Commuting Codazzi tensors and the Ribaucour transformation for submanifolds

Marcos Dajczer & Ruy Tojeiro

The notion of Ribaucour transformation was extended in [3] 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 carrying a principal system of coordinates. As a main application, it was used to obtain a transformation theory for constant curvature submanifolds, for which the existence of principal coordinates is automatic in some important cases. It also proved to be useful in the study of Lagrangian submanifolds with constant sectional curvature of complex space forms developed in [4] and [7], where holonomicity is also automatic. In the study of reducibility of Dupin submanifolds (see [2]), however, it soon became clear the need for a more general theory, which could be applied to a larger class of submanifolds.

The starting goal of this paper was to develop such a theory, so as to make it suitable for a wider range of applications. In the process, a beautiful correspondence between Ribaucour transforms of a submanifold and Codazzi tensors that “commute” with its second fundamental form was revealed. In fact, the Ribaucour transformation may be seen as a transformation between pairs (M^{n}, Φ), where M^{n} is a Riemannian submanifold of a pseudo-Riemannian space form and Φ is a commuting Codazzi tensor defined on it. Moreover, the transformation induces an explicit isomorphism between the spaces of commuting Codazzi tensors on both submanifolds that also commute with the given tensors, modulo a certain equivalence relation.

The Ribaucour transforms of a submanifold include those submanifolds that are obtained by a conformal transformation of the ambient space or, more generally, by a Lie transformation; these are the Ribaucour transforms that are determined by the “trivial” commuting Codazzi tensors. Holonomic submanifolds carry an abundance of nontrivial commuting Codazzi tensors, and this explains why such submanifolds show up naturally in the study of the Ribaucour transformation. However, we prove that the class of submanifolds carrying nontrivial commuting Codazzi tensors, the proper setting of our general theory, is much larger than that of holonomic submanifolds.

We derive several basic relations between a submanifold and any of its Ribaucour transforms, which should be useful in further applications. We also study the permutability properties of the Ribaucour transformation. We show that a pair of commuting Codazzi tensors on a submanifold, which also commute with each other, gives rise to a one-parameter family of Ribaucour transforms. This family admits a conjugate one all of whose elements are Ribaucour transforms of any submanifold of the former. Moreover, any element of the latter can be parametrized just by algebraic computations involving the data associated to the starting submanifolds. Finally, the permutability results are used to establish the invariance of the Ribaucour transformation under Lie transformations.

Commuting Codazzi tensors

Given an n–dimensional Riemannian manifold M^{n}, we denote by S(M) the real vector space of smooth sections of the vector bundle of symmetric endomorphisms of the tangent bundle of M^{n}. Recall that Φ ∈ S(M) belongs to the real vector space C(M) of Codazzi tensors on M^{n} if it satisfies the differential equation

(∇_{X}Φ)Y = (∇_{Y} Φ)X for all X, Y ∈ T M.  (1)

Given an isometric immersion f: M^{n} → ℝ_{s}^{n+p} into the standard flat pseudo–Riemannian Euclidean space form of index s, we say that Φ ∈ S(M) belongs to the subspace S(f) of commuting tensors if

α_{f}(X, ΦY) = α_{f}(ΦX, Y) for all X, Y ∈ T M,  (2)

where α_{f}: T M × T M → T_{f}^{⊥}M stands for the second fundamental form of f. We denote C(f) = C(M) ∩ S(f). A map F: M^{n} → ℝ_{s}^{n+p} is called a Combescure transform of f determined by Φ ∈ S(M) when

F_{∗} = f_{∗} ∘ Φ.

Proposition 1. If f: M^{n} → ℝ_{s}^{n+p} is an isometric immersion and F a Combescure transform of f determined by Φ ∈ S(M), then Φ ∈ C(f). Conversely, if M^{n} is simply connected then any Φ ∈ C(f) gives rise to a Combescure transform of f.

Proof: Regard ω = f_{∗} ∘ Φ as a one–form on M^{n} with values in ℝ_{s}^{n+p}. Then,

dω(X, Y) = f_{∗}(∇_{X}ΦY − ∇_{Y} ΦX − Φ[X, Y]) + α_{f}(X, ΦY) − α_{f}(ΦX, Y).

The following result extends the observation in [5] and [6] that any Codazzi tensor on a simply connected open subset U ⊂ ℝ^{n+p} can be given as Φ = Hess φ for some φ ∈ C^{∞}(U). We denote by A_{δ}^{f}: T M → T M the shape operator of f in the direction δ ∈ T_{f}^{⊥}M.

Proposition 2. Let f: M^{n} → ℝ_{s}^{n+p} be an isometric immersion of a simply connected Riemannian manifold. Then any tensor Φ ∈ C(f) and the corresponding Combescure transform F of f may be given as

Φ = Φ_{φ,β} = Hess φ − A_{β}^{f} and F = C_{φ,β}(f) = f_{∗}∇φ + β,  (3)

where φ ∈ C^{∞}(M) and β ∈ T_{f}^{⊥}M satisfy

α_{f}(∇φ, X) + ∇_{X}^{⊥}β = 0 for all X ∈ T M.  (4)

Conversely, given φ and β satisfying (4), let Φ and F be defined by (3). Then Φ ∈ C(f) and F is a Combescure transform of f.

Proof: Write F = f_{∗}Z + β for some Z ∈ T M and β ∈ T_{f}^{⊥}M. Then,

⟨∇_{X}Z, Y ⟩ = ⟨ΦX, Y ⟩ + ⟨α_{f}(X, Y), β⟩.

Symmetry of Φ is equivalent to ⟨∇_{X}Z, Y ⟩ = ⟨∇_{Y} Z, X⟩. Hence, there exists φ ∈ C^{∞}(M) such that Z = ∇φ, and thus

F_{∗}X = f_{∗}(Hess φ − A_{β}^{f})X + α_{f}(∇φ, X) + ∇_{X}^{⊥}β.

We denote by L(f) the real vector space of all pairs (φ, β) satisfying (4). Hence, Proposition 2 yields a linear map from L(f) onto C(f) given by

(φ, β) ↦ Φ_{φ,β}  (5)

Examples 3. (i) Given P_{0}, v ∈ ℝ_{s}^{n+p} with ⟨v, v⟩ = ϵ = ±1, set φ_{0} = ⟨f − P_{0}, v⟩ and let β_{0} = v^{⊥} be the normal vector field obtained by projecting v pointwise onto T_{f}^{⊥}M. Then (φ_{0}, β_{0}) ∈ L(f), C_{φ_{0},β_{0}}(f) = v and Φ_{φ_{0},β_{0}} = 0.

(ii) Given P_{0} ∈ ℝ_{s}^{n+p} and b ≠ 0, define 2φ_{1} = ⟨f − P_{0}, f − P_{0}⟩ − b and let β_{1} = (f −P_{0})^{⊥} be the normal vector field obtained by projecting the position vector f − P_{0} pointwise onto T_{f}^{⊥}M. Then (φ_{1}, β_{1}) ∈ L(f), C_{φ_{1},β_{1}}(f) = f − P_{0} and Φ_{φ_{1},β_{1}} = I.

(iii)Assume that f admits a nonzero parallel normal section ξ. Set φ_{2} =c ∈ ℝ and β_{2} = −ξ. Then (φ_{2}, β_{2}) ∈ L(f), C_{φ_{2},β_{2}}(f) = −ξ and Φ_{φ_{2},β_{2}} = A_{ξ}^{f}.

The subspace of L(f) spanned by pairs of one of the types in Examples 3 will be denoted by L_{0}(f). Thus, the image C_{0}(f) ⊂ C(f) of L_{0}(f) under (5) is spanned by the identity endomorphism and the shape operators in the direction of parallel normal vector fields. We call trivial commuting Codazzi tensors the elements of C_{0}(f).

