The isotropic geometry introduced by Strubecker in [13], [14] and [15], and developed by several authors, study of the properties invariant by the action of the 6-parameter group G₆ in R³.

where a, b, c, c₁, c₂, φ ∈ R.

In other word, G₆ is the group of rigid motions. Notice that on the xy-plane this geometry looks exactly like the plane Euclidean geometry R². The projection of a vector u = (u₁,u₂,u₃) on the xy-plane is the top view of u and we shall denote it by u^- = (u₁,u₂,0). The top view concet plays a fundamental role in the isotropic space I³, since the z-direction is preserved by the action of G₆. A line with this direction is called an isotropic line and a plane that contained an isotropic line is said to be an isotropic plane. One may introduce a isotropic inner product between two vectors u, v as

from which the isotropic distance is defined as:

The inner product and distance above are just the plane Euclidean counterparties of the top views u and e ve. In addition, since the isotropic metric is degenerate, the distance from a point A = (a₁, a₂, a₃) to v = (b1, b₂, b₃) is zero if . In this cases, one may define a codistance by:

Cd_z(A, B) = |b₃-a₃|,

which is then preserved by G₆. To study the geometry of a surface in isotropic space, it is considered a surface as a graph of a function, given by:

X(u₁, u₂) = (u₁ , u₂ , h(u₁ , u₂)).

The isotropic Gauss map, is given by NI = (−h,1 , −h,2 , 1).

The coefficients of the first and second fundamental forms are defined by:

In this work motivated by isotropic geometry, we generalize the idea of isotropic distance. Let Rⁿ⁺¹ be the Euclidean space with the usual metric <, > and consider M a hypersurface fixed in Rⁿ⁺¹, with Gauss map N. We introduce the space relative to M, as being

We define the space relative to M of the type I, as being the space , with the distance defined by

1.1

Where is the distance between p₁ and p₂, considering M as a metric space.

On the other hand, let q = p + tN(p) be a point in , we can consider , and we define the space relative to M of the type II, as being the space , with the metric defined by

1.2

where denotes the orthogonal projection of V_q on T_pM.

Let M be a hypersurface in , locally, M can be parametrized by X : U ⊂ Rⁿ →

1.3

where h : U ⊂ Rⁿ → R is a real function and Y : U → M is a local parameterization of the M. In this case, the function h is called the height function.

Let N be the Gauss map of the M. We define the relative Gauss map by

1.4

Considering M to be a hypersurface in the relative space of the type I, the coefficients of the first and second fundamental forms of the M relative to M are given by.

1.5

Moreover, we define the relative Weingartem matrix W of the , where B is the matrix of coefficients of the second fundamental form and G₁ is the matrix of coefficients of the first fundamental form.

Analogously, considering M to be a hypersurface in the relative space of the type II, the coefficients of the first and second fundamental forms of the M relative to M are given by

1.6

We define the relative Weingartem matrix W of the , where B and G₂ are, respectively, the matrices of the coefficients of the first and second fundamental forms of the M relative to M.

We observe that in the two relative spaces, M has the same relative normal map and the same second fundamental form, what are different are its first fundamental forms.

Ribaucour transformations for hypersurfaces, parametrized by lines of curvature, were classically studied by Bianchi [4]. They can be applied to obtain surfaces of constant Gaussian curvature and surfaces of constant mean curvature, from a given such surface, respectively, with constant Gaussian curvature and constant mean curvature. The first application of this method to minimal and cmc surfaces in R³ was obtained by Corro, Ferreira, and Tenenblat in [6]-[8].

Dupin’s surfaces in Euclidean space are classified. There are several equivalent definitions of Dupin cyclides, for example, in Euclidean space, they can be defined as any inversion of a torus, cylinder or double cone, i.e, Dupin cyclide is invariant under Mobius transformations. Classically the cyclides of Dupin were characterized by the property that both sheets of the focal set are curves. Another equivalent definition says that such surfaces can also be given as surfaces that are the envelope of two families at 1-parameter spheres (including planes as degenerate spheres). For more on Dupin cyclides see [2] and [3].

