Pseudo-parallel surfaces of $\mathbb {S}_c^n \times \mathbb {R}$ and $\mathbb {H}_c^n \times \mathbb {R}$

Abstract

In this work we give a characterization of pseudo-parallel surfaces in $\mathbb {S}_c^n \times \mathbb {R}$ and $\mathbb {H}_c^n\times \mathbb {R}$, extending an analogous result by Asperti-Lobos-Mercuri for the pseudo-parallel case in space forms. Moreover, when $n=3$, we prove that any pseudo-parallel surface has flat normal bundle. We also give examples of pseudo-parallel surfaces which are neither semi-parallel nor pseudo-parallel surfaces in a slice. Finally, when $n\ge 4$ we give examples of pseudo-parallel surfaces with non vanishing normal curvature.

1 Introduction

In the theory of submanifolds of a space form, Asperti-Lobos-Mercuri introduced in Asperti et al. (1999) pseudo-parallel submanifolds as a direct generalization of semi-parallel submanifolds in the sense of Deprez (1985), which in turn, are a generalization of parallel submanifolds (extrinsically symmetric in Ferus’ terminology) Ferus (1980) (in particular, of umbilical and totally geodesic submanifolds), and as extrinsic analogues of pseudo-symmetric spaces in the sense of Deszcz (1992). They studied pseudo-parallel surfaces of a space form in Asperti et al. (2002), Lobos (2002), and proved that they are surfaces with flat normal bundle or $\lambda$-isotropic surfaces in the sense of O’Neill (1965) (i.e. surfaces whose ellipse of curvature in any point is a circle). In particular, they proved that pseudo-parallel surfaces of space forms with non vanishing normal curvature in codimension 2 are superminimal surfaces in the sense of Bryant (1982) (i.e. surfaces which are minimal and $\lambda$-isotropic).

An isometric immersion $f:M^m \rightarrow \tilde{M}^n$ is said to be pseudo-parallel if its second fundamental form $\alpha$ satisfies the following condition:

$$\begin{aligned} \tilde{R}(X,Y)\cdot \alpha = \phi (X \wedge Y)\cdot \alpha , \end{aligned}$$

for some smooth real-valued function $\phi$ on $M^m$, where $\tilde{R}$ is the curvature tensor corresponding to the Van der Waerden-Bortolotti connection $\tilde{\nabla }$ of f and $X\wedge Y$ denotes the endomorphism defined by

$$\begin{aligned} (X\wedge Y) Z =\langle Y,Z\rangle X -\langle X,Z\rangle Y. \end{aligned}$$

Considering the product space $\mathbb {Q}^n_c \times \mathbb {R}$ as the ambient space, the first studies of pseudo-parallel submanifolds were started in Lin and Yang (2014) and Lobos and Tassi (2019), where a classification of its hypersurfaces was given, generalizing the classification of parallel and semi-parallel hypersurfaces in Calvaruso et al. (2010) and Van der Veken and Vrancken (2008).

In this work we started the study of pseudo-parallel surfaces in $\mathbb {Q}_{c}^n \times \mathbb {R}$ (with $c\ne 0$). We begin by observing that any isometric immersion $f: M^2 \rightarrow \mathbb {Q}^n_c \times \mathbb {R}$ with flat normal bundle is pseudo-parallel (see Proposition 2.2). So, we state the main result of this work:

Theorem 1.1

Let $f:M^2 \rightarrow \mathbb {Q}_{c}^n \times \mathbb {R}$ be a pseudo-parallel surface which does not have flat normal bundle on any open subset of $M^2$. Then $n \ge 4$, f is $\lambda$-isotropic and

$$\begin{aligned}&\displaystyle K > \phi , \end{aligned}$$

(1.1)

$$\begin{aligned}&\displaystyle \lambda ^2 = 4K - 3\phi + c(\Vert T \Vert ^2 - 1) > 0, \end{aligned}$$

(1.2)

$$\begin{aligned}&\displaystyle \Vert H \Vert ^2 = 3K - 2\phi + c(\Vert T \Vert ^2 - 1) \ge 0, \end{aligned}$$

(1.3)

where K is the Gaussian curvature, $\lambda$ is a smooth real-valued function on $M^2$, H is the mean curvature vector field of f and T is the tangent part of $\frac{\partial }{\partial t}$, the canonical unit vector field tangent to the second factor of $\mathbb {Q}_c^n \times \mathbb {R}$.

Conversely, if f is $\lambda$-isotropic then f is pseudo-parallel.