If f has flat normal bundle, that is, the normal curvature tensor R_{f}^{⊥} of f vanishes, then Proposition 2 implies that any Φ ∈ C(f) gives rise to φ ∈ C^{∞}(M) such that Hess φ ∈ S(f). We show next that the converse also holds.

Proposition 4. Let f: M^{n} → ℝ_{s}^{n+p} be an isometric immersion with flat normal bundle of a simply connected Riemannian manifold. If φ ∈ C^{∞}(M) satisfies Hess φ ∈ S(f), then there exists β ∈ T_{f}^{⊥}M such that Φ_{φ,β} ∈ S(f).

Proof: It follows easily by taking the inner product with ∇φ of the Codazzi equation (∇_{X}A_{ξ_{r}}^{f})Y = (∇_{Y} A_{ξ_{r}}^{f})X that there exist β_{r} ∈ C^{∞}(M) such that A_{ξ_{r}}^{f}∇φ = −∇β_{r}, 1 ≤ r ≤ p, where {ξ_{1}, …, ξ_{p}} is a parallel orthonormal normal frame. Thus (φ, β) ∈ L(f), where β = ∑_{r=1}^{p}⟨ξ_{r}, ξ_{r}⟩β_{r}ξ_{r}.

Flatness of the normal bundle must hold for an isometric immersion f: M^{n} → ℝ_{s}^{n+p} carrying a tensor Φ ∈ S(f) that has distinct eigenvalues on a dense subset of M^{n}, as follows from the next result.

Proposition 5. If f: M^{n} → ℝ_{s}^{n+p} is an isometric immersion and Φ ∈ S(f), then R_{f}^{⊥} vanishes on planes spanned by eigenvectors of Φ correspondent to distinct eigenvalues.

Proof: It follows from (2) that α_{f}(X, Y) = 0 if X, Y are eigenvectors of Φ correspondent to distinct eigenvalues. In particular, the eigenspaces of Φ are invariant by A_{ξ}^{f} for any ξ ∈ T_{f}^{⊥}M. The Ricci equation concludes the proof.

Proposition 6. Let Φ̄ ∈ C(ℝ_{s}^{n+p}) and let f: M^{n} → ℝ_{s}^{n+p} be a Φ̄–invariant isometric immersion, that is, there exists Φ ∈ S(M) such that Φ̄∘f_{∗} = f_{∗}∘Φ. Then Φ ∈ C(f).

Proof: Denote by \overline{∇} the connection in ℝ_{s}^{n+p}. Then

(\overline{∇}_{X} Φ̄) f_{∗}Y + Φ̄α_{f}(X, Y) = f_{∗}(∇_{X}Φ)Y + α_{f}(X, ΦY).

Since Φ̄ is a Codazzi tensor, the left hand side of the equation is symmetric, hence the same holds for the right hand side.

For an isometric embedding with flat normal bundle we show next that any commuting Codazzi tensor arises as in the preceding result.

Proposition 7. Let f: M^{n} → ℝ_{s}^{n+p} be an isometric embedding with flat normal bundle of a simply connected Riemannian manifold. If Φ ∈ C(f), then there exist an open subset U ⊂ ℝ_{s}^{n+p} containing f(M) and φ̄ ∈ C^{∞}(U) such that

f_{∗} ∘ Φ = Hess φ̄ ∘ f_{∗}.

Moreover, the corresponding Combescure transform of f is F = ∇φ̄ ∘ f.

Proof: Let V be an open neighborhood of the zero section in T_{f}^{⊥}M that is mapped diffeomorphically onto U ⊂ ℝ_{s}^{n+p} by the normal exponential map. Write Φ = Hess φ − A_{β}^{f} as in (3) and set β = ∑_{r=1}^{p}⟨ξ_{r}, ξ_{r}⟩β_{r}ξ_{r}, where {ξ_{1}, …, ξ_{p}} is a parallel orthonormal normal frame. Define φ̄ ∈ C^{∞}(U) by

φ̄(exp_{x} ∑_{r=1}^{p} t_{r}ξ_{r}(x)) = φ(x) + ∑_{r=1}^{p} t_{r}β_{r}(x).

Then φ̄ ∘ f = φ and ∇φ̄ = f_{∗}∇φ + β along f.

The following result shows that nontrivial commuting Codazzi tensors are abundant on holonomic submanifolds.

Proposition 8. Let M^{n} be a Riemannian manifold and let Φ ∈ S(M). If (u_{1}, …, u_{n}) are coordinates on M^{n} such that Φ ∂/∂u_{i} = ϕ_{i}∂/∂u_{i}, 1 ≤ i ≤ n, then Φ ∈ C(M) if and only if the functions ψ_{i} = v_{i}ϕ_{i} are solutions of

∂ψ_{i}/∂u_{j} = h_{ji}ψ_{j}, i ≠ j,  (6)

where v_{i} = |∂/∂u_{i}| and h_{ij} = v_{i}^{−1}∂v_{j}/∂u_{i}. The integrability conditions of (6) are satisfied if the curvature tensor of M^{n} has the property that

R(∂/∂u_{i}, ∂/∂u_{j})∂/∂u_{k} = 0, i ≠ j ≠ k ≠ i.  (7)

Moreover, if f: M^{n} → ℝ_{s}^{n+p} is a holonomic isometric immersion and the coordinates are principal for f, then (7) is always satisfied. In particular, C(f) depends on n functions of one variable.

Proof: If X_{1}, …, X_{n} is any orthonormal diagonalyzing frame for Φ ∈ S(M), then Φ ∈ C(M) if and only if

\{ \begin{matrix}X_{j}ϕ_{i} + Γ_{ij}^{i}(ϕ_{i} − ϕ_{j}) = 0,  i ≠ j, \\ Γ_{ij}^{k}(ϕ_{j} − ϕ_{k}) = Γ_{ji}^{k}(ϕ_{i} − ϕ_{k}), i ≠ j ≠ k ≠ i, \end{matrix}  (8)

where Γ_{ij}^{k} = ⟨∇_{X_{i}}X_{j}, X_{k}⟩. Moreover, it is a classical elementary fact that if X_{i} = v_{i}^{−1}∂/∂u_{i} for an orthogonal system of coordinates, then the Christoffel symbols Γ_{ij}^{k} satisfy h_{ji} = v_{i}Γ_{ij}^{i}, i ≠ j, and Γ_{ij}^{k} = 0, i ≠ j ≠ k ≠ i. Then (8) reduces to (6), whereas (7) becomes ∂h_{ki}/∂u_{j} = h_{kj}h_{ji}, i ≠ j ≠ k ≠ i. Therefore, the integrability conditions

(∂h_{ji}/∂u_{k} − h_{jk}h_{ki})ψ_{j} = (∂h_{ki}/∂u_{j} − h_{kj}h_{ji})ψ_{k}, i ≠ j ≠ k ≠ i,

are trivially satisfied if (7) holds. Moreover, by the Gauss equations this is always the case if (u_{1}, …, u_{n}) are principal coordinates for f.

Hypersurfaces with distinct principal curvatures that are “as far as possible” from being holonomic admit only trivial commuting Codazzi tensors, as shown by the following result.

Proposition 9. Let f: M^{n} → ℝ^{n+1}, n ≥ 3, be a hypersurface with n distinct principal curvatures such that the Christoffel symbols Γ_{ij}^{k} with respect to a principal frame are nowhere vanishing for i ≠ j ≠ k ≠ i. Then C(f) = C_{0}(f).

Proposition 9 is a consequence of the following more general result.