We consider M a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface M, we define a relative Dupin hypersurface M of type I and type II, we provide necessary and sufficient conditions for a relative hypersurface M to be relative Dupin parametrized by relative lines of curvature, in both spaces. Moreover, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative M. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S¹×Rⁿ⁻¹ and S²×Rⁿ⁻², in both spaces. This work is organized as follows. In section 1, we provide the basic local theory of the Ribaucour transformation and Dupin hypersurface definition. In section 2, we provide a local characterization of the hypersurfaces relatives, to a fixed hypersurface parametrized by lines of curvature, in both types. Moreover, we provide the relative Weingarten matrix and a necessary and sufficient condition for a relative hypersurface M to be a relative Dupin parametrized by lines of relative curvature, in both types. In section 3, we highlight the type I relative Dupin hypersurfaces and we generate families of type I Dupin hypersurface relative to a hyperplane, a torus, S¹ × Rⁿ⁻¹ and S² × Rⁿ⁻². In section 4, we highlight the type II relative Dupin hypersurfaces, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative to M. Moreover, we generate families of type II Dupin hypersurface relative to a hyperplane, a torus, S¹ × Rⁿ⁻¹ and S² × Rⁿ⁻².

This section contains definitions and basic concepts that will be used in later sections.

A sphere congruence is an n-parameter family of spheres whose centers lie on an n-dimensional manifold M₀ contained in Rⁿ⁺¹. Locally, we may condider M₀ parametrized by X₀ : U ⊂ Rⁿ → Rⁿ⁺¹. For each u ∈ U, we consider a sphere centered at X₀(u) with radius r(u), where r is a differentiable real function. Two hypersurfaces M and are said to be associated by a sphere congruence if there is a difeomorphism ψ : M → , such that at corresponding points p and ψ(p) the manifolds are tangent to the same sphere of the sphere congruence. A special case occurs when ψ preserves lines of curvature.

Let M and be orientable hypersurfaces of Rⁿ⁺¹. We denote by N and Ne their Gauss map. We say that M and are associated by a Ribaucour transformation, if and only if, there exists a differentiable function h defined on M and a diffeomorphism.

(a) for all p ∈ M, p + h(p)N(p) = ψ(p) + h(p) (ψ(p)), where is the Gauss map of

(b) The subset p + h(p)N(p), p ∈ M, is a n-dimensional submanifold.

(c) ψ preserves lines of curvature.

We say that M and are locally associated by a Ribaucour transformation if, for all p ∈ M, there exists a neighborhood of p in M which is associated by a Ribaucour transformation to an open subset of . Similarly, one may consider the notion of parametrized hypersurfaces locally associated by a Ribaucour transformation.

A hypersurface M ⊂ Rⁿ⁺¹ is a Dupin submanifold if its principal curvatures are constant along the corresponding lines of curvature. Whenever, the principal curvatures are constant, M is a called an isoparametric submanifold.

2.1, 2.2, 2.3, 2.4, 2.5

The Christoffel symbols in terms of the metric (2.1) are given by

2.6, 2.7

In this section, we start with a local characterization of a relative hypersurface of the type I and (or) type II to a fixed hypersurface in Rⁿ⁺¹. We provide the relative Weingarten matrix and a necessary and sufficient condition for a relative hypersurface M has a parameterization by lines of relative curvature.

Theorem 3.1. Let M be an orientable hypersurface in Rⁿ⁺¹, N its Gauss map and suppose that M has an orthogonal parameterization by lines of curvature Y : U ⊂ Rⁿ → M, with principal curvatures −λi, 1 ≤ i ≤ n. Let M be a hypersurface in of the type I or type II. Then M can be parameterized by

3.1

with the relative normal N_R, given by

3.2

where Lrr = <Y,r , Y,r >. Moreover, the type I (or type II) relative Weingarten matrix of is given by

3.3

where are given by (2.6) and g_jj are the coefficients of the first fundamental form of X given by (1.5), if type I and by (1.6), if type II.

Proof.

Let M be a hypersurface in . Since that Y : U ⊂ Rⁿ → M is a parameterization by lines of curvature for M, we have that M can be parametrized by X = Y + hN, where N is a vector field normal to Y . Differentiating X with respect to ui and uj 1 ≤ i, j ≤ n, we get:

3.4, 3.5

where λ_i, 1 ≤ i ≤ n are the principal curvatures of the M.

In order, we will consider N the unit vector field normal to M given by

3.6

where

Since <X,_i , N_i> = 0, for all 1 ≤ i ≤ n, using (3.4) we get

Substituting in (3.6), we have

Therefore the relative normal

is given by (3.2).

