′ − ττ′).

Proof: Set f̃ = R_{w}(f), and let w̄ = [(φ̄, γ̄, β̄)] ∈ L̄(f̃) be given by (24). Then any element f̄ of the associated family R_{R_{w}(ℓ)}(R_{w}(f)) is given by

F̄ = F̃ − 2φ̄ν̄F̄,

where

F̄ = ∑_{i=1}^{n} γ̄_{i}F̃_{∗}X̃_{i} + ∑_{r=1}^{p} β̄_{r}P(ξ_{r}) + c̃φ̄F̃ = P(ντ′F + F′).

In particular,

ϑ̄ = ν̄^{−1} = ⟨F̄, F̄⟩ = ντ^{′2} + 2ντ′Θ + ϑ′,  (27)

where Θ = ⟨F, F′⟩. Using that P(F) = −F and P(F′) = F′ − 2νΘF yields

F̄ = F − 2νφF + \frac{2(ντ′φ + φ′)}{ντ^{′2} + 2ντ′Θ + ϑ′} (ντ′F − F′ + 2νΘF),

and (26) follows easily. Now (25) follows from the symmetry of (26) with respect to w and w′.

We call the one–parameter family of immersions (R_{ℓ}(f))^{c} given by (25) the conjugate family to the associated family R_{ℓ}(f).

Proposition 17. The following facts hold:

(i) Any element of (R_{ℓ}(f))^{c} is a Ribaucour transform of any one on R_{ℓ}(f);

(ii) ((R_{ℓ}(f))^{c})^{c} = R_{ℓ}(f);

(iii) For a fixed x ∈ M^{n}, the sets {R_{ℓ}(f)(x)} and {(R_{ℓ}(f))^{c}(x)} parametrize a common circle.

Proof: Assertion (i) is immediate from (25) and (iii) follows by a straightforward computation. Now, ((R_{ℓ}(f))^{c})^{c} is the associated family to Rw̃(R_{w}(ℓ)). Assertion (ii) will follow once we have shown that Rw̃(R_{w}(ℓ)) = ℓ. Any ŵ = [(φ̂, γ̂, β̂)] ∈ Rw̃(R_{w}(ℓ)) is given by

φ̂ = ν̃τ̃φ̃ + φ̄, γ̂_{i} = ν̃τ̃γ̃_{i} + γ̄_{i} and β̂_{r} = ν̃τ̃β̃_{r} + β̄_{r},

where

dτ̃ = −2 ∑_{i=1}^{n} B̃_{i}γ̄_{i}du_{i}  (28)

and

φ̄ = ντ′φ + φ′, γ̄_{i} = ντ′γ_{i} + γ′_{i} and β̄_{r} = ντ′β_{r} + β′_{r},

for some fixed w′ = [(φ′, γ′, β′)] ∈ L̄(f) and τ′ with dτ′ = −2 ∑_{i=1}^{n} B_{i}γ′_{i}du_{i}. We need only prove that (φ̂, γ̂, β̂) = (φ′, γ′, β′) for some choice of τ̃ satisfying (28). The preceding equality is easily seen to be satisfied by choosing τ̃ = ντ′. Now, it is easy to verify that B̃_{i} = νB_{i}, and hence

∂(ντ′)/∂u_{i} = −2B_{i}γ′_{i}ν − 2τ′ν^{2}γ_{i}B_{i} = −2νB_{i}γ̄_{i} = −2B̃_{i}γ̄_{i}.

We now derive a permutability result for the class of isometric immersions f: M^{n}(c) → ℚ_{s}^{n+p}(c̃) that satisfy the assumptions of either Corollary 3 or Corollary 4 according to c = c̃ or c ≠ c̃, respectively. By Theorem 13, if w = [(φ, γ, β)] ∈ L̄(f) and f̃ = R_{w}(f) is a Ribaucour transform of f whose induced metric has also constant sectional curvature c, then (φ, γ, β) satisfies (17) and (18) for some constant C ≠ 0. We say for short that f̃ is a R_{C}-transform of f.

