K-isothermic hypersurfaces

We consider n dimensional hypersurfaces in the Euclidean space and introduce the k-isothermic hypersurfaces, with k < n, as hypersurfaces that locally admit orthogonal parametrization by curvature lines with k distinct coefficients of the first fundamental form. It easy to check that the transformations: isometries, dilations and invertions, preserve k-isothermic hypersurfaces. We prove that there are no k-isothermic hypersurface of dimension n with distinct principal curvatures for n ≥ k + 3. We introduced two ways to generate a (k + 1)-isothermic hypersurface from a k-isothermic hypersurfaces, which we will call 2-reducible. Moreover, we provide a local characterization of Dupin 2-isothermic hypersurfaces and include explicit examples of 2-irreducible Dupin 2-isothermic hypersurfaces.

In this paper, we define the n dimensional k-isothermic hypersurfaces in the eu-clidean space, with k < n, as hypersurfaces that locally admit orthogonal parametrization by curvature lines with k distinct coefficients of the first quadratic form. Such hypersurfaces are preserved by isometries, dilations and invertions. We prove that there are no n dimensional k-isothermic hypersurface with distinct principal curvatures for n ≥ k + 3. We introduced two ways to generate a (k + 1)-isothermic hypersurface from a k-isothermic hypersurface, which we will call 2-reducible. Moreover, we provide a local characterization of Dupin 2-isothermic hypersurface in R 4 and R 5 and include explicit examples of 2-isothermic hypersurface irreducible. The paper is organized as follows. In sect. 2, we define k-isothermic hypersurfaces in a local coordinate system and we give some properties of hypersurfaces parametrized by lines of curvature, with distinct principal curvatures, that will be used in the next section. In sect. 3, we show that there is no k-isothermic hypersurface in R n+1 parametrized by lines of curvature, if n ≥ k + 3. In sect. 4, we give a local characterization of Dupin hypersurfaces in R 4 and R 5 that are 2-isothermic. We finish by presenting examples of such a 2-isothermic Dupin hypersurfaces in R 5 which are irreducible and have nonconstant Lie curvature.

Properties of hypersurfaces with distinct principal curvatures
An immersion X : Ω ⊆ R n → R n+1 is a k-isothermic hypersurface if the first fundamental form is given by with α 1 = 0 and for 2 ≤ l ≤ k, α l = l−1 j=1 m j , where m i are called multiplicity of An hypersurface M is a k-isothermic hypersurface if locally can be parametrized by an immersion k-isothermic.
Let X : Ω ⊆ R n → R n+1 , X(x 1 , ..., x n ) be a hypersurface parametrized by lines of curvature, with distinct principal curvatures, −λ i , 1 ≤ i ≤ n, and let N : Ω ⊆ R n → R n+1 be a unit normal vector field of X. Then where <, > denotes the Euclidean metric on R n+1 .
where Γ i ij are the Christoffel symbols, given in terms of the metric (2) by where i, j, k are distinct. It follows from (5) that From (2) and (6), we get For a hypersurfaces with distinct principal curvatures, the Lie curvature is defined by Proposition 1. [13]. Let X : Ω ⊆ R n → R n+1 , be a hypersurface parametrized by lines of curvature, whose principal curvatures −λ i , 1 ≤ i ≤ n, are distinct. Then the Gauss equation for the immersion X is given by NEXUS Mathematicae, Goiânia, v. 3, 2020, e20004. where Using (5), we obtain immediately Proposition 2. Let X : Ω ⊆ R n → R n+1 , be a k-isothermic hypersurface parametrized by lines of curvature, with n distinct principal curvatures. Then the Christoffel symbols (5) satisfy for α l + 1 ≤ i, j ≤ α l + m l , α r + 1 ≤ a, b ≤ α r + m r and 1 ≤ s ≤ n, where α l , α r , m l and m r with 1 ≤ l = r ≤ k, are given by (1).
An immersion X : Ω ⊆ R n → R n+1 is a Dupin hypersurface if each principal curvature is constant along its corresponding line ( or surface ) of curvature. If the multiplicity of the principal curvatures is constant then the Dupin submanifold is said to be proper.

K-isothermic hypersurfaces
In this section, we show that there is no k-isothermic hypersurface parametrized by lines of curvature, if n ≥ k + 3. This is achieved by the following theorem.
Theorem 3. There is no M n ⊆ R n+1 k-isothermic hypersurface, with n distinct principal curvatures for n ≥ k + 3.
Proof: Suppose that M n ⊆ R n+1 be a k-isothermic hypersurface. Let X : Ω ⊆ R n → R n+1 be a parametrization by lines of curvature for M and denote by −λ i the distinct principal curvatures. As k < n, it follows from (1) that there is 1 < m i 0 < n, such that the first fundamental form of X, is given by We will prove that 1 < m i 0 ≤ 3. In fact, suppose m i 0 ≥ 4, then using (9) and (10), for Subtracting, respectively, in (11) the first and second equation, and the third and last equation, we obtain Therefore we obtained one contradiction, because all principal curvatures are distinct. Therefore 1 < m i 0 ≤ 3.
Now we will prove that for j 0 = i 0 , m j 0 = 1. Suppose m j 0 > 1, then rewritten the first fundamental form of X as α i +m i and using (9) and (10), for we have Again, subtracting the two last equations and two first equations, using (10) and expression of L rs given by (9), we obtain (λ i −λ j )(λ l −λ k ) = 0, which is a contradiction since all principal curvatures are pairwise distinct. Therefore m j 0 = 1.
Therefore, for the existence of M, we must have m i 0 = 2 or m i 0 = 3 and for all Thus If m i 0 = 2, then for all j = i 0 , m j = 1 and so n = 2 + k − 1 = 1 + k.
If m i 0 = 3, then for all j = i 0 , m j = 1 and so n = 3 + k − 1 = 2 + k. Thus, for M k-isothermic to exist, we must have n = k + 1 or n = k + 2. Therefore, if n ≥ k + 3, then there is no M k-isothermic.
Let M n ⊆ R n+1 be a k-isothermic hypersurface, with coefficients L i , 1 ≤ i ≤ k not constant. It is then easy to check that the following constructions lead to a (k+1)-isothermic hypersurface M in R n+2 : (2) Project M stereographically onto a hypersurface M ⊆ S n+1 . Let M be the cone R. M over M . We define a 2-reducible k-isothermic hypersurface as a k-isothermic hypersurface obtained by one of the two above constructions. More generally, all locally k-isothermic hypersurfaces obtained by isometries, dilations and inversions in a hypersurface will also be called 2-reducible. A hypersurface is a 2-irreducible k-isothermic hypersur-face if it is not 2-reducible.
Remark 4. We observed that in [18] the author presents two more constructions that preserve Dupin hypersurfaces.
(3) Take an n dimensional linear subspace R n ⊆ R n+1 and consider the rotations φ t of R n+2 that leave R n pointwise fixed. Let M be the hypersurface of R n+2 generated by M under the rotations φ t .
A hypersurface is said to be reducible if it is obtained by one of the four above constructions, and every Dupin hypersurface that is locally Lie equivalent to such a hypersurface. The third and fourth construction does not generate a (k + 1)isothermic hypersurface.