Lemma 10. Let Φ ∈ C(M) have distinct eigenvalues ϕ_{1}, …, ϕ_{n}, and let {X_{1}, …, X_{n}} be an orthonormal frame of corresponding eigenvectors. Assume that Γ_{ij}^{k} ≠ 0 everywhere for all i ≠ j ≠ k ≠ i in some subset A ⊂ {1, …, n}, and set D = span {X_{j}: j ∈ A}. If Φ′ ∈ C(M) satisfies [Φ′, Φ] = 0, then there exist a, b ∈ C^{∞}(M) and Φ′′ ∈ S(M) such that

Φ′ = aΦ + bI + Φ′′,

where the gradients ∇a, ∇b ∈ D^{⊥} and D ⊂ ker Φ′′

Proof: Let ϕ′_{1}, …, ϕ′_{n} be the eigenvalues of Φ′. It follows from the second set of equations in (8) that there exist a, b ∈ C^{∞}(M) such that ϕ′_{i} = aϕ_{i} + b for any i ∈ A. The first set of equations in (8) then yields X_{j}(a)ϕ_{i} + X_{j}(b) = 0 for j ≠ i ∈ A. Hence, X_{j}(a) = X_{j}(b) = 0 for all j ∈ A, because the ϕ_{i} are pairwise distinct.

Given a hypersurface f: M^{3} → ℝ^{4} with distinct principal curvatures, it follows from Proposition 9 that either f is holonomic or C(f) = C_{0}(f). We conclude this section showing that for higher dimensions this is no longer true.

We consider hypersurfaces f: M^{4} → ℝ^{5} with distinct principal curvatures λ_{1}, …, λ_{4} in the intermediate case where neither f is holonomic nor the assumption of Proposition 9 on the Γ_{ij}^{k} is satisfied. For an orthonormal frame of principal directions {X_{1}, …, X_{4}}, we assume that Γ_{12}^{3} ≠ 0 everywhere (hence, Γ_{ij}^{k} ≠ 0 if {i, j, k} = {1, 2, 3} from (8)), and that Γ_{ij}^{4} = 0 if 1 ≤ i ≠ j ≤ 3 (hence, Γ_{ij}^{k} = 0 if i ≠ j ≠ k ≠ i and 4 ∈ {i, j, k}). In particular, the distribution D = span{X_{1}, X_{2}, X_{3}} is integrable. Since our results are local in nature we may assume that the leaves of D are simply connected.

Lemma 11. (i) There exists a unique function μ ∈ C^{∞}(M) such that along each leaf L of D the gradient in L satisfies

∇^{L} log(μ|_{L}) = ∇_{X_{4}}X_{4}|_{L}  (9)

once an initial condition along one integral curve γ_{0} of span{X_{4}} is fixed.

(ii) There exist a_{j} ∈ C^{∞}(M) such that ∇a_{j} ∈ span{X_{4}} for 1 ≤ j ≤ 2, and

Γ_{ii}^{4} = a_{1}λ_{i} + a_{2}, 1 ≤ i ≤ 3.  (10)

Proof: (i) To see that ∇_{X_{4}}X_{4}|_{L} is a gradient in each L, we have to show that

⟨∇_{X_{j}}∇_{X_{4}}X_{4}, X_{i}⟩ = ⟨∇_{X_{i}}∇_{X_{4}}X_{4}, X_{j}⟩ if 1 ≤ i ≠ j ≤ 3.  (11)

From ⟨R(X_{j}, X_{4})X_{4}, X_{i}⟩ = 0, ∇_{X_{j}}X_{4} = −Γ_{jj}^{4}X_{j} and ∇_{X_{4}}X_{j} = −Γ_{44}^{j}X_{4} if 4 ≠ i ≠ j ≠ 4, we obtain that

⟨∇_{X_{j}}∇_{X_{4}}X_{4}, X_{i}⟩ = ⟨∇_{X_{4}}∇_{X_{j}}X_{4}, X_{i}⟩ + ⟨∇_{[X_{j},X_{4}]}X_{4}, X_{i}⟩ = Γ_{44}^{j}Γ_{44}^{i},

and (11) follows.

(ii) Since X_{4} is a parallel normal vector field along each leaf L of D, it follows using our assumptions that A^{f}|_{T L}, A_{X_{4}}^{f|_{L}} ∈ C(f|_{L}) and commute. By Lemma 10, there exist functions a_{1}, a_{2} ∈ C^{∞}(M) satisfying (10) such that ∇a_{j} ∈ span{X_{4}}.

Proposition 12. Let f: M^{4} → ℝ^{5} be as before and let a_{3}, a_{4} ∈ C^{∞}(M) be a solution of

X_{4}(a_{j+2}) = −a_{j}μ, 1 ≤ j ≤ 2,  (12)

where μ, a_{1}, a_{2} are given by Lemma 11. Then the tensor Φ ∈ S(f) given by

Φ = a_{3}A^{f} + a_{4}I + μX_{4}^{∗}X_{4}  (13)

belongs to C(f). Conversely, any Φ ∈ C(f) can be given this way.

Proof: System (8) for Φ as in (13) reduces to

X_{i}(μ) = μΓ_{44}^{i}  (14)

and

X_{4}(a_{3})λ_{i} + X_{4}(a_{4}) = −μΓ_{ii}^{4}, 1 ≤ i ≤ 3,  (15)

which follow from (9), (10) and (12).

We now prove the converse. If Φ ∈ C(f), by Lemma 10 there exist a_{3}, a_{4}, μ ∈ C^{∞}(M) with ∇a_{3}, ∇a_{4} ∈ span{X_{4}} such that

Φ = a_{3}A^{f} + a_{4}I + μX_{4}^{∗}X_{4}.

Moreover, a_{3}, a_{4}, μ satisfy (14) and (15). Therefore, μ satisfies (9) from (14) and it follows from (15) that (12) holds for a_{1}, a_{2} ∈ C^{∞}(M) given by (10).

Corollary 13. Let f: M^{4} → ℝ^{5} be a rotation hypersurface over a nowhere holonomic submanifold h: N^{3} → ℝ^{4} with distinct principal curvatures. Then any Φ ∈ C(f) has the form

Φ = aA^{f} + bI + μZ^{∗}Z, for a, b ∈ ℝ and μ(t, x) = ψ(t)h_{1}^{−1}(x),

where h(x) = (h_{1}, h_{2}, h_{3})(x), the parameter t belongs to the circles of rotation, Z = h_{1}^{−1}(x)∂/∂t ∈ T M and ψ(t) is an arbitrary smooth function.

Proof: In this case the leaves L = N^{3} of D are totally geodesic submanifolds of M^{4}, and hence the functions a_{3}, a_{4} in (13) are constant by (12), because the functions a_{1}, a_{2} vanish by (10). Moreover, we have from (9) that

∂μ/∂x_{j} = μ|∂/∂t|^{−2}⟨∇_{∂/∂t}∂/∂t, ∂/∂x_{j}⟩,

where |∂/∂t| = h_{1} > 0 and ⟨∇_{∂/∂t}∂/∂t, ∂/∂x_{j}⟩ = −h_{1}∂h_{1}/∂x_{j}.

The Ribaucour transformation

We start with a definition of the Ribaucour transformation that applies to arbitrary submanifolds of ℝ_{s}^{n+p}.

Definition 14. Given an isometric immersion f: M^{n} → ℝ_{s}^{n+p}, we say that an immersion f̃: M^{n} → ℝ_{s}^{n+p} is a Ribaucour transform of f when |f − f̃| ≠ 0 everywhere and there exist a vector bundle isometry P: f^{∗}Tℝ_{s}^{n+p} → f̃^{∗}Tℝ_{s}^{n+p}, a tensor D ∈ S(M) and a nowhere vanishing smooth vector field δ ∈ f^{∗}Tℝ_{s}^{n+p} such that

(a) P(Z) − Z = ⟨δ, Z⟩(f − f̃) for all Z ∈ f^{∗}Tℝ_{s}^{n+p},

and

(b) f̃_{∗} = P ∘ f_{∗} ∘ D

Geometrically, condition (a) means that for any Z ∈ T_{f(x)}ℝ_{s}^{n+p} with ⟨δ, Z⟩ ≠ 0 the lines in ℝ_{s}^{n+p} through f(x) and f̃(x) tangent to Z and P(Z), respectively, intersect at a point that is at a common distance |Z|/⟨δ, Z⟩ to f(x) and f̃(x). When ⟨δ, Z⟩ = 0 the lines are parallel.