Finally, let be the type I (or type II) relative Weingarten matrix of X. Thus

where g_jj are the coefficients of the first fundamental form of X given by (1.5), if type I and by (1.6), if type II. Using (3.2) and (3.5), we have (3.3).

Let M be an orientable hypersurface in Rn+1 and consider M a hypersurface in of the type I

(or type II). For each p ∈ M there exists a type I (or type II) relative orthonormal basis of TpM such that . The functions are called the type I (or type II) relative principal curvatures at p and the corresponding directions, that is, eⁱ _R are called type I (or type II) relative principal directions at p. We say that a hypersurface M in is parametrized by lines of relative curvature X of the type I (or type II), if for each , are type I (or type II) relative principal directions.

Theorem 3.2 . Let M be an orientable hypersurface in Rⁿ⁺¹, N its Gauss map and suppose that M has an orthogonal parameterization by lines of curvature Y : U ⊂ Rⁿ → M, with principal curvatures −λ_i, 1 ≤ i ≤ n. Then (3.1) is an orthogonal parameterization by lines of relative curvature of the type I (or type II) for a hypersurface M relative to M of the type I (or type II), if and only if, there exists nonvanishing functions Ω, Ωⁱ and W, where h = Ω/W , such that

3.7

With and W(W + λ_iΩ) /= 0. Moreover X given by (3.1) becomes

3.8

Proof. From Theorem 1, the type I (or type II) relative Weingarten matrix of , is given by

(3.3). Therefore, if V is diagonal, then X is a parameterization by lines of relative curvature of the type I (or typeII). Thus, for V_ij = 0, i/=j, we get h,

3.9

From Proposition 2.3 of [6], h is a solution of (3.9), if and only if, there exists nonvanishing functions Ω,

Ωi and W, where , which satisfy

3.10

with and W(W + λ_iΩ) /= 0.

Remark 3.1.

Let M be a hypersurface in of the type I (or type II), parametrized by lines of relative curvature of the type I (or type II), as in Theorem 2. Then the type I (or type II) relative Weingarten matrix, given by in the Theorem 1, can be rewritten as follows V_ij = 0, 1 ≤ i /= j ≤ n and

3.11

where W and Ω satisfies (3.10).

Remark 3.2.

Let M be a hypersurface in of the type I (or type II), parametrized by lines of relative curvature of the type I (or type II). Then, the relative principal curvatures of M of the type I (or type II) λR_i , are given by λR_i = V_ii, 1 ≤ i ≤ n.

Definition 3.1. A hypersurface is a relative Dupin submanifold of the type I (or type II) if its relative principal curvatures of type I ( or type II) are constant along the corresponding relative lines of curvature of type I ( or type II). Whenever, the relative principal curvatures of type I ( or type II) are constant, M is a called a relative isoparametric submanifold of type I ( or type II).

Using Remark 2 and Definition 1, we immediately get the corollary.

Corollary 3.1. Let M be a Dupin hypersurface in Rⁿ⁺¹ and suppose that M has an orthogonal parameterization by lines of curvature Y : U ⊂ Rⁿ → M. Let M be a ypersurface in Rⁿ⁺¹ of the type I (or type II), and consider the relative Weingarten matrix of the type I (or type II), given by (3.11). Then M is a Dupin hypersurface in of the type I (or type II) if, and only if, Vii,i = 0.

Remark 3.3. If M is the hyperplane Rⁿ, then the hypersurface M relative to M of the type I and type II is an isotropic hypersurface.

In this section, we highlight the relative Dupin hypersurfaces type I. We start by providing a relationship between the Dupin hypersurfaces locally associated to Rⁿ by a Ribaucour transformation and the type I Dupin hypersurfaces relative to Rn. We will generate families of type I Dupin hypersurfaces relative to a hyperplane, a torus, S¹ × Rⁿ⁻¹ and S² × Rⁿ⁻².

Let M be an orientable hypersurface in Rⁿ⁺¹, N its Gauss map and suppose that M has an orthogonal parameterization by lines of curvature Y : U ⊂ Rn → M, with principal curvatures −λi, 1 ≤ i ≤ n. Let M be a hypersurface in of the type I. Then M can be parametrized by

where h is a differentiable real function defined on M. Moreover, for (1.5), the coefficients of the first and second fundamental forms of X are given by

where the normal relative N_R is given by (3.2).