Theorem 18. Let f: M^{n}(c) → ℚ_{s}^{n+p}(c̃) satisfy the assumptions of either Corollary 3 or Corollary 4 according to c = c̃ or c ≠ c̃, respectively. Given R_{C_{k}}-transforms f̃_{k}: M̃_{k}^{n}(c) → ℚ_{s}^{n+p}(c̃) of f, 1 ≤ k ≤ 2, C_{1} ≠ C_{2}, determined by solutions (φ_{k}, γ^{k}, β^{k}) of R_{2}, let R_{ℓ}(f) be their associated family. Then in the conjugate family there exists exactly one element f̄: M̄^{n}(c) → ℚ_{s}^{n+p}(c̃), which is a R_{C_{k}}-transform of f̃_{ℓ}, ℓ ≠ k. Moreover, f̄ is explicitly given by

F̄ = F − Γ(ϑ_{2}φ_{1} + τ_{1}φ_{2})F_{1} − Γ(ϑ_{1}φ_{2} + τ_{2}φ_{1})F_{2},

where

ϑ_{k} = C_{k}(∑_{r} ϵ_{r}(β_{r}^{k})^{2} − (c − c̃)φ_{k}^{2}),  (29)

τ_{k} = \frac{2C_{ℓ}}{(C_{k} − C_{ℓ})}(∑_{i} γ_{i}^{1}γ_{i}^{2} + (1−C_{k}) ∑_{r} ϵ_{r}β_{r}^{1}β_{r}^{2} + (C_{k}(c−c̃) + c̃)φ_{1}φ_{2}), (30)

Γ = 2/(ϑ_{1}ϑ_{2} − τ_{1}τ_{2}) and F_{k} = ∑_{i} γ_{i}^{k}F_{∗}X_{i} + ∑_{r} β_{r}^{k}ξ_{r} + c̃φ_{k}F.

Proof: Any element in the conjugate family (R_{ℓ}(f))^{c} is a Ribaucour transform f̄ = Rw̄(f̃_{1}) of f̃_{1}, where w̄ = [(φ̄, γ̄, β̄)] ∈ L̄(f̃_{1}) is given by

φ̄ = ν_{1}τ_{2}φ_{1} + φ_{2}, γ̄_{i} = ν_{1}τ_{2}γ_{i}^{1} + γ_{i}^{2} and β̄_{r} = ν_{1}τ_{2}β_{r}^{1} + β_{r}^{2},

and

dτ_{2} = −2 ∑_{i=1}^{n} B_{i}γ_{i}^{2}du_{i}.  (31)

By Theorem 13, f̄ is a R_{C_{2}}-transform of f̃_{1} if and only if

∂γ̄_{i}/∂u_{i} = − ∑_{j≠i} h̃_{ji}^{1}γ̄_{j} + (1 − C_{2}) ∑_{r} β̄_{r}Ṽ_{ir}^{1} − (C_{2}(c − c̃) + c̃)ṽ_{i}^{1}φ̄ (32)

and

ϑ̄ − C_{2}(∑_{r} ϵ_{r}β̄_{r}^{2} − (c − c̃)φ̄^{2}) = 0,  (33)

where (ṽ^{1}, h̃_{ij}^{1}, Ṽ_{ir}^{1}) is the triple associated to f̃_{1}. Using (18) for ϑ_{1} and ϑ_{2}, and (27), it follows that the left-hand-side of (33) is equal to

ν_{1}τ_{2}(τ_{2}C_{1}^{−1}(C_{1} − C_{2}) + 2(∑_{i} γ_{i}^{1}γ_{i}^{2} + (1−C_{2}) ∑_{r} ϵ_{r}β_{r}^{1}β_{r}^{2} + (C_{2}(c−c̃) + c̃)φ_{1}φ_{2}),

and hence (33) holds if and only if

τ_{2} = \frac{2C_{1}}{(C_{2} − C_{1})}(∑_{i} γ_{i}^{1}γ_{i}^{2} + (1−C_{2}) ∑_{r} ϵ_{r}β_{r}^{1}β_{r}^{2} + (C_{2}(c−c̃) + c̃)φ_{1}φ_{2}). (34)

It now follows from straightforward computations that τ_{2} defined by (34) does satisfy (31) and (32). Finally, symmetry with respect to the indices shows that f̄ is also a R_{C_{1}}-transform of f̃_{2}.