By Proposition 8 in [3], if f̃: M^{n} → ℝ_{s}^{n+p} is a Ribaucour transform of a holonomic submanifold f: M^{n} → ℝ_{s}^{n+p} as defined therein, then f̃ is also a Ribaucour transform of f in the sense of the preceding definition. Conversely, it follows from Corollary 21-(vi) below that a holonomic submanifold and any of its Ribaucour transforms in the sense of Definition 14 admit common principal coordinates. Moreover, Corollary 21-(iii) shows that the restriction of the vector bundle isometry P to the normal bundle is parallel in the normal connections, and thus the requirements of Definition 7 in [3] are satisfied, except for the additional technical assumption that the normal component of δ is nowhere a principal normal of f, which is not needed under our definition here.

The following basic result was proved in [3] for holonomic submanifolds.

Theorem 15. Let f: M^{n} → ℝ_{s}^{n+p} be an isometric immersion of a simply connected Riemannian manifold and let f̃: M^{n} → ℝ_{s}^{n+p} be a Ribaucour transform of f with data (P, D, δ). Then, there exists (φ, β) ∈ L(f) such that

f̃ = f − 2νφF,  (16)

where F = C_{φ,β}(f) and ν^{−1} := ϑ = ⟨F, F⟩. Moreover,

P = I − 2νFF^{∗}, D = I − 2νφΦ_{φ,β} and δ = −φ^{−1}F.  (17)

Conversely, given (φ, β) ∈ L(f) such that φϑ ≠ 0 everywhere, let P, D and δ be given by (17) on an open subset U ⊂ M^{n} where D is invertible. Then f̃: U → ℝ_{s}^{n+p} given by (16) is a Ribaucour transform of f|_{U} with data (P, D, δ).

Proof: Define μ ∈ C^{∞}(M) and ζ ∈ f^{∗}Tℝ_{s}^{n+p} with ⟨ζ, ζ⟩ = ϵ = ±1, by

f − f̃ = μζ.  (18)

Then, P(Z) = Z + μ⟨δ, Z⟩ζ. Thus,

μ⟨δ, Z⟩ = −2ϵ⟨ζ, Z⟩ and PZ = Z − 2ϵ⟨ζ, Z⟩ζ.  (19)

Define Z_{0} ∈ T M by f_{∗}Z_{0} = μ^{−1}(f_{∗}∇μ − 2ϵζ^{⊤}). An easy computation yields

\begin{matrix}⟨∇_{X}Z_{0}, Y ⟩ = Hess log μ(X, Y) − 2ϵμ^{−1}⟨α_{f}(X, Y), ζ⟩ \\ +2ϵμ^{−2}(X(μ)⟨ζ, f_{∗}Y ⟩ − μ⟨ ˜∇_{X}ζ, f_{∗}Y ⟩). \end{matrix}  (20)

On one hand,

⟨(f −μζ)_{∗}X, Pf_{∗}Y ⟩ = ⟨f̃_{∗}Ψ_{∗}X, Pf_{∗}Y ⟩ = ⟨Pf_{∗}DX, Pf_{∗}Y ⟩ = ⟨DX, Y ⟩. (21)

On the other hand, using (19) we have

\begin{matrix}⟨(f − μζ)_{∗}X, Pf_{∗}Y ⟩ = ⟨X, Y ⟩ − 2ϵ⟨ζ, f_{∗}X⟩⟨ζ, f_{∗}Y ⟩ \\ +X(μ)⟨ζ, f_{∗}Y ⟩ − μ⟨ ˜∇_{X}ζ, f_{∗}Y ⟩. \end{matrix}  (22)

It follows from (20), (21), (22) and D ∈ S(M) that ⟨∇_{X}Z_{0}, Y ⟩ = ⟨∇_{Y} Z_{0}, X⟩. Thus, there is ρ ∈ C^{∞}(M) such that Z_{0} = ∇ log ρ. Using (19) we obtain from ⟨(f − μζ)_{∗}X, Pξ⟩ = 0 for all ξ ∈ T_{f}^{⊥}M that

μ⟨ ˜∇_{X}ζ, ξ⟩ = ⟨ζ, ξ⟩(X(μ) − 2ϵ⟨ζ, f_{∗}X⟩).

Now, using that

μ⟨Z_{0}, X⟩ = X(μ) − 2ϵ⟨ζ, f_{∗}X⟩,  (23)

it follows that F = ρ^{−1}ζ satisfies ⟨F_{∗}X, ξ⟩ = 0. Set φ = ϵμ/2ρ. We obtain from (23) that ⟨F, f_{∗}X⟩ = X(φ). Thus F = f_{∗}∇φ + β is a Combescure transform of f. Formulas (16) and the last two in (17) follow immediately from (18), (19) and the definitions of φ and F. Differentiating (16) we easily see that f̃_{∗}Ψ_{∗}X = Pf_{∗}DX with D given by (17). The converse is straightforward.

By Theorem 15 any Ribaucour transform of f is determined by a pair (φ, β) ∈ L(f) such that φϑ ≠ 0. Moreover, it is easily seen that two pairs (φ, β) and (φ′, β′) yield the same Ribaucour transforms if and only if (φ′, β′) = λ(φ, β) for some λ ≠ 0. We denote by L̄(f) the set of equivalence classes of such pairs under the equivalence relation defined by the preceding equation. The Ribaucour transform of f determined by w ∈ L̄(f) will be denoted by R_{w}(f).

Examples 16. For w ∈ L̄_{0}(f) as in Examples 3 we have, respectively, that

(i) R_{w}(f) = R_{v} ∘ f, where

R_{v}(X) = X − 2ϵ⟨X − P_{0}, v⟩v  (24)

is the reflection with respect to the hyperplane H_{v} of ℝ_{s}^{n+p} through P_{0} orthogonal to v.

(ii) R_{w}(f) = I_{b} ∘ f, where

I_{b}(X) = P_{0} + b⟨X − P_{0}, X − P_{0}⟩^{−1}(X − P_{0})  (25)

is an inversion with respect to S_{b}(P_{0}) = {X ∈ ℝ_{s}^{n+p}: ⟨X − P_{0}, X − P_{0}⟩ = b}.

(iii) R_{w}(f)=f + 2c⟨ξ, ξ⟩^{−1}ξ is parallel to f in the direction of ξ.

We call a local diffeomorphism Ψ̄: U → V between open subsets U, V ⊂ ℝ_{s}^{n+p} a Ribaucour local diffeomorphism if |X − Ψ̄(X)| ≠ 0 everywhere and there exist D̄ ∈ S(U), a vector bundle isometry \overline{P}: T U → T V covering Ψ̄ and a nowhere vanishing smooth vector field δ̄ ∈ T U such that Ψ̄_{∗} = \overline{P} ∘ D̄ and

\overline{P}(X)Z − Z = ⟨δ̄, Z⟩(X − Ψ̄(X)) for all X ∈ U, Z ∈ T_{X}U.

An equivalent version of the following consequence of Theorem 15 was proved by Bianchi ([1]) in the context of n–orthogonal systems in Euclidean space.