The first theorem provides a relationship between the Dupin hypersurfaces locally associated to Rⁿ by a Ribaucour transformation and the type I Dupin hypersurfaces relative to Rⁿ.

Theorem 4.1. Let Rⁿ be a hyperplane parametrized by Y (u₁, ..., u_n) = (u₁, ..., u_n, 0). Consider Mf the hypersurface locally associated to Rn by a Ribaucour transformation. Let M be a type I hypersurface relative to Rⁿ, then M is a type I Dupin hypersurface relative to Rⁿ, if and only if, Mf is a Dupin hypersurface.

Proof: From Corollary 1, M is a type I Dupin hypersurface relative to Rn if, and only if, V_ii,i = 0, where

with functions W and Ω satisfying (3.10).

On the other hand, from [9], Mf locally associated to Rn by a Ribaucour transformation, is a Dupin hypersurface, if and only if, T_i,i = 0, where

with functions W and Ω satisfying (3.10).

Since the principal curvatures of Y are λ_i = 0 and the metric L_ij = δ_ij, for 1 ≤ i,j ≤ n, it follows from equation (3.10) that

where f_i(u_i) are differentiable functions. Therefore, V_ii,i = 0, if and only if, T_i,i = 0.

Remark 4.1. When M is the hyperplane Rⁿ, the geometry of coincides with the isotropic geometry. Then in the theorem 3, we show that the hypersurface Mf locally associated to Rn by a Ribaucour transformation is a Dupin hypersurface, if and only if, the hypersurface M is an isotropic Dupin hypersurface. Moreover, M is the hypersurface of center of the Ribaucour transformation.

In the next results, we provide families of type I Dupin hypersurfaces relative to a hyperplane, a torus, S¹ × Rⁿ⁻¹ and S² × R^n−2.

Proposition 4.1. Consider the hyperplane in the Euclidean space Rⁿ⁺¹, parametrized by Y (u₁,...,u_n) = (u1,...,u_n,0). Then M is a type I Dupin hypersurface relative to Rⁿ⁺¹, if and only if, M can be parametrized by

4.1

Proof. Since the principal curvatures of Y are λi = 0 and the metric L_ij = δ_ij, for 1 ≤ i,j ≤ n, it follows from equation (3.10) that

where f_i(u_i) are differentiable functions. In order, to obtain type I Dupin hypersurface relative to Rⁿ, we consider V_ii given by (3.11),

From Corollary 1, M parametrized by , where e_n+1 = (0,0,...,0,1) is a unit vector field normal to Rⁿ, is a type I Dupin hypersurface relative to Rⁿ, if and only if, V_ii,i = 0. Therefore, fi(ui) = c_i2u² _i + c_i1u_i + c_i0, with c_i2, c_i1, c_i0 ∈ R and from (3.8), X is given by (5.2).

Proposition 4.2. Consider the torus in R³, parametrized by

Then M is a Type I Dupin hypersurface relative to torus, if and only if, M can be parametrized by

4.2

where B_i, A_i, A and B are real constants.

Proof. The principal curvatures of the torus and coefficients of the metric of the torus are

Using (3.10), we obtain

where A, B are constants and f₁, f₂ are differentiable functions of u₁ and u₂, respectively.

Consider V_ii given by (3.11). Thus

4.3, 4.4

From Corollary 1, M parametrized by (3.8) is a type I Dupin hypersurface relative to torus, if and only if, V_ii,i = 0 for all 1 ≤ i ≤ 2.

Since and , we conclude that V_ii,i = 0, if and only if,

4.5, 4.6

If , then we have V_11,1 = 0. Then suppose Since (4.6), we get

This last equation can be rewritten as

4.7

Differentiating with respect to u₁, we get

As W, and W, , then if , we get Substituting in (4.7) and using that W, , we obtain a contradiction. Therefore, we have . Thus

4.8

Substituting (4.8) in (4.5), we obtain

4.9

Thus

This last equation can be rewritten as

4.10

Differentiating with respect to u₂, we get

As W, and W, , then if , we get and

Hence , since (4.8). Thus , which is a

contradiction, since (4.10 and W = −cos u₂ f₁ − f₂ + B. Therefore, we have . Substituting in (4.10), we get , since W,_{1 6}= 0. Thus

4.11

Figure 4.1 Figure 4.1 On the surface above we have a type I Dupin surface relative to torus, with a = 4, r = 1, A = 3, B = −2, A2 = B2 = 1, B1 = −1 and A1 = B = −2.