Theorem 18 has the following analytical consequence.

Corollary 19. Let (v, h, V) be a solution of (II) such that the augmented matrix V̂ ∈ 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n), and let (φ_{k}, γ^{k}, β^{k}), 1 ≤ k ≤ 2, be solutions of R_{2} with constants C_{1} ≠ C_{2}. Then (v̄, h̄, V̄) defined by

v̄_{i} = v_{i} + D_{i}^{1}Γ(ϑ_{2}φ_{1} + τ_{1}φ_{2}) + D_{i}^{2}Γ(ϑ_{1}φ_{2} + τ_{2}φ_{1}),
h̄_{ij} = h_{ij} + D_{j}^{1}Γ(ϑ_{2}γ_{i}^{1} + τ_{2}γ_{i}^{2}) + D_{j}^{2}Γ(ϑ_{1}γ_{i}^{2} + τ_{1}γ_{i}^{1}),
V̄_{ir} = V_{ir} − D_{i}^{1}Γ(ϑ_{2}β_{r}^{1} + τ_{2}β_{r}^{2}) + D_{i}^{2}Γ(ϑ_{1}β_{r}^{2} + τ_{1}β_{r}^{1}),

where ϑ_{k}, τ_{k} are given by (29), (30) and D_{i}^{k} = C_{k}(∑_{r} V_{ir}β_{r}^{k} + (c − c̃)v_{i}φ_{k}), is a new solution of (II) such that ˆ V̄ ∈ 𝕆_{s+ϵ_{o}}^{t}((p + 1) × n).

The examples

In this section we compute explicit examples of isometric immersions f: M^{n}(c)→ℝ^{2n−1}, c≠0, which are nowhere compositions when c > 0.

Let (v, h, V) be the trivial solution of (II) with V̂ =I as augmented matrix and h = 0. Solving the system

\{(i) \frac{∂f}{∂u_{i}} = δ_{in}X_{n}, (ii) \frac{∂X_{i}}{∂u_{j}} = δ_{ij}ξ_{i}, i ≠ n, \frac{∂X_{n}}{∂u_{n}} = 0, (iii) \frac{∂ξ_{r}}{∂u_{i}} = −δ_{ir}X_{i},

we find that f = (0, …, 0, u_{n}),

X_{i} = (0, …, cos u_{i}, sin u_{i}, …, 0), i ≠ n, X_{n} = (0, …, 0, 1),
ξ_{r} = (0, …, − sin u_{r}, cos u_{r}, …, 0).

Although f is not an immersion, the Ribaucour transformation may still be applied. System R_{2} takes the form

R_{3} = \{ \begin{matrix}(i) \frac{∂φ}{∂u_{i}} = δ_{in}γ_{n}, (ii) \frac{∂γ_{j}}{∂u_{i}} = δ_{ij}(1 − C)β_{i}, i ≠ n, \frac{∂γ_{n}}{∂u_{n}} = −Ccφ, \\ (iii) \frac{∂β_{r}}{∂u_{i}} = −δ_{ir}γ_{i}, \end{matrix}

whereas the first integral (18) becomes

∑_{i=1}^{n} γ_{i}^{2} + (1 − C) ∑_{r=1}^{n−1} β_{r}^{2} + Ccφ^{2} = 0.  (35)

Next we consider separately the spherical and hyperbolic cases. We assume for simplicity that c = ±1.

The spherical case. By (35), either C < 0 or C > 1.