Proposition 17. Let φ̄ ∈ C^{∞}(U) be nowhere vanishing on an open subset U ⊂ ℝ_{s}^{n+p}. Suppose that |∇φ̄| ≠ 0 and that D̄φ̄ = I − 2ν̄φ̄Hess φ̄, where ν̄ = ⟨∇φ̄, ∇φ̄⟩^{−1}, is invertible everywhere. Then Ψ̄φ̄: U → ℝ_{s}^{n+p} given by

Ψ̄φ̄ = I − 2ν̄φ̄∇φ̄

is a Ribaucour local diffeomorphism for \overline{P} = I − 2ν̄∇φ̄dφ̄ and δ̄ = −φ̄^{−1}∇φ̄. Conversely, any Ribaucour local diffeomorphism between open subsets of ℝ_{s}^{n+p} may be given this way.

Examples 18. With the notations of Examples 16, we have:

(i) Set U = ℝ_{s}^{n+p} − H_{v} and define φ̄ ∈ C^{∞}(U) by

φ̄(X) = ⟨X − P_{0}, v⟩.

Then Ψ̄φ̄: U → ℝ_{s}^{n+p} is the reflection R_{v} given by (24).

(ii) Set U = ℝ_{s}^{n+p} − (S_{b}(P_{0}) ∪ S_{1}(0)) and define φ̄ ∈ C^{∞}(U) by

2φ̄(X) = ⟨X − P_{0}, X − P_{0}⟩ − b.

Then Ψ̄φ̄: U → ℝ_{s}^{n+p} is the inversion I_{b} given by (25).

Ribaucour diffeomorphisms can be used to construct Ribaucour transforms of submanifolds, as follows from the next observation.

Proposition 19. Let Ψ̄: U → V be a Ribaucour diffeomorphism with data (D̄, \overline{P}, δ̄). Suppose that f: M^{n} → U is a D̄–invariant isometric immersion, and let D ∈ S(M) be defined by D̄ ∘f_{∗} = f_{∗} ∘D. Set P = \overline{P} ∘f and δ = δ̄∘f. Then f̃ = Ψ̄ ∘ f: M^{n} → V is a Ribaucour transform of f with data (D, P, δ).

Examples 16 (i) and (ii) are obtained as in Proposition 19 from the Ribaucour diffeomorphisms Ψ̄φ̄ given by Examples 18 (i) and (ii) respectively. We have that (i) Hess φ̄ = 0 or (ii) Hess φ̄ = I, and thus any isometric immersion f: M^{n} → U is trivially D̄φ̄–invariant.

The following immediate consequence of Proposition 7 shows that any Ribaucour transform of an isometric embedding with flat normal bundle arises as in Proposition 19.

Corollary 20. Let f: M^{n} → ℝ_{s}^{n+p} be an isometric embedding with flat normal bundle, and let f̃: M^{n} → ℝ_{s}^{n+p} be a Ribaucour transform of f with data (P, D, δ). Then there exists a Ribaucour local diffeomorphism

Ψ̄φ̄: U ⊃ f(M) → V ⊂ ℝ_{s}^{n+p}

with data (\overline{P}, D̄, δ̄) such that f̃ = Ψ̄φ̄ ∘ f, P = \overline{P} ∘ f, D̄ ∘ f_{∗} = f_{∗} ∘ D, and δ = δ̄ ∘ f.

Next, we establish some basic relations between an isometric immersion f: M^{n} → ℝ_{s}^{n+p} and one of its Ribaucour transforms f̃: M^{n} → ℝ_{s}^{n+p}. We denote by M̃^{n} the manifold M^{n} endowed with the metric induced by f̃ and mark with a tilde any object that refers to f̃.

Corollary 21. The following facts hold:

i) The metrics of M^{n} and M̃^{n} are related by

⟨X, Y ⟩^{∼} = ⟨DX, DY ⟩ for all X, Y ∈ T M.

ii) The gradients of ϕ ∈ C^{∞}(M) are related by

˜∇ϕ = D^{−2}∇ϕ.  (26)

iii) The restriction of the isometry P to the normal bundle is parallel in the normal connections, that is,

˜∇_{X}^{⊥}Pξ = P∇_{X}^{⊥}ξ for all X ∈ T M and ξ ∈ T_{f}^{⊥}M.  (27)

iv) The normal curvature tensors of f and f̃ are related by

R̃⊥ ; f̃(X, Y)Pξ = PR_{f}^{⊥}(X, Y)ξ for all X, Y ∈ T M and ξ ∈ T_{f}^{⊥}M.

v) Connections and curvature tensors of M^{n} and M̃^{n} are related by

D ˜∇_{X}Y = ∇_{X}DY + 2ν(∇φ ∧ ΦX)DY  (28)

and

R̃(X, Y, Z, W)=R(X,Y, DZ, DW)+2ν⟨Φ ∧T (X, Y)DZ, DW ⟩,  (29)

where A ∧ B(X, Y) = BX ∧ AY − BY ∧ AX and T = A_{β} + ν⟨β, β⟩Φ.

vi) The second fundamental forms of f and f̃ are related by

Ã_{Pξ} = D^{−1} (A_{ξ} + 2ν⟨β, ξ⟩Φ)  (30)

or, equivalently,

α̃(X, Y) = P (α(DX, Y) + 2ν⟨ΦX, DY ⟩β) .  (31)

Proof: Assertion (i) is immediate from Definition 14-(b) and (ii) follows easily from (i). We have,

−f̃_{∗}Ã_{Pξ}X + ˜∇_{X}^{⊥}P(ξ) = \overline{∇}_{X}Pξ = \overline{∇}_{X} (ξ − 2ν⟨F, ξ⟩F) .  (32)

From X(ν) = −ν^{2}X(ν^{−1}) = −2ν^{2}⟨F_{∗}X, F⟩ = −2ν^{2}⟨ΦX, ∇φ⟩, we have

∇ν = −2ν^{2}Φ∇φ.  (33)

Using (4) and (33), we obtain from (32) that

−f̃_{∗}Ã_{Pξ}X + ˜∇_{X}^{⊥}P(ξ) = −Pf_{∗} (A_{ξ} + 2ν⟨β, ξ⟩Φ) X + P∇_{X}^{⊥}ξ,

which implies (27) and (30).

Assertion (iv) follows immediately from (27). To prove (28) and (31), we have on one hand that

\begin{matrix}\overline{∇}_{X} f̃_{∗}Y = \overline{∇}_{X}Pf_{∗}DY = \overline{∇}_{X} (f_{∗}DY − 2ν⟨F, f_{∗}DY ⟩F) \\ = Pf_{∗} (∇_{X}DY +2ν(∇φ ∧ ΦX)DY)+P (α(X, DY)+2ν⟨ΦX, DY ⟩β) . \end{matrix}

On the other hand,

\overline{∇}_{X} f̃_{∗}Y = f̃_{∗} ˜∇_{X}Y + α̃(X, Y) = Pf_{∗}D ˜∇_{X}Y + α̃(X, Y),

and the conclusion follows. Finally, we obtain (29) from (31) and the Gauss equation.

In the next result we obtain the data that determine the inverse of a Ribaucour transformation.

Corollary 22. Let f̃ = R_{w}(f): M^{n} → ℝ_{s}^{n+p} be the Ribaucour transform determined by w = [(φ, β)] ∈ L̄(f) of an isometric immersion f: M^{n} → ℝ_{s}^{n+p}. Then f = Rw̃(f̃), where w̃ = [(φ̃, β̃)] ∈ L̄(f̃) is given by

φ̃ = −νφ and β̃ = −νPβ.  (34)

Moreover, F̃ = C_{φ̃,β̃}(f̃) and F = C_{φ,β}(f) are related by

F̃ = νF,  (35)

and Φ̃ = Φ̃_{φ̃,β̃} ∈ C(f̃) is given by

Φ̃ = D^{−1} ∘ νΦ.  (36)

Proof: By (33) we have that

∇(νφ) = νD∇φ.  (37)

It follows from (26) and (37) that

˜∇φ̃ = −νD^{−1}∇φ.  (38)

Using (31), (33), (34) and (38) we obtain that

α̃( ˜∇φ̃, X) + ˜∇_{X}^{⊥} β̃ = −νP (α(∇φ, X) + ∇_{X}^{⊥}β) = 0,

and hence (φ̃, β̃) ∈ L(f̃). Moreover, (38) and Definition 14-(b) yield

F̃ = f̃_{∗} ˜∇φ̃ + β̃ = −νPF = νF.