We remark that Theorem 1.1 extends for $\mathbb {Q}^n_c\times \mathbb {R}$ a similar result for pseudo-parallel surfaces into space forms given by Asperti-Lobos-Mercuri in Asperti et al. (2002).

However, the class of pseudo-parallel surfaces in $\mathbb {Q}_c^3\times \mathbb {R}$ is not empty. In the last section we give examples of semi-parallel surfaces which are not parallel as well as examples of pseudo-parallel surfaces in $\mathbb {S}_c^3\times \mathbb {R}$ and $\mathbb {H}_c^3\times \mathbb {R}$ which are neither semi-parallel nor pseudo-parallel surfaces in a slice.

Finally, we remark that pseudo-parallel surfaces in $\mathbb {Q}^n_c \times \mathbb {R}$ with $n\ge 4$ and non vanishing normal curvature do exist, as shown in Examples 4.3, 4.5 and 4.6 in the last section.

2 Preliminaries

First of all, we establish the notation that we use along this work. Let $f: M^m \rightarrow \tilde{M}^n$ be an isometric immersion. We decompose the tangent bundle $T\tilde{M}$ of $\tilde{M}^n$ in its tangent and normal parts, as a sum $T\tilde{M} = TM \oplus N_fM$, where TM and $N_fM$ are the tangent bundle of $M^m$ and the normal bundle of f, respectively. Using this notation we consider $\tilde{\nabla } =\nabla \oplus \nabla ^\perp$ the Van der Waerden-Bortolotti connection of f and $\tilde{R} = R \oplus R^\perp$ its curvature tensor. The second fundamental form of f is the symmetric 2-tensor denoted by $\alpha :TM \times TM \rightarrow N_fM$. For any $\xi \in N_fM$ the correspondent Weingarten operator in the $\xi$-direction is denoted by $A_{\xi }$, that is,

$$\begin{aligned} \langle \alpha (X,Y), \xi \rangle = \langle A_\xi X,Y \rangle , \quad \text{ for } \text{ all } \quad X,Y \in TM, \quad \text{ and } \quad \xi \in N_fM. \end{aligned}$$

The mean curvature vector field of f is denoted by H. Finally, we say that f has flat normal bundle (or vanishing normal curvature) if $R^\perp = 0$.

An isometric immersion $f:M^m \rightarrow \tilde{M}^n$ is said to be:

• 1. Totally geodesic if

  $$\begin{aligned} \alpha (X,Y) = 0; \end{aligned}$$

  (2.1)

• 2. Umbilical if the mean curvature vector field H of f satisfies

  $$\begin{aligned} \alpha (X,Y) = \langle X,Y \rangle H; \end{aligned}$$

  (2.2)

• 3. Locally parallel if

  $$\begin{aligned} (\tilde{\nabla }_X\alpha )(Y,Z) = 0; \end{aligned}$$

  (2.3)

• 4. Semi-parallel if

  $$\begin{aligned} (\tilde{R}(X,Y)\cdot \alpha )(Z,W) = 0; \end{aligned}$$

  (2.4)

• 5. Pseudo-parallel if

  $$\begin{aligned} (\tilde{R}(X,Y)\cdot \alpha )(Z,W) = \phi [(X \wedge Y)\cdot \alpha ](Z,W), \end{aligned}$$

  (2.5)

  for some smooth real-valued function $\phi$ on $M^m$ and for any vector X,Y,Z and W tangents to M.

Here the notation means

$$\begin{aligned} (\tilde{\nabla }_X\alpha )(Y,Z)&= \nabla ^\perp _X\alpha (Y,Z) - \alpha (\nabla _XY,Z) - \alpha (Y,\nabla _XZ),\\ (\tilde{R}(X,Y)\cdot \alpha )(Z,W)&= R^\perp (X,Y)[\alpha (Z,W)] - \alpha (R(X,Y)Z,W) \\&\quad - \alpha (Z,R(X,Y)W),\\ [(X \wedge Y)\cdot \alpha ](Z,W)&= - \alpha ((X \wedge Y)Z,W) - \alpha (Z,(X \wedge Y)W). \end{aligned}$$

A space form $\mathbb {Q}^n_c$ is a simply connected, complete, n-dimensional Riemannian manifold with constant sectional curvature c. Namely, $\mathbb {Q}^n_c$ is the n-dimensional sphere $\mathbb {S}^n_c$ or the n-dimensional hyperbolic space $\mathbb {H}^n_c$, respectively given by

$$\begin{aligned}&\mathbb {S}^n_c = \left\{ (x_1,x_2, \dots , x_{n+1}) \in \mathbb {R}^{n+1}; \sum _{i=1}^{n+1}x_i^2 = \frac{1}{c}\right\} , \quad \text{ if } c> 0,\\&\mathbb {H}^n_c = \left\{ (x_1,x_2, \dots , x_{n+1}) \in \mathbb {L}^{n+1}; -x_1^2 + \sum _{i=2}^{n+1}x_i^2 = \frac{1}{c}, x_1 > 0\right\} , \quad \text{ if } c < 0, \end{aligned}$$

where $\mathbb {L}^{n+1}$ is the $(n+1)$-dimensional Minkowski space, that is, the $(n+1)$-dimensional euclidean space $\mathbb {R}^{n+1}$ endowed with the inner product

$$\begin{aligned} \langle (x_1,x_2,\ldots ,x_{n+1}),(y_1,y_2,\ldots ,y_{n+1})\rangle =-x_1y_1 + \sum _{i=2}^{n+1}x_iy_i. \end{aligned}$$

This work is devoted to the study of these classes of surfaces with $\mathbb {Q}_c^n \times \mathbb {R}$ as the ambient space and we always assume $c \ne 0$. Thus, let $\frac{\partial }{\partial t}$ be the canonical unit vector field tangent to the second factor of $\mathbb {Q}_c^n \times \mathbb {R}$. For a given isometric immersion $f:M^2 \rightarrow \mathbb {Q}_c^n \times \mathbb {R}$, it is convenient to consider the following decomposition of $\frac{\partial }{\partial t}$ in its tangent and normal parts:

$$\begin{aligned} \dfrac{\partial }{\partial t}= f_{*}T + \eta , \end{aligned}$$

(2.6)

for some $T \in TM$ and some $\eta \in N_fM$.

Another tools we make use are the Fundamental Equations for a surface $f:M^2 \rightarrow \mathbb {Q}_{c}^n \times \mathbb {R}$ and now we recall them. Let $\{e_1,e_2\}$ be an orthonormal local frame for $M^2$ and set $\alpha _{ij} = \alpha (e_i,e_j)$. By $\delta _{ij}$ we mean the Kronecker’s Delta. From Mendonça and Tojeiro (2013) we have the following equations:

Gauss:

$$\begin{aligned} R(e_1,e_2)e_k&= c(\delta _{2k}e_1 - \delta _{1k}e_2 - \langle e_2,T \rangle \langle e_k,T \rangle e_1 + \delta _{1k}\langle e_2,T \rangle T \nonumber \\&\quad - \delta _{2k}\langle e_1,T \rangle T + \langle e_1,T \rangle \langle e_k,T \rangle e_2) + A_{\alpha _{2k}}e_1 - A_{\alpha _{1k}}e_2. \end{aligned}$$

(2.7)

Codazzi:

$$\begin{aligned} (\tilde{\nabla }_{e_1} \alpha )(e_2,e_k) - (\tilde{\nabla }_{e_2} \alpha )(e_1,e_k) = c(\delta _{1k} \langle e_2,T \rangle - \delta _{2k} \langle e_1,T \rangle ) \eta \end{aligned}$$

(2.8)

Ricci:

$$\begin{aligned} R^{\perp }(e_1,e_2)\xi = \alpha (e_1, A_{\xi }e_2) - \alpha (A_{\xi }e_1, e_2). \end{aligned}$$

(2.9)

It follows from the Ricci equation that

$$ R^{\perp }(e_1,e_2)\xi \in \text{ span }\{\alpha (X,Y); X,Y \in TM\}, \quad \text{ for } \text{ all } \quad \xi \in N_fM(x). $$

Thus, the equation (2.9) is equivalent to the following equation:

$$\begin{aligned} R^\perp (e_1,e_2)\alpha _{ij}&= \langle \alpha _{12},\alpha _{ij} \rangle (\alpha _{11}-\alpha _{22}) + \langle \alpha _{22} - \alpha _{11},\alpha _{ij} \rangle \alpha _{12}. \end{aligned}$$

(2.10)

On the other hand, the pseudo-parallelism condition is equivalent to the following two equations:

$$\begin{aligned} R^\perp (e_1,e_2)\alpha _{ii}&= (-1)^i2(K-\phi )\alpha _{12}, \end{aligned}$$

(2.11)

$$\begin{aligned} R^\perp (e_1,e_2)\alpha _{12}&= (K - \phi )(\alpha _{11} - \alpha _{22}), \end{aligned}$$

(2.12)

where

$$\begin{aligned} K = c(1 - \Vert T \Vert ^2) + \langle \alpha _{11}, \alpha _{22}\rangle - \Vert \alpha _{12} \Vert ^2 \end{aligned}$$

(2.13)

is the Gaussian curvature of $M^2$. As a consequence, we have the next lemma.

Lemma 2.1

Let $f:M^2 \rightarrow \mathbb {Q}_{c}^n \times \mathbb {R}$ be a pseudo-parallel surface. Then $R^{\perp }(X,Y)H = 0$, for all $X,Y \in TM$.

Proof

Immediate by equation (2.11), since $H = \frac{1}{2}(\alpha _{11} + \alpha _{22})$. $\square$

Proposition 2.2

Let $f:M^2 \rightarrow \mathbb {Q}_{c}^n \times \mathbb {R}$ be a surface with flat normal bundle. Then f is a pseudo-parallel immersion.

Proof

Since f has flat normal bundle, by equations (2.11) and (2.12) we conclude that f is $\phi$-pseudo-parallel by taking $\phi = K$, where K is the Gaussian curvature of $M^2$. $\square$

In the following, we have two propositions that is useful to construct examples of pseudo-parallel surfaces.

Proposition 2.3

Let $f: M^m \rightarrow \mathbb {Q}^n_c$ be an isometric immersion and let $j:\mathbb {Q}^n_c \rightarrow \mathbb {Q}^{n}_c\times \mathbb {R}$ be a totally geodesic immersion. If f is $\phi$-pseudo-parallel, then $j \circ f$ is $\phi$-pseudo-parallel.

Proof

In this proof, we denote the second fundamental form of f and $j \circ f$ respectively by $\alpha ^f$ and $\alpha ^{j \circ f}$. In the same way, we denote the normal curvature tensors of f and $j \circ f$ respectively by $R^{\perp }_f$ and $R^{\perp }_{j \circ f}$. Since j is a totally geodesic immersion, we have the following relations:

$$\begin{aligned} \alpha ^{j \circ f}(Z,W)&= j_*\alpha ^f(Z,W),\\ R_{j \circ f}^\perp (X,Y)\alpha ^{j \circ f}(Z,W)&= j_*R_f^\perp (X,Y)\alpha ^f(Z,W), \end{aligned}$$

Therefore, applying Definition 2.5 we obtain

$$\begin{aligned} (\tilde{R}(X,Y)\cdot \alpha ^{j \circ f})(Z,W)&= R_{j \circ f}^\perp (X,Y)\alpha ^{j \circ f}(Z,W) - \alpha ^{j \circ f}(R(X,Y)Z,W) \\&\quad - \alpha ^{j \circ f}(Z,R(X,Y)W)\\&= j_*R_f^\perp (X,Y)\alpha ^f(Z,W) - j_*\alpha ^f(R(X,Y)Z,W)\\&\quad - j_*\alpha ^f(Z,R(X,Y)W)\\&= \phi \{- j_*\alpha ^f((X \wedge Y)Z,W) - j_*\alpha ^f(Z,(X \wedge Y)W)\}\\&= \phi \{-\alpha ^{j \circ f}((X \wedge Y)Z,W) - \alpha ^{j \circ f}(Z,(X \wedge Y)W)\} \\&= \phi [(X \wedge Y) \cdot \alpha ^{j \circ f}](Z,W). \end{aligned}$$

$\square$

Proposition 2.4

Let $f: M^m \rightarrow \mathbb {Q}_{c}^n \times \mathbb {R}$ be an isometric immersion and let $j:\mathbb {Q}_{c}^n \times \mathbb {R}\rightarrow \mathbb {Q}^{n+l}_c\times \mathbb {R}$ be a totally geodesic immersion. If f is $\phi$-pseudo-parallel, then $j \circ f$ is $\phi$-pseudo-parallel.

Proof

Is analogous to that of Proposition 2.3. $\square$

3 Proof of the main theorem

Before we give a proof of Theorem 1.1 we recall that $f: M^2 \rightarrow \mathbb {Q}_{c}^n \times \mathbb {R}$ is a $\lambda$-isotropic surface if, for each $x \in M$, the ellipse of curvature $\{\alpha (X,X) \in N_fM(x); X \in T_xM \quad \text{ with } \quad \Vert X \Vert = 1\}$ is a sphere with radius $\lambda (x)$, where $\lambda : M^2 \rightarrow \mathbb {R}$ is a smooth function. The following result, due to Sakaki in Sakaki (2015) plays a vital role in the proof of Theorem 1.1. Its statement is:

Theorem 3.1

(see Sakaki (2015)) Let $f:M^2 \rightarrow \mathbb {Q}_c^3 \times \mathbb {R}$ be a minimal surface with $c \ne 0$. If f is $\lambda$-isotropic, then f is totally geodesic.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1

Let us suppose that $f: M^2 \rightarrow \mathbb {Q}^n_c \times \mathbb {R}$ is pseudo-parallel with non vanishing normal curvature. Combining equations (2.10) to (2.13) we get

$$\begin{aligned} \langle \alpha _{12},\alpha _{ii} \rangle (\alpha _{11} - \alpha _{22}) + \{2(-1)^{i+1}(K - \phi ) + \langle \alpha _{ii},\alpha _{22} - \alpha _{11} \rangle \}\alpha _{12} = 0, \end{aligned}$$

(3.1)

$$\begin{aligned} \{\Vert \alpha _{12} \Vert ^2 + (\phi - K)\}(\alpha _{11} - \alpha _{22}) + \langle \alpha _{22} - \alpha _{11}, \alpha _{12} \rangle \alpha _{12} = 0. \end{aligned}$$

(3.2)

Next, we prove that $\{\alpha _{12}, \alpha _{11} - \alpha _{22}\}$ is linearly independent. We can suppose $\phi \ne K$. Otherwise, since $\langle R^{\perp }(e_1,e_2)\xi ,\zeta \rangle = - \langle R^{\perp }(e_1,e_2)\zeta ,\xi \rangle$, by the equations (2.11) and (2.12) we would have $R^{\perp } = 0$, which is a contradiction. Notice that $\alpha _{12} \ne 0$ and $\alpha _{11} \ne \alpha _{22}$. In fact, if $\alpha _{12} = 0$ then $R^\perp (e_1,e_2) \alpha _{12} = 0$ which implies by equation (2.12) that $\alpha _{11} = \alpha _{22}$, and in this case f is umbilical and has flat normal bundle, a contradiction. If $\alpha _{11} - \alpha _{22} = 0$, then $R^\perp (e_1,e_2) (\alpha _{11} - \alpha _{22}) = 0$, which implies by equation (2.11) that $\alpha _{12} = 0$.

Assume that there exist $\lambda , \mu \in \mathbb {R}$ such that $\lambda \alpha _{12} + \mu (\alpha _{11} - \alpha _{22}) = 0$. Then, by equations (2.11) and (2.12) we get $\lambda (\alpha _{11} - \alpha _{22}) - 4\mu \alpha _{12}= 0$. If $\mu \ne 0$ then $(\alpha _{11} - \alpha _{22}) = \frac{-\lambda }{\mu }\alpha _{12}$ and thus $\left( \frac{-\lambda ^2}{\mu } - 4\mu \right) \alpha _{12} = 0$, which lead us to $\lambda ^2 = -4\mu ^2 < 0$, a contradiction. So $\mu = 0$, and therefore $\lambda = 0$.

Using this and equations (3.1) and (3.2) we obtain

$$\begin{aligned} \langle \alpha _{12}, \alpha _{11}\rangle&= \langle \alpha _{12}, \alpha _{22}\rangle = 0, \end{aligned}$$

(3.3)

$$\begin{aligned} \langle \alpha _{22} - \alpha _{11}, \alpha _{ii}\rangle&= (-1)^i2(K - \phi ), \end{aligned}$$

(3.4)

$$\begin{aligned} \Vert \alpha _{12} \Vert ^2&= K - \phi > 0. \end{aligned}$$

(3.5)

From the equation (2.13) we get

$$\begin{aligned} \langle \alpha _{11}, \alpha _{22}\rangle&= 2K - \phi + c(\Vert T \Vert ^2 - 1), \end{aligned}$$

(3.6)

$$\begin{aligned} \Vert \alpha _{11} \Vert ^2 = \Vert \alpha _{22} \Vert ^2&= 4K - 3\phi + c(\Vert T \Vert ^2 - 1) > 0, \end{aligned}$$

(3.7)

$$\begin{aligned} \Vert \alpha _{11} - \alpha _{22} \Vert ^2&= 4(K - \phi ) > 0, \end{aligned}$$

(3.8)

$$\begin{aligned} \Vert H \Vert ^2&= 3K - 2\phi + c(\Vert T \Vert ^2 - 1). \end{aligned}$$

(3.9)

In particular, f is $\lambda$-isotropic with $\lambda ^2 = 4K - 3\phi + c(\Vert T \Vert ^2 - 1)$.

Now, we prove that $n \ge 4$. Suppose that $n=3$. Since f has non flat normal bundle, for any $x \in M^2$ we have that $R^{\perp }(x)(e_1,e_2): N_fM(x) \rightarrow N_fM(x)$ is a non zero antisymmetric linear operator, defined in a two-dimensional vector space. Thus, by Lemma 2.1 we conclude that $H(x) = 0$. But from Theorem 3.1, we conclude that f is totally geodesic and in particular, $R^{\perp }(x) = 0$, which is a contradiction.

Conversely, let us assume that f is $\lambda$-isotropic. Set $X = \cos \theta e_1 + \sin \theta e_2$. Then

$$\begin{aligned} \lambda ^2&= \Vert \alpha (X,X) \Vert ^2\\&= (\cos ^4\theta + \sin ^4\theta ) \lambda ^2 + 2\sin ^2 \theta \cos ^2 \theta \langle \alpha _{11}, \alpha _{22} \rangle \\&\quad + 4\sin ^3\theta \cos \theta \langle \alpha _{22}, \alpha _{12}\rangle + 4\sin \theta \cos ^3\theta \langle \alpha _{11}, \alpha _{12} \rangle \\&\quad + 4 \sin ^2\theta \cos ^2\theta \Vert \alpha _{12} \Vert ^2. \end{aligned}$$

Since $\lambda$ does not depend on $\theta$, taking the derivative with respect to $\theta$ we get

$$\begin{aligned} 0&= \left. \dfrac{d \lambda ^2}{d\theta }\right| _{\theta = 0} = \dfrac{d}{d\theta }(\Vert \alpha (X,X)\Vert ^2)\vert _{\theta = 0} = 4\langle \alpha _{11}, \alpha _{12}\rangle ,\\ 0&= \left. \dfrac{d \lambda ^2}{d\theta }\right| _{\theta = \frac{\pi }{2}} = \dfrac{d}{d\theta }(\Vert \alpha (X,X)\Vert ^2)\vert _{\theta = \frac{\pi }{2}} = -4\langle \alpha _{22}, \alpha _{12}\rangle . \end{aligned}$$

On the other hand, with $Y = \dfrac{1}{\sqrt{2}}(e_1 + e_2)$ we get

$$\begin{aligned} \lambda ^2&= \Vert \alpha (Y,Y) \Vert ^2 \\&= \frac{1}{4} \{2\lambda ^2 + 4\Vert \alpha _{12} \Vert ^2 + 2\langle \alpha _{11},\alpha _{22}\rangle \}, \end{aligned}$$

that is,

$$ \lambda ^2 = 2\Vert \alpha _{12} \Vert ^2 + \langle \alpha _{11},\alpha _{22}\rangle . $$

Using this and the Gauss equation we get

$$\Vert \alpha _{12} \Vert ^2 =\frac{1}{3}\{ \lambda ^2 - K + c(1 - \Vert T \Vert ^2) \}.$$

From the Ricci equation $R^{\perp }(e_1,e_2)\alpha _{ii} = \langle \alpha _{22} - \alpha _{11}, \alpha _{ii} \rangle \alpha _{12}, i = 1,2$, we obtain

$$\begin{aligned} R^{\perp }(e_1,e_2)\alpha _{ii}&= \langle \alpha _{22} - \alpha _{11}, \alpha _{ii} \rangle \alpha _{12} \\&= (-1)^i2\Vert \alpha _{12}\Vert ^2\alpha _{12} \\&= (-1)^i\frac{2}{3}\{\lambda ^2 - K + c(1 - \Vert T \Vert ^2)\}\alpha _{12}, \end{aligned}$$

Using the Ricci equation once more we obtain

$$\begin{aligned} R^{\perp }(e_1,e_2)\alpha _{12}&= \Vert \alpha _{12} \Vert ^2 (\alpha _{11} - \alpha _{22}) \\&= \frac{1}{3}\{\lambda ^2 - K + c(1 - \Vert T \Vert ^2)\}(\alpha _{11} - \alpha _{22}). \end{aligned}$$

Therefore, taking $\phi = \frac{4K - \lambda ^2 + c(\Vert T \Vert ^2 - 1)}{3}$, we conclude that f is pseudo-parallel according to equations (2.11) and (2.12). $\square$

4 Some Examples

We now introduce the first examples of semi-parallel and pseudo-parallel surfaces of $\mathbb {Q}^3_c \times \mathbb {R}$ which are not locally parallel and semi-parallel, respectively, and that are not just inclusions of surfaces of $\mathbb {Q}^3_c$ into $\mathbb {Q}^3_c \times \mathbb {R}$.

Example 4.1

A general construction of submanifolds of $\mathbb {Q}^n_c \times \mathbb {R}$ with flat normal bundle and T as a principal direction can be found in Mendonça and Tojeiro (2014), by Mendonça-Tojeiro. For our purpose, based on this work, the construction becomes: let $g: J \rightarrow \mathbb {Q}^3_{c}$ be a regular curve and $\{\xi _1, \xi _2\}$ an orthonormal set of vector fields normal to g. Put

$$\begin{aligned}&\tilde{g} = i \circ j \circ g, \\&\tilde{\xi } _k = i_*j_*\xi _k, \quad \text{ for } \quad k \in \{1, 2\}, \\&\tilde{\xi }_0 = \tilde{g}, \quad \tilde{\xi }_3 = i_*\frac{\partial }{\partial t}, \end{aligned}$$

where $j: \mathbb {Q}^3_{c} \rightarrow \mathbb {Q}^3_{c} \times \mathbb {R}$ and $i: \mathbb {Q}^3_{c} \times \mathbb {R}\rightarrow \mathbb {E}^{5}$ are the canonical inclusions. If $\alpha = (\alpha _0, \alpha _1, \alpha _2, \alpha _3): I \rightarrow \mathbb {Q}^{2}_{c} \times \mathbb {R}$ is a smooth regular curve with $\alpha ' _3(s) \ne 0, \quad \forall s \in I$, we have the following isometric immersion $f: M^2 = J \times I \rightarrow \mathbb {Q}^3_c \times \mathbb {R}$ given by

$$\begin{aligned} \tilde{f}(x,s) = (i \circ f)(x,s) = \sum _{k=0}^{3}\alpha _k(s)\tilde{\xi }_k(x). \end{aligned}$$

(4.1)

At regular points, f is an isometric immersion with flat normal bundle and T as a principal direction. Conversely, if $f: M^2 \rightarrow \mathbb {Q}^3_c \times \mathbb {R}$ is an isometric immersion with flat normal bundle and T as a principal direction, then f is given by (4.1) for some isometric immersion $g: \mathbb {Q}^3_c \times \mathbb {R}$ and some smooth regular curve $\alpha : I \rightarrow \mathbb {Q}_c^{2} \times \mathbb {R}$ whose its last coordinate has non vanishing derivative.

Geometrically, f is obtained by parallel transporting a curve in a product submanifold $\mathbb {Q}^{2}_{c} \times \mathbb {R}$ of a fixed normal space of $\tilde{g}$ with respect to its normal connection.

In particular, when dealing with pseudo-parallel surfaces in $\mathbb {Q}^3_c \times \mathbb {R}$, at least those that have T as a principal direction are fully described by this method.

We now construct two simple examples. Let us define

$$ C_{c}(s) = \left\{ \begin{array}{lll} \cos (s),\quad \text{ if }\quad c> 0\\ \cosh (s),\quad \text{ if }\quad c< 0 \end{array} \right. \quad \text{ and } \quad S_{c}(s) = \left\{ \begin{array}{lll} \sin (s),\quad \text{ if }\quad c > 0\\ \sinh (s),\quad \text{ if }\quad c < 0. \end{array} \right. $$

By taking

$$\begin{aligned}&\tilde{g}(x) = (C_{c}(\theta (x)), S_{c}(\theta (x)), 0, 0, 0),\\&\tilde{\xi }_1(x) = (0, 0, 1, 0, 0), \quad \tilde{\xi }_2(x) = (0, 0, 0, 1, 0), \\&\alpha _0(s) = \sqrt{1 - {{\mathrm{\mathrm {sgn}}}}(c) d^2}, \quad \alpha _1(s) = d \cos s, \quad \alpha _2(s) = d \sin s, \quad \alpha _3(s) = s. \end{aligned}$$

where $0< d < 1$, if $c > 0$, or $d > 0$, if $c < 0$, and $\theta : \mathbb {R}^2 \rightarrow \mathbb {R}$ is the smooth function given by

$$\begin{aligned} \theta (u)=\dfrac{u}{\sqrt{1 - {{\mathrm{\mathrm {sgn}}}}(c)d^2}}, \end{aligned}$$

we obtain a semi-parallel surface in $\mathbb {Q}_c^3 \times \mathbb {R}$ that is not locally parallel.

Another example can be obtained by taking $0< d < 1$ and

$$\begin{aligned}&\tilde{g}(x) = (0, \cos (x), \sin (x), 0, 0, 0),\\&\tilde{\xi }_1(x) = (1, 0, 0, 0, 0), \quad \tilde{\xi }_2(x) = (0, 0, 0, 1, 0), \\&\alpha _0(s) = d S_{c}(s), \quad \alpha _1(s) = d C_{c}(s), \quad \alpha _2(s) = \sqrt{{{\mathrm{\mathrm {sgn}}}}(c)(1-d^2)}, \quad \alpha _3(s) = s, \end{aligned}$$

where the surface obtained is pseudo-parallel in $\mathbb {Q}_c^3 \times \mathbb {R}$ but not semi-parallel since its Gaussian curvature does not vanish. Also, notice that it is not contained in a totally geodesic slice of the form $\mathbb {Q}^3_c \times \{t\}$, for some $t \in \mathbb {R}$.

Question 4.2

Are there other examples, up to isometries, of pseudo-parallel surfaces in $\mathbb {Q}^3_c \times \mathbb {R}$ ($c \ne 0$), for which T is not a principal direction?

The next three examples show us that for $n > 3$ there exists pseudo-parallel surfaces with non vanishing normal curvature.

Example 4.3

Let $f: \mathbb {S}^2_{1/3} \rightarrow \mathbb {S}^4_{1}$ be the classical Veronese surface, given by

$$\begin{aligned} f(x,y,z) = \left( \frac{1}{\sqrt{3}}xy,\frac{1}{\sqrt{3}}xz,\frac{1}{\sqrt{3}}yz, \frac{1}{2\sqrt{3}}(x^2 - y^2), \frac{1}{6}(x^2 + y^2 -2z^2)\right) , \end{aligned}$$

which is a locally parallel, minimal and $\lambda$-isotropic immersion (as we can see in Chern et al. (1970), Itoh and Ogiue (1973) and Sakamoto (1977)) in $\mathbb {S}^4_1$ with non vanishing normal curvature. If $i: \mathbb {S}^4_1 \rightarrow \mathbb {S}^4_1 \times \mathbb {R}$ is the totally geodesic inclusion given by $i(x) = (x,0)$, then by Proposition 2.3 we have that $i \circ f$ is a pseudo-parallel immersion in $\mathbb {S}^4_1 \times \mathbb {R}$ with non vanishing normal curvature.

Conjecture 4.4

The only minimal pseudo-parallel surface in$\mathbb {Q}^4_c \times \mathbb {R}$with non vanishing normal curvature and constant$\phi$are these of Example4.3.

Example 4.5

It’s known by Chern in Chern (1970) that: “Any minimal immersion of a topological 2-sphere $\mathbb {S}^2$ into $\mathbb {S}^4_c$ is a superminimal immersion”. So, by Theorem 1.1, we have that any minimal immersion of a topological 2-sphere into a slice of $\mathbb {S}^4_c \times \mathbb {R}$ whit non vanishing normal curvature is pseudo-parallel with $\phi = \frac{4K - c - \lambda ^2}{3}$. Moreover, if the Gaussian curvature is not constant, the immersion is not semi-parallel.

Example 4.6

Let $f:\mathbb {R}^2 \rightarrow \mathbb {S}^5_c$ be the surface given by

$$\begin{aligned} f(x,y)&= \frac{2}{\sqrt{6c}}\left( \cos u \cos v, \cos u \sin v, \frac{\sqrt{2}}{2}cos(2u), \sin u \cos v, \right. \\&\left. \quad \sin u \sin v, \frac{\sqrt{2}}{2} \sin (2u)\right) , \end{aligned}$$

where $u = \sqrt{\frac{c}{2}}x$, $v = \frac{\sqrt{6c}}{2}y$.

This example, that appears in Sakamoto (1989), is a minimal $\lambda$-isotropic flat torus with $\lambda = \sqrt{\frac{c}{2}}$ and non vanishing normal curvature. In particular, f is a pseudo-parallel immersion in $\mathbb {S}^5_c$ with $\phi = \frac{-c}{2}$.

Thus, if $i: \mathbb {S}^5_c \rightarrow \mathbb {S}^5_c \times \mathbb {R}$ is the totally geodesic inclusion given by $i(x) = (x,0)$, by Proposition 2.3 we have that $i \circ f$ is a pseudo-parallel immersion in $\mathbb {S}^5_c \times \mathbb {R}$ with non vanishing normal curvature.