1) Case C < 0. Set C = −a^{2}, b = \sqrt{1 + a^{−2}}, and let ∑_{j=1}^{n−1} λ_{j}^{2} = 1. The solution of R_{3} satisfying (35) is

\{ \begin{matrix}φ(u_{n}) = cosh au_{n}, β_{i}(u_{i}) = λ_{i}b^{−1} sin abu_{i}, \\ γ_{i}(u_{i}) = −λ_{i}a cos abu_{i}, i ≠ n, γ_{n}(u_{n}) = a sinh au_{n}. \end{matrix}

The corresponding Ribaucour transforms f̃ of f are given by

f̃_{1} = Gλ_{1}(b cos abu_{1} cos u_{1} + a^{−1} sin ab sin u_{1})
f̃_{2} = Gλ_{1}(a^{−1} sin abu_{1} cos u_{1} − b cos abu_{1} sin u_{1})
⋮
f̃_{2n−3} = Gλ_{n−1}(b cos abu_{n−1} cos u_{n−1} + a^{−1} sin abu_{n−1} sin u_{n−1})
f̃_{2n−2} = Gλ_{n−1}(a^{−1} sin abu_{n−1} cos u_{n−1} − b cos abu_{n−1} sin u_{n−1})
f̃_{2n−1} = u_{n} − bG sinh au_{n},

where G(u_{1}, …, u_{n}) = 2a^{−1}b cosh au_{n}/(b^{2} cosh^{2} au_{n} − ∑_{j=1}^{n−1} λ_{j}^{2} sin^{2} abu_{j}).

2) Case C > 1. Set C = a^{2}, b = −\sqrt{1 − a^{−2}}, and let ∑_{j=1}^{n−1} λ_{j}^{2} = 1. The solution of R_{3} satisfying (35) is

\{ \begin{matrix}φ(u_{n}) = sin au_{n}, β_{i}(u_{i}) = λ_{i}b^{−1} cosh abu_{i}, \\ γ_{i}(u_{i}) = −λ_{i}a sinh abu_{i}, i ≠ n, γ_{n}(u_{n}) = a cos au_{n}. \end{matrix}

The corresponding Ribaucour transforms f̃ of f are given by

f̃_{1} = Gλ_{1}(b sinh abu_{1} cos u_{1} + a^{−1} cosh ab sin u_{1})
f̃_{2} = Gλ_{1}(a^{−1} cosh abu_{1} cos u_{1} − λ_{1}b sinh abu_{1} sin u_{1})
⋮
f̃_{2n−3} = Gλ_{n−1}(b sinh abu_{n−1} cos u_{n−1} + a^{−1} cosh abu_{n−1} sin u_{n−1})
f̃_{2n−2} = Gλ_{n−1}(a^{−1} cosh abu_{n−1} cos u_{n−1} − b sinh abu_{n−1} sin u_{n−1})
f̃_{2n−1} = u_{n} − bG cos au_{n},

where G(u_{1}, …, u_{n}) = 2a^{−1}b sin au_{n}/(b^{2} sin^{2} au_{n} − ∑_{j=1}^{n−1} λ_{j}^{2} cosh^{2} abu_{j}). The hyperbolic case. By (35), we have that C > 0.

1) Case C < 1. Set C = a^{2}, b = \sqrt{a^{−2} − 1}, and let ∑_{j=1}^{n−1} λ_{j}^{2} = 1. The solution of R_{3} satisfying (35) is

\{ \begin{matrix}φ(u_{n}) = cosh au_{n}, β_{i}(u_{i}) = λ_{i}b^{−1} sin abu_{i}, \\ γ_{i}(u_{i}) = −aλ_{i} cos abu_{i}, i ≠ n, γ_{n}(u_{n}) = a sinh au_{n}. \end{matrix}