Finally, considering the unit vector field normal to Y

and substituting f₁, f₂, Ω and W in X = Y + WΩ N we obtain (5.3).

Proposition 4.3. Consider the submanifold S² × Rⁿ⁻² in Rⁿ⁺¹, parametrized by

Then M is a type I Dupin hypersurface relative to Y , if and only if, M can be parametrized by

where

with C, Ai, B_i, C_j2, C_j1 and C_j0 are real constants.

Proof. The principal curvatures and coefficients of the metric of the of the S² × Rⁿ⁻² are

Using (3.10), we obtain

where C is constant and f_i are differentiable functions of u_i, 1 ≤ i ≤ n.

Consider V_ii given by (3.11). Thus

4.12, 4.13, 4.14

From Corollary 1, M parametrized by (3.8) is a type I Dupin hypersurface relative to S² × Rⁿ⁻² , if and only if, V_ii,i = 0 for all 1 ≤ i ≤ n.

Proceeding similarly to the proof of Proposition 2, we obtain that V_ii,i = 0, 1 ≤ i ≤ 2, if and only if, f₁ and f₂ are given by

4.15

Figure 4.2

On the surface above we have a type I Dupin surface relative to S², with B₂ = 0, B₁ = 2, A1 = −1 and C = 1.

In this section, we highlight the relative Dupin hypersurfaces of the type II. We start by providing a relationship between the Dupin hypersurfaces locally associated to a fixed Dupin hypersurface M by a Ribaucour transformation and the type II Dupin hypersurfaces relative M. We will generate families of type II Dupin hypersurfaces relative to a hyperplane, a torus, S¹ × Rⁿ⁻¹ and S² × Rⁿ⁻².

Let M be an orientable hypersurface in Rⁿ⁺¹, N its Gauss map and suppose that M has an orthogonal parameterization by lines of curvature Y : U ⊂ Rⁿ → M, with principal curvatures −λ_i, 1 ≤ i ≤ n. Let

M be a hypersurface in of the type II. Then M can be parametrized by

where h is a differentiable real function defined on M. From (1.6), the coefficients of the first and second fundamental forms of X are given by

where δ_ijL_ii =<Y,i , Y,j > and the normal relative N_R is given by (3.2). Moreover, since that X is a parameterization by lines of relative curvature, then the relative Weingarten matrix of is given by V_ij = 0, 1 ≤ i /= j ≤ i, and

5.1

where Ω and W satisfies (3.10).

Theorem 5.1. Let M be a Dupin hypersurface and suppose that it has a parameterization by lines of curvature Y : U ⊂ Rⁿ → M, with principal curvatures −λ_i, 1 ≤ i ≤ n. Consider the hypersurface locally associated to M by a Ribaucour transformation. Let M be a type II hypersurface relative to M, then M is a type II Dupin hypersurface relative to M, if and only if, Mf is a Dupin hypersurface.

Proof. From Corollary 1, M is a type II Dupin hypersurface relative to M if, and only if, V_ii,i = 0, where

with functions W and Ω satisfying (3.10).

On the other hand, from [9], Mf locally associated to M by a Ribaucour transformation, is a Dupin hypersurface, if and only if, T_i,i = 0, where

with functions W and Ω satisfying (3.10).

Since M is a Dupin hypersurface, we have (W + λ_iΩ),i = 0. Therefore, V_ii,i = 0, if and only if, T_i,i = 0.

In the next results, we provides families of type II Dupin hypersurfaces relative to a hyperplane, a torus, S¹ × Rⁿ⁻¹ and S² × Rⁿ⁻².

Proposition 5.1. Consider the hyperplane in the Euclidean space Rⁿ⁺¹, parametrized by Y (u₁,...,_un) = (u₁,...,u_n,0). Then M is a type II Dupin hypersurface relative to Rⁿ⁺¹, if and only if, M can be parametrized by

5.2

where f_i(u_i) = c_i2u² _i + c_i1ui + c_i0, 1 ≤ i ≤ n, and c /= 0, c_i2, c_i1, c_i0 ∈ R.