In particular, ν̃ = ν^{−1}. We conclude that w̃ = [(φ̃, β̃)] ∈ L̄(f̃) and that

R_{φ̃,β̃}(f̃) = f̃− 2ν̃φ̃F̃ = f.

Finally, we obtain (36) from

F̃_{∗}X = −2ν^{2}⟨f_{∗}ΦX, F⟩F + νf_{∗}ΦX = νPf_{∗}ΦX

and F̃_{∗}X = f̃_{∗}Φ̃X = Pf_{∗}DΦ̃X.

We notice the following consequence of (35) for later use.

Corollary 23. Let f̃ = R_{w}(f): M^{n} → ℝ_{s}^{n+p} be the Ribaucour transform determined by w ∈ L̄(f) of an isometric immersion f: M^{n} → ℝ_{s}^{n+p}, and let w̃ ∈ L̄(f̃) be such that f = Rw̃(f̃). Then, for any (φ, β) ∈ w and (φ̃, β̃) ∈ w̃, we have that

C_{φ̃,β̃}(R_{w}(f)) = ± I(C_{φ,β}(f)),

for an inversion I with pole at the origin.

Given an isometric immersion f: M^{n} → ℝ_{s}^{n+p} and (φ, β)∈L(f), define

L_{φ,β}(f) = {(φ′, β′) ∈ L(f) : [Φ_{φ′,β′}, Φ_{φ,β}] = 0}.

Notice that L_{0}(f) ⊂ L_{φ,β}(f) and that the image under (5) of L_{φ,β}(f) is

C(f; Φ_{φ,β}) = {Φ′ ∈ C(f) : [Φ′, Φ_{φ,β}] = 0}.

Denote by L̂_{φ,β}(f) the quotient space of L_{φ,β}(f) under the equivalence relation defined by

(φ′′, β′′) ∼ (φ′, β′) if (φ′′, β′′) − (φ′, β′) = λ(φ, β) for some λ ∈ ℝ,

and by {(φ′, β′)} the equivalence class of (φ′, β′).

Theorem 24. Let f̃ = R_{w}(f): M^{n} → ℝ_{s}^{n+p} be the Ribaucour transform determined by w = [(φ, β)] ∈ L̄(f) of an isometric immersion f: M^{n} → ℝ_{s}^{n+p} of a simply connected Riemannian manifold. Then, for any (φ′, β′) ∈ L_{φ,β}(f) there exists τ′ ∈ C^{∞}(M) such that

∇τ′ = −2Φ_{φ,β}∇φ′.  (39)

Furthermore, if

φ̄ = ντ′φ + φ′, β̄ = P(ντ′β + β′),  (40)

and w̃ = [(φ̃, β̃)] ∈ L̄(f̃) is such that f = Rw̃(f̃), then (φ̄, β̄) ∈ L_{φ̃,β̃}(f̃), and the linear map R_{φ,β}: L̂_{φ,β}(f) → L̂_{φ̃,β̃}(f̃) given by

R_{φ,β}({(φ′, β′)}) = {(φ̄, β̄)}

is an isomorphism whose inverse is R_{φ̃,β̃}. Moreover, R_{φ,β}(L̂_{0}(f)) = L̂_{0}(f̃).

Proof: The first assertion follows easily using that Φ := Φ_{φ,β} ∈ C(f) and that [Φ, Φ′] = 0, where Φ′ = Φ_{φ′,β′}. Using (26), (37) and (39) we obtain that

˜∇φ̄ = D^{−1}(ντ′∇φ + ∇φ′).  (41)

It follows from (27), (31) and (41) that

\begin{matrix}α̃(X, ˜∇φ̄) + ˜∇_{X}^{⊥} β̄ = ντ′P(α(X, ∇φ) + ∇_{X}^{⊥}β) + P(α(X, ∇φ′) + ∇_{X}^{⊥}β′) \\ +2ν⟨ΦX, ντ′∇φ + ∇φ′⟩ + X(ντ′)Pβ. \end{matrix}

The first two terms on the right hand side vanish because (φ, β) and (φ′, β′) belong to L(f). Using (33) and (39), it follows that the last two terms cancel out, and thus (φ̄, β̄) ∈ L(f̃). Now, a long but straightforward computation using (28), (30), (33), (39) and (41) shows that

Φ̄ := Φ̄_{φ̄,β̄} = H̃ess φ̄ − Ãβ̄ = D^{−1}(ντΦ + Φ′),

where τ = −τ′ − 2⟨F, F′⟩. Using (36), we have

[Φ̄, Φ̃] = ν[D^{−1}Φ, D^{−1}(ντΦ + Φ′)] = 0,

where Φ̃ = Φ̃_{φ̃,β̃}, because [Φ′, Φ] = 0. Thus (φ̄, β̄) ∈ L_{φ̃,β̃}(f̃). We show that

R_{φ̃,β̃}(R_{φ,β}({φ′, β′})) = {(φ′, β′)}.

We have that R_{φ,β}({φ′, β′}) = {(φ̄, β̄)}, where (φ̄, β̄) is given by (40), and that R_{φ̃,β̃}({(φ̄, β̄)}) = {(φ̂, β̂)}, where

φ̂ = ν̃τ̃′φ̃ + φ̄, β̂ = P^{−1}(ν̃τ̃′β̃ + β̄),

and

˜∇τ̃′ = −2Φ̃ ˜∇φ̄.  (42)

We need only show that (φ̂, β̂) = (φ′, β′) for some τ̃′ satisfying (42). This equality is clearly satisfied for τ̃′ = ντ′. That (42) holds for this choice of τ̃′ follows from an easy computation using (26), (33), (36) and (39).

We prove the last assertion. First notice that, by Definition 14-(b) and (41), we have

F̄ = C_{φ̄,β̄}(f̃) = P(ντ′F + F′).  (43)

Now let (φ_{i}, β_{i}) ∈ L_{0}(f), 0 ≤ i ≤ 2, be as in Examples 3, and let (φ̄_{i}, β̄_{i}) be similarly defined by φ̄_{0} = ⟨f̃− P_{0}, v⟩, β̄_{0} = v^{⊥}, 2φ̄_{1} = ⟨f̃ − P_{0}, f̃ − P_{0}⟩ − b, β̄_{1} = (f̃− P_{0})^{⊥}, and φ̄_{2} = c, β̄_{2} = P(ξ). It suffices to prove that

R_{φ,β}({(φ_{i}, β_{i})}) = {(φ̄_{i}, β̄_{i})}.

We give the proof for the case i = 1; the others are similar. Let

φ̄ = ντ_{1}φ + φ_{1} and β̄ = P(ντ_{1}β + β_{1}),

where τ_{1} is defined up to a constant by ∇τ_{1} = −2Φ_{φ,β}∇φ_{1}. We need only show that (φ̄, β̄) = (φ̄_{1}, β̄_{1}) for some choice of this constant. An easy computation using that f_{∗}∇φ_{1} is the tangent component of f − P_{0} proves that

∇τ_{1} = 2∇(φ − ⟨f − P_{0}, F⟩).

Choosing τ_{1} = 2(φ−⟨f −P_{0}, F⟩) we easily obtain that φ̄ = φ̄_{1}. On the other hand, by (43) we have that F̄ = P(ντ_{1}F + F_{1}). Using that P(F) = −F easily yields F̄ = f̃− P_{0}, and hence β̄ = β̄_{1}.

Denote by Ĉ(f; Φ) the quotient space of C(f; Φ) under the equivalence relation Φ′′ ∼ Φ′ if Φ′′ − Φ′ = λΦ for some λ ∈ ℝ, and by {Φ′} the equivalence class of Φ′.

Corollary 25. Under the assumptions of Theorem 24, given Φ′ ∈ C(f; Φ) define

Φ̄ = D^{−1} ∘ (ντΦ + Φ′),  (44)

where Φ := Φ_{φ,β} and ∇τ = −2Φ′∇φ. If Φ̃ := Φ̃_{φ̃,β̃}, then Φ̄ ∈ C(f̃; Φ̃) and R_{Φ}: Ĉ(f; Φ) → Ĉ(f̃; Φ̃) defined by R_{Φ}{Φ′} = {Φ̄} is an isomorphism whose inverse is RΦ̃. Moreover, R_{Φ}(Ĉ^{0}(f)) = Ĉ^{0}(f̃).

Theorem 26. Given an isometric immersion f: M^{n} → ℝ_{s}^{n+p}, set f̃ = R_{w}(f) and f′ = R_{w′}(f), where w = [(φ, β)] and w′ = [(φ′, β′)] belong to L̄(f) and (φ′, β′) ∈ L_{φ,β}(f). If (φ̄, β̄) ∈ R_{φ,β}({(φ′, β′)}) and (φ̄′, β̄′) ∈ R_{φ′,β′}({(φ, β)}) are given by Theorem 24, then

τ + τ′ + 2⟨F, F′⟩ = k ∈ ℝ,  (45)

where F := F_{φ,β} and F′ := F_{φ′,β′}. Choosing k = 0 in (45), we have that

Rw̄(f̃) = R_{}_{w̄′}(f′) = f − Γ(ϑ′φ + τφ′)F − Γ(ϑφ′ + τ′φ)F′,  (46)

where w̄ = [(φ̄, β̄)], w̄′ = [(φ̄′, β̄′)] and Γ = 2/(ϑϑ′ − ττ′).

Proof: The proof of (45) is straightforward. We have that

Rw̄(f̃) = f̃− 2φ̄ν̄F̄,

where F̄ = C_{φ̄,β̄}(f̃) = f̃_{∗} ˜∇φ̄ + β̄. We obtain from (43) that

ϑ̄ = ν̄^{−1} = ⟨F̄, F̄⟩ = ντ^{′2} + 2ντ′Θ + ϑ′,  (47)

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

Rw̄(f̃) = f − 2νφF + \frac{2(ντ′φ + φ′)}{ντ^{′2} + 2ντ′Θ + ϑ′} (ντ′F − F′ + 2νΘF)
= f − Γ(ϑ′φ + τφ′)F − Γ(ϑφ′ + τ′φ)F′.

The first equality in (46) now follows from the symmetry of the preceding expression with respect to w and w′.

Given w = [(φ, β)] and w′ = [(φ′, β′)] in L̄(f) with (φ′, β′) ∈ L_{φ,β}(f), we define the projective line ℓ = ℓ_{w,w′} by

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

We call the associated family to ℓ and denote by R_{ℓ}(f) the one-parameter family of Ribaucour transforms of f determined by the elements of ℓ.

Corollary 27. Given an isometric immersion f: M^{n} → ℝ_{s}^{n+p}, let ℓ = ℓ_{w,w′} be the projective line through w, w′ ∈ L̄(f). Then the associated family R_{ℓ}(f) has a conjugate family (R_{ℓ}(f))^{c} all of whose elements is a Ribaucour transform of any element of R_{ℓ}(f). Moreover, ((R_{ℓ}(f))^{c})^{c} = R_{ℓ}(f).

Proof: Let (φ, β) ∈ w and (φ′, β′) ∈ w′, and let {(φ̄, β̄)} = R_{φ,β}({(φ′, β′)}). Then {(φ̄, β̄)} determines a projective line through w̃ ∈ L̄(f̃), where we have Rw̃(R_{w}(f)) = f, which we denote by R_{w}(ℓ). Set (R_{ℓ}(f))^{c} = R_{R_{w}(ℓ)}(R_{w}(f)). By Theorem 24, we also have that (R_{ℓ}(f))^{c} = R_{R_{w′}(ℓ)}(R_{w′}(f)) for any w′ = [(φ′, β′)] ∈ ℓ, and hence any element of (R_{ℓ}(f))^{c} is a Ribaucour transform of any one in R_{ℓ}(f). Finally, the last assertion follows from the fact that ((R_{ℓ}(f))^{c})^{c} is the associated family to Rw̃(R_{w}(ℓ)) = ℓ.

Remarks 28. (i) For a fixed x ∈ M^{n}, it is easily seen that the sets {R_{ℓ}(f)(x)} and {(R_{ℓ}(f))^{c}(x)} parametrize a common circle.

(ii) Given w = [(φ, β)] ∈ L̄(f) and a parallel normal vector field ξ along f, set φ′ = φ, β′ = β +ξ, w′ = [(φ′, β′)] and ℓ = ℓ_{w,w′}. Then, it is not difficult to check that the associated family R_{ℓ}(f) to ℓ and its conjugate family (R_{ℓ}(f))^{c} coincide. Therefore, any two elements of R_{ℓ}(f) are Ribaucour transforms of each other. Moreover, each circle {R_{ℓ}(f)(x)} intersects any element of R_{ℓ}(f) orthogonally. For surfaces in ℝ^{3}, this gives rise to a so–called cyclic system. In fact, it was shown by Bianchi that any cyclic system in ℝ^{3} arises this way. A far–reaching generalization of this fact, with applications to a local description of all holonomic Dupin submanifolds, is developed in a forthcoming paper (see [2]).

We now show that the Ribaucour transformation is invariant under inversions. A more general result will be proved in Proposition 31 below.

Proposition 29. Let f̃ = R_{w}(f) be the Ribaucour transform of an isometric immersion f: M^{n} → ℝ_{s}^{n+p} determined by w = [(φ, β)] ∈ L̄(f). Then I(f̃) is a Ribaucour transform of I(f) for any inversion I.

Proof: Let I be an inversion with pole P_{1} and ratio r_{1}. Then I(f) = R_{w_{1}}(f), where w_{1} = [(φ_{1}, β_{1})] is given by 2φ_{1} = ⟨f − P_{0}, f − P_{0}⟩ − b, β_{1} = (f − P_{0})^{⊥}. Moreover, it follows from the proof of the last assertion in Theorem 24 that I(f̃) = Rw̄(f̃), where w̄ = [(φ̄, β̄)] is given by

φ̄ = ντ_{1}φ + φ_{1}, β̄ = P(ντ_{1}β + η̄_{1}),

with τ_{1} = 2(φ − ⟨f − P_{1}, F⟩). We conclude that I(f̃) = R_{}_{w̄′}(I(f)) from Theorem 26, where w̄′ = [(φ̄′, β̄′] is given by

φ̄′ = ν_{1}τφ_{1} + φ, β̄ = P(ν_{1}τβ_{1} + β),

with τ = −τ_{1} − 2⟨F_{1}, F⟩ = −2φ.

Given an isometric immersion f: M^{n} → ℝ_{s}^{n+p}, we say that an immersion L(f): M^{n} → ℝ_{s}^{n+p} is a Lie transform of f if it can be obtained from f by a composition of conformal transformations of the ambient space and translations by parallel normal vector fields. If f̃: M^{n} → ℝ_{s}^{n+p} is a Ribaucour transform of f, to each Lie transform L(f) of f we associate a Lie transform L̃(f̃) of f̃ as follows:

(i) L̃(f̃) = T ∘ f̃ when L(f) = T ∘ f and T is a conformal map of ℝ^{n+p};

(ii) L̃(f̃) = f̃+ Pξ when L(f) = f + ξ is a parallel translation;

(iii) L̃(f̃) = (L̃_{1} ∘ … ∘ L̃_{k})(f̃) when L(f) = (L_{1} ∘ … ∘ L_{k})(f) and L_{1}, …, L_{k} are as in items (i) or (ii).

Corollary 30. Let f: M^{n} → ℝ_{s}^{n+p} be an isometric immersion. If w = [(φ, β)] ∈ L̄_{0}(f) then f̃ = R_{w}(f) is a Lie transform of f. More precisely, there exist a Lie transform L(f) of f and an inversion I with pole at the origin such that

L̃(f̃) = ± I ∘ L(f).

Proof: Let (φ, β) = (φ_{0} + φ_{1} + φ_{2}, β_{0} + β_{1} + β_{2}), where φ_{0} = ⟨f − Q_{0}, v⟩, β_{0} = v^{⊥}, 2φ_{1} = a(⟨f − P_{0}, f − P_{0}⟩ − b), β_{1} = (f − P_{0})^{⊥}, φ_{2} = c and β_{2} = ξ, with P_{0}, Q_{0}, v ∈ ℝ_{s}^{n+p} and a, c, r ∈ ℝ. Then, C_{φ,β}(f) = a(f − P_{0}) + ξ + v. Moreover, for w̃ ∈ L̄(f̃) such that Rw̃(f̃) = f, it is easily seen that we can choose (φ̃, β̃) ∈ w̃ such that C_{φ̃,β̃}(f̃) = a(f̃ − P_{0}) + P(ξ) + v. If a ≠ 0, then L(f) := C_{φ,β}(f) is a Lie transform of f and C_{φ̃,β̃}(f̃) = L̃(f̃), and thus the conclusion follows from Corollary 23. Otherwise, let I_{1} be an inversion with pole P_{1} and ratio r_{1}. Then, the proof of Proposition 29 shows that I_{1}(f̃) = Rw̄_{1}(I_{1}(f)) for w̄_{1} = [(φ̄_{1}, β̄_{1})] given by

φ̄_{1} = −2φν_{1}φ_{1} + φ, β̄_{1} = P_{1}(−2φν_{1}β_{1} + β),

where P_{1} = I − 2ν_{1}F_{1}^{∗}F_{1} = I − 2⟨f − P_{1}, f − P_{1}⟩^{−1}(f − P_{1})^{∗}(f − P_{1}). An easy computation shows that

C_{φ̄_{1},β̄_{1}}(I_{1}(f)) = P_{1}(−2φν_{1}F_{1} + F) = ã(I_{1}(f) − P_{1}) + P(ξ) + v,

where b_{1}^{2}ã = 2(φ − ⟨f − P_{1}, v⟩). Hence w̄_{1} ∈ L̄_{0}(I_{1}(f)) and we may choose P_{1} so that ã ≠ 0. Applying the first part of the proof to I_{1}(f) and I_{1}(f̃) = Rw̄_{1}(I_{1}(f)), it follows that there exist a Lie transform L_{1}(I_{1}(f)) of I_{1}(f) and an inversion I with pole at the origin such that L̃_{1}(I_{1}(f̃)) = ± I ∘ L_{1}(I_{1}(f)). We conclude the proof by setting L = L_{1} ∘ I_{1}.

We now establish the invariance of the Ribaucour transformation under Lie transformations.

Proposition 31. Let f̃ = R_{w}(f): M^{n} → ℝ_{s}^{n+p} be the Ribaucour transform of f: M^{n} → ℝ_{s}^{n+p} determined by w = [(φ, β)] ∈ L̄(f), and let L(f) = R_{w′}(f) be the Lie transform of f determined by w′ = [(φ′, β′)] ∈ L̄_{0}(f). Then L̃(f̃) is a Ribaucour transform of L(f). In particular, I(f̃) is a Ribaucour transform of I(f) for any inversion I.

Proof: It follows from the proof of the last assertion in Theorem 24 that there exists (φ̄, β̄) ∈ R_{w}({(φ′, β′)}) such that L̃(f̃) = Rw̄(f̃), where w̄ = [(φ̄, β̄)]. For w̄′ ∈ L̄(L(f)) as in Theorem 26, we obtain R_{}_{w̄′}(L(f)) = Rw̄(f̃) = L̃(f̃).

We conclude the paper observing that the theory developed so far can be easily extended to submanifolds of nonflat pseudo–Riemannian space forms. Given an isometric immersion f: M^{n} → ℚ_{s}^{n+p}(c) where c ≠ 0, we set F = i ∘ f: M^{n} → ℝ_{s+ϵ_{o}}^{n+p+1}, where ϵ_{o} = 0 or 1, according to c > 0 or c < 0, respectively, and i: ℚ_{s}^{n+p}(c) → ℝ_{s+ϵ_{o}}^{n+p+1} is the umbilical inclusion. Then, one can check that many of the results in the first section, after the obvious extensions of the basic definitions therein to the nonflat case, remain valid or have a corresponding version simply by noticing that C(f) = C(F). The latter equality is an immediate consequence of the fact that the second fundamental forms of f and F are related by

α_{F} (X, Y) = α_{f}(X, Y) − c⟨X, Y ⟩F.  (48)

In particular, defining L(f) as before, it is easily seen that any Φ ∈ C(f) can now be written as Φ = Φ_{φ,β}^{c} = Hess φ + cφI − A_{β}^{f}, where (φ, β) ∈ L(f).

We say that f̃: M̃^{n} → ℚ_{s}^{n+p}(c) is a Ribaucour transform of f with data (P, D, δ) if F̃: M^{n} → ℝ_{s+ϵ_{o}}^{n+p+1} is a Ribaucour transform of F with data (P̂, D, δ̂), where δ̂ = δ − cF and P̂: F^{∗}T M → F̃^{∗}T M is the extension of P defined by setting P̂(F) = F̃. With this definition, the results on the Ribaucour transformation can also be easily extended to nonflat pseudo– Riemannian space forms. For instance, Theorem 15 now reads as follows.

Theorem 32. Let f: M^{n} → ℚ_{s}^{n+p}(c) be an isometric immersion of a simply connected Riemannian manifold and let f̃: M̃^{n} → ℚ_{s}^{n+p}(c) be a Ribaucour transform of f with data (P, D, δ). Then there exists (φ, β) ∈ L(f) such that

F̃ = F − 2νφG,  (49)

where G = F_{∗}∇φ + β + cφF and ν = ⟨G, G⟩^{−1}. Moreover,

P̂ = I − 2νGG^{∗}, D = I − 2νφΦ_{φ,β}^{c} and δ̂ = −φ^{−1}G.  (50)

Conversely, given (φ, β) ∈ L(f) such that φν ≠ 0 everywhere, let U ⊂ M^{n} be an open subset where the tensor D given by (50) is invertible, and let F̃: U → ℝ_{s+ϵ_{o}}^{n+p+1} be defined by (49). Then F̃ = i ∘ f̃, where f̃ is a Ribaucour transform of f.

References

[1] L. Bianchi, “Lezioni di Geometria Differenziale”, Bologna, 1927.

[2] M. Dajczer, L Florit and R. Tojeiro, Reducibility of Dupin submanifolds, Preprint.

[3] M. Dajczer and R. Tojeiro, An extension of the classical Ribaucour transformation, Preprint.

[4] M. Dajczer and R. Tojeiro, The Ribaucour transformation for flat Lagrangian submanifolds, J. Geom. Anal. 10 (2000), 269–280.

[5] D. Ferus, A remark on Codazzi tensors in constant curvature spaces, Lect. Notes in Math. 838, Berlin 1981, 247.

[6] V. Schulikovski, “Classical Differential Geometry”, Moscow 1963 (in Russian); cf. Zbl. Math. 122, 393.

[7] R. Tojeiro, Lagrangian submanifolds with constant sectional curvature and their Ribaucour transformation, Bull. Belgian Math. Soc. 8 (2001), 29-46.

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