The corresponding Ribaucour transforms f̃ of f are given by f̃_{1} = Gλ_{1}(b cos abu_{1} cos u_{1} + a^{−1} sin abu_{1} sin u_{1})

f̃_{2} = Gλ_{1}(a^{−1} sin abu_{1} cos u_{1} − b cos abu_{1} sin u_{1})
⋮

f̃_{2n−3} = Gλ_{n−1}(b cos abu_{n−1} cos u_{n−1} + a^{−1} sin abu_{n−1} sin u_{n−1}) f̃_{2n−2} = Gλ_{n−1}(a^{−1} sin abu_{n−1} cos u_{n−1} − b cos abu_{n−1} sin u_{n−1})

f̃_{2n−1} = u_{n} − Gb sinh au_{n},

where G = 2a^{−1}b cosh au_{n}/(b^{2} cosh^{2} au_{n} + ∑_{j=1}^{n−1} λ_{j}^{2} sinh^{2} abu_{j}).

2) Case C =1. Let ∑_{j=1}^{n−1} λ_{j}^{2} = 1. The solution of R_{3} satisfying (35) is

\{ \begin{matrix}φ(u_{n}) = cosh u_{n}, β_{i}(u_{i}) = −λ_{i}u_{i}, \\ γ_{i} = λ_{i}, i ≠ n, γ_{n}(u_{n}) = sinh u_{n}. \end{matrix}

The corresponding Ribaucour transforms are given by f̃_{1} = Gλ_{1}(cos u_{1} + u_{1} sin u_{1})

f̃_{2} = Gλ_{1}(sin u_{1} − u_{1} cos u_{1})
⋮
f̃_{2n−3} = Gλ_{n−1}(cos u_{n−1} + u_{n−1} sin u_{n−1})
f̃_{2n−2} = Gλ_{n−1}(sin u_{n−1} − u_{n−1} cos u_{n−1})
f̃_{2n−1} = u_{n} − G sinh u_{n},

where G(u_{1}, …, u_{n}) = 2 cosh u_{n}/(∑_{j=1}^{n−1} λ_{j}^{2}u_{j}^{2} + cosh^{2} u_{n}).

3) Case C > 1. Set C = a^{2}, b = \sqrt{1 − a^{−2}}, and let ∑_{j=1}^{n−1} λ_{j}^{2} = 1. The solution of R_{3} satisfying (35) is

\{ \begin{matrix}φ(u_{n}) = sinh au_{n}, β_{i}(u_{i}) = λ_{i}b^{−1} cosh abu_{i}, \\ γ_{i}(u_{i}) = −aλ_{i} sinh abu_{i}, i ≠ n, γ_{n}(u_{n}) = a cosh au_{n}. \end{matrix}

The corresponding Ribaucour transforms f̃ of f are given by

f̃_{1} = Gλ_{1}(b sinh abu_{1} cos u_{1} + a^{−1} cosh abu_{1} sin u_{1})
f̃_{2} = Gλ_{1}(a^{−1} cosh abu_{1} cos u_{1} − b sinh abu_{1} sin u_{1})
⋮
f̃_{2n−3} = Gλ_{n−1}(b sinh abu_{n−1} cos u_{n−1} + a^{−1} cosh abu_{n−1} sin u_{n−1})
f̃_{2n−2} = Gλ_{n−1}(a^{−1} cosh abu_{n−1} cos u_{n−1} − b sinh abu_{n−1} cosh u_{n−1})
f̃_{2n−1} = u_{n} − bG cosh au_{n},

where G = 2a^{−1}b sinh au_{n}/(b^{2} sin^{2} au_{n} + ∑_{j=1}^{n−1} λ_{j}^{2} cosh^{2} abu_{j}).

Finally, observe that all of the above examples have the property that the curvature lines u_{i} = constant, i ≠ n, are contained in 2–dimensional planes through the axis generated by the vector (0, …, 0, 1), while all the others are contained in 2–dimensional spheres with centers in that axis.

Remark 20. Further examples may be easily constructed by taking triples (φ_{k}, γ^{k}, β^{k}), 1 ≤ k ≤ 2, correspondent to different values of C and applying Theorem 18.

To conclude we compute a Ribaucour transform of the n-dimensional pseudosphere in ℝ^{2n−1} parametrized by

f(u_{1}, …, u_{n}) = sech u_{n} ∑_{j=1}^{n−1} λ_{j}Z_{j} + (u_{n} − tanh u_{n})Z_{n},

where ∑_{j=1}^{n−1} λ_{j}^{2} = 1 and

Z_{j} = (0, …, 0, cos u_{j}, sin u_{j}, 0, …, 0), 1 ≤ j ≤ n − 1, Z_{n} = (0, …, 0, 1).

An orthonormal tangent frame is given by

X_{j} = (0, …, 0, sin u_{j}, cos u_{j}, 0, …, 0), 1 ≤ j ≤ n − 1,
X_{n} = −sech u_{n} ∑_{j=1}^{n−1} λ_{j}Z_{j} + tanh u_{n}Z_{n}.

The normal bundle is spanned by the parallel orthonormal frame

ξ_{1} = tanh u_{n} ∑_{j=1}^{n−1} λ_{j}Z_{j} + sech u_{n}Z_{n}, ξ_{j} = ∑_{k=1}^{n−1} a_{j}^{k}Z_{k}, 2 ≤ j ≤ n − 1,

where (λ_{1}, …, λ_{n−1}) and (a_{j}^{1}, …, a_{j}^{n−1}), 2 ≤ j ≤ n − 1, form an orthonormal basis for ℝ^{n−1}. The induced metric is ds^{2} = sech^{2}u_{n} ∑_{j=1}^{n−1} λ_{j}^{2} du_{j}^{2} + tanh^{2} u_{n} du_{n}^{2}. Hence,

v_{i} = λ_{i}sech u_{i}, v_{n} = tanh u_{n}, 1 ≤ i ≤ n − 1,
V_{i1} = λ_{i} tanh u_{n}, V_{ni} =−sech u_{n}, V_{ir} = a_{r}^{i}, V_{nr} = 0, 2 ≤ r ≤ n−1,
h_{ij} = 0, h_{ni} = −λ_{i} sech u_{n}, 1 ≤ j ≤ n, j ≠ i.

(36)

A solution of R_{3} for, say, C = 1, is

φ = cosh u_{n} + sech u_{n} ∑_{j=1}^{n−1} λ_{j}^{2}u_{j}^{2},
γ_{j} = 2λ_{j}u_{j}, 1 ≤ j ≤ n − 1, γ_{n} = cosh u_{n} − sech u_{n} ∑_{j=1}^{n−1} λ_{j}^{2}u_{j}^{2},
β_{1} = u_{n} − tanh u_{n} ∑_{j=1}^{n−1} λ_{j}^{2}, β_{r} = − ∑_{j=1}^{n−1} a_{r}^{j}λ_{j}u_{j}^{2}, 2 ≤ r ≤ n − 1,

and the corresponding Ribaucour transform is given by

f̃ = f − 2νφ(∑_{j=1}^{n} γ_{j}X_{j} + ∑_{r=1}^{n−1} β_{r}ξ_{r}), ν^{−1} = ∑_{j=1}^{n} γ_{j}^{2} + ∑_{r=1}^{n−1} β_{r}^{2},

which becomes a rather long expression even for n = 2.

Remark 21. The above examples also yield explicit solutions (ṽ, h̃, Ṽ) of the Generalized sine–Gordon and sinh–Gordon equations given by (23), where (φ, γ, β) is the corresponding solution of R_{3} and (v, h, V) is the trivial solution, except for the last one, for which (v, h, V) is given by (36).

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IMPA – Estrada Dona Castorina, 110 Universidade Federal de São Carlos 22460-320 – Rio de Janeiro – Brazil 13565-905 – São Carlos – Brazil E-mail: marcos@impa.br E-mail: tojeiro@dm.ufscar.br