Class of 2-isothermic Dupin hypersurfaces parametrized by lines of curvature
In the previous section, we show that there is no k-isothermic hypersurface parametrized by lines of curvature, if n ≥ k + 3. Using the 2-reducible definition we can construction examples of 2-isothermic Dupin hypersurfaces.
Example 5. Let M 2 ⊆ R 3 be a Dupin isothermic surface, with two non zero distinct principal curvatures. Then the cylinder M 2 × R ⊆ R 4 is a 2-isothermic Dupin hypersurface with three distinct principal curvatures, two of them given by the principal curvatures of M 2 and the third one is given by the null function.
We characterize the Dupin hypersurfaces that are 2-isothermic in R 4 and R 5 . Using the Theorem 3, where for simplicity of notation we make i 0 = 1 we have two cases, which without loss of generality are: Case 1: m 1 = 2 i.e, the Dupin hypersurface that is 2-isothermic in R 4 has the first fundamental form Case 2: m 1 = 3 i.e, the Dupin hypersurface that is 2-isothermic in R 5 has the first NEXUS Mathematicae, Goiânia, v. 3, 2020, e20004.
fundamental form We will show that there exist such hypersurfaces by presenting examples.
The following lemma provides a characterization of the principal curvatures of a Dupin hypersurface from Case 1.

Lemma 6.
Let M 3 ⊆ R 4 , be a proper Dupin hypersurface that is 2-isothermic of the Case 1 type and that has three distinct principal curvatures. Then the principal curvatures are given by Moreover, there is a change in each coordinate separately such that the coefficients L 1 and L 2 of the first fundamental form (13) are given by Where f i and h j are, respectively, differentiable functions of x i and x j , Proof: We will first show that each λ i is given by (15). Using (10), we have In this equation, using (4), we obtain Moreover, differentiating λ 3 −λ 2 λ 3 −λ 1 , with respect to x 1 and x 2 , we get Using (17) and (18), we obtain where f 1 and f 2 are, respectively, differentiable functions of x 1 and x 2 . From (19), we have Differentiating this equation with respect to x 3 , we get Now, differentiating (21) with respect to x 2 and x 1 , respectively, we obtain Therefore integrating this two equations and using (21) and (19), we obtain (15). We show (16) by comparing (4) and (5), we get and therefore where f 3 is an arbitrary function of the variable x 3 and f 23 is an arbitrary function NEXUS Mathematicae, Goiânia, v. 3, 2020, e20004.
of the variables x 2 and x 3 .
The following lemma provides a characterization of the principal curvatures of a Dupin hypersurface from case 2.
Lemma 7. Let M 4 ⊆ R 5 , be a proper Dupin hypersurface that is 2-isothermic of the Case 2 type and that has four distinct principal curvatures. Then the principal curvatures are given by Moreover, there is a change in each coordinate separately such that the coefficients L 1 and L 2 of the first fundamental form (14) are given by Where f i and h j are, respectively, differentiable functions of x i and x j , 1 ≤ i ≤ 3, 1 ≤ j ≤ 4, and c = 1 is a constant.
Theorem 8. Let M r ⊆ R r+1 , 3 ≤ r ≤ 4, be a 2-isothermic proper Dupin hypersurface that is parametrized by lines of curvature and has r distinct principal curvatures −λ i given by (15) if r = 3, and by (26) if r = 4. Then M can be parametrized by where G i and H j 1 ≤ i ≤ 3, 1 ≤ j ≤ 4 are, respectively, vector valued functions of R 4 and R 5 .
Proof: We will show case r = 4. The case r = 3 is similar.
According to Pinkall [18], a proper Dupin submanifold M n−1 ⊆ Λ 2n−1 is reducible if, and only if, it has a curvature sphere map K that lies in a linear subspace of codimension two in P n+2 . The curvature sphere maps of this example are given by at the point (0, 0, 0, 0) is nonzero. Hence, for any i, the curvature sphere map K i does not lie in a linear subspace of codimension two in P 7 . Therefore, X is an irreducible Dupin hypersurface.
Remark 10. The futures works will be in the directions of obtaining examples of 2-irredutible hypersurfaces that are not necessarily Dupin hypersurfaces.