Proof. Since the principal curvatures of Y are λ_i = 0 and the metric L_ij = δij, for 1 ≤ i,j ≤ n, it follows from equation (3.10) that

where f_i(u_i) are differentiable functions. In order, to obtain type II Dupin hypersurface relative to Rⁿ⁺¹, we consider Vii given by (5.1),

From Corollary 1, M parametrized by , where e_n+1 = (0,0,...,0,1) is a unit vector field normal to Rⁿ, is a type II Dupin hypersurface relative to Rn, if and only if, V_ii,i = 0. Therefore, f_i(u_i) = c_i2u2i + c_i1u_i + c_i0, with c_i2, c_i1, c_i0 ∈ R and from (3.8), X is given by (5.2).

Remark 5.1. In Proposition 5, one observes that the type II Dupin hypersurface X relative to Rⁿ is an isotropic Dupin hypersurface.

Proposition 5.2. Consider the torus in R³, parametrized by

Then M is a type II Dupin hypersurface relative to Y , if and only if, M can be parametrized by

5.3

where B_i, A_i, A and B are real constants.

Proof. The principal curvatures of the torus and coefficients of the metric of the torus are

Using (3.10), we obtain

where A, B are constants and f₁, f₂ are differentiable functions of u₁ and u₂, respectively. Consider Vii given by (5.1). Thus

5.4, 5.5

From Corollary 1, M parametrized by (3.8) is a type II Dupin hypersurface relative to torus, if and only if, V_ii,i = 0 for all 1 ≤ i ≤ 2.

Since , we conclude from V_ii,i = 0 that the functions fi are given by f₁(u₁) =A₁ cosu₁+A₂sinu₁+A₃ , f₂ =B₁cosu₂+B₂sinu₂+B₃.

Finally, considering the unit vector field normal to Y

and substituting f₁, f₂, Ω and we obtain (5.3).

Figure 5.1 Figure 5.1 On the surfaces above we have a type II Dupin surface relative to torus with a=4, r=1, A=10, B= −3, A2= B2=0, B1=B3=1, A1=−1 and A3=−2

Proposition 5.3. Consider the submanifold S² × Rⁿ⁻² in Rⁿ⁺¹, parametrized by

Then M is a type II Dupin hypersurface relative to Y , if and only if, M can be parametrized by

where

Using (3.10), we obtain

where C is constant and fi are differentiable functions of u_i, 1 ≤ i ≤ n.

Consider V_ii given by (5.1). Thus

5.6, 5.7, 5.8

From Corollary 1, M parametrized by (3.8) is a type II Dupin hypersurface relative to S² × Rⁿ², if and only if, V_ii,i = 0 for all 1 ≤ i ≤ n.

Since and W_,r = 0, 3 ≤ r ≤ n we conclude from V_ii,i = 0 that the functions fi are given by

where A_i, B_i, C_i, C_j2, C_j1 and C_j0 are real constants.

Proposition 5.4. Consider the submanifold S¹ × Rⁿ⁻¹ in Rⁿ⁺¹, parametrized by

Then M is a type II Dupin hypersurface relative to Y , if and only if, M can be parametrized by

Where

Figure 5.2 Figure 5.2 On the surfaces above we have a type II Dupin surface relative to S2, with A2 = B2 = 0, B1 =2, A3 = 4, A1 = −1, B3 = −2 and C = 1.

and fj(uj) = C_j2u² _j +C_j1u_j + C_j0, 2 ≤ j ≤ n, with C, A₁, B₁, C₁, C_j2, C_j1 and Cj0 are real constants.

Proof: The principal curvatures and coefficients of the metric of the of the S¹ × Rⁿ⁻¹ are

Using (3.10), we obtain

where C is constant and f_i are differentiable functions of ui, 1 ≤ i ≤ n.

Consider V_ii given by (5.1). Thus

5.9, 5.10

From Corollary 1, M parametrized by (3.8) is a type II Dupin hypersurface relative to S¹ x Rⁿ⁻¹, if and only if, V_ii,i = 0 for all 1 ≤ i ≤ n.

Since and W_,r = 0, 2 ≤ r ≤ n we conclude from Vii,i = 0 that the functions fi are given by

where A₁, B₁, C₁, C_j2, C_j1 and C_j0 are real constants.

Figure 5.3 Figure 5.3 On the surface above we have a type II Dupin surface relative to cylinder S1 × R, with C21 =B1 = 0, C1 = 3, A1 = −2, C22 = −1, C20 = 2 and C = 1.

From the results obtained in this work we can make the following conclusions: For each fixed hypersurface M in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry.