[Paper Review] On rotational surfaces in pseudo-Euclidean space $\mathbb{E}^4_t$ with pointwise 1-type Gauss map

Burcu Bektaş Demirci, Elif Özkara Canfes+1 more|arXiv (Cornell University)|Aug 13, 2015

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper classifies rotational surfaces in pseudo-Euclidean 4-space $\mathbb{E}^4_t$ with pointwise 1-type Gauss maps, focusing on surfaces in $\mathbb{E}^4_1$ (Minkowski space) and $\mathbb{E}^4_2$ (signature (2,2)). It proves that non-planar timelike double rotational surfaces in $\mathbb{E}^4_1$ with zero mean curvature and pointwise 1-type Gauss map of the second kind exist only if the profile curve satisfies $y(s) = b_0(w(s))^{\pm b}$, and fully characterizes such surfaces in $\mathbb{E}^4_2$ with zero mean curvature and second-kind Gauss map.

ABSTRACT

In this work, we study some classes of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}^4_t$ with profile curves lying in 2-dimensional planes. First, we determine all such surfaces in the Minkowski 4-space $\mathbb{E}^4_1$ with pointwise 1-type Gauss map of the first kind and second kind. Then, we obtain rotational surfaces in $\mathbb{E}^4_2$ with zero mean curvature and having pointwise 1--type Gauss map of second kind.

Motivation & Objective

• To classify rotational surfaces in pseudo-Euclidean 4-space $\mathbb{E}^4_t$ with pointwise 1-type Gauss map.
• To determine all such surfaces in $\mathbb{E}^4_1$ with pointwise 1-type Gauss map of the first and second kind.
• To identify rotational surfaces in $\mathbb{E}^4_2$ with zero mean curvature and pointwise 1-type Gauss map of the second kind.
• To establish conditions under which non-planar timelike or spacelike rotational surfaces in $\mathbb{E}^4_1$ and $\mathbb{E}^4_2$ admit pointwise 1-type Gauss maps.

Proposed method

• Parametrizes rotational surfaces in $\mathbb{E}^4_t$ using profile curves in 2D planes, defining two families: $M_1(b)$ and $M_2(b)$.
• Computes the Gauss map and its Laplacian using the induced connection and second fundamental forms in the orthonormal frame.
• Applies the condition for pointwise 1-type Gauss map: $\Delta\nu = f(\nu + C)$, where $f$ is a smooth function and $C$ is a constant vector.
• Uses the structure equations and curvature relations to derive differential equations for the profile curves.
• Analyzes the system of equations from the Laplacian expression and the pointwise 1-type condition to classify solutions.
• Distinguishes between first-kind ($C=0$) and second-kind ($C\neq0$) Gauss maps via the vanishing of components in the normal bundle.

Experimental results

Research questions

• RQ1 Which rotational surfaces in $\mathbb{E}^4_1$ with pointwise 1-type Gauss map of the first kind exist?
• RQ2 Under what conditions does a non-planar timelike double rotational surface in $\mathbb{E}^4_1$ have a pointwise 1-type Gauss map of the second kind?
• RQ3 What rotational surfaces in $\mathbb{E}^4_2$ with zero mean curvature admit a pointwise 1-type Gauss map of the second kind?
• RQ4 Can rotational surfaces in $\mathbb{E}^4_2$ with zero mean curvature have pointwise 1-type Gauss map of the first kind?

Key findings

• All non-planar rotational surfaces in $\mathbb{E}^4_1$ with pointwise 1-type Gauss map of the first kind are classified, and such surfaces exist only if the profile curve satisfies a specific differential equation.
• There exists no non-planar timelike double rotational surface in $\mathbb{E}^4_1$ with zero mean curvature and pointwise 1-type Gauss map of the second kind unless the profile curve is $y(s) = b_0(w(s))^{\pm b}$.
• For $b=1$, the timelike surface $M_1(1)$ in $\mathbb{E}^4_2$ has pointwise 1-type Gauss map of the second kind if the profile curve satisfies $(x(s)+z(s))^2 + \lambda_0(x(s)-z(s))^2 = \mu_0$ for constants $\lambda_0 \neq 0$, $\mu_0$.
• For $b \neq 1$, the spacelike surface $M_2(b)$ in $\mathbb{E}^4_2$ has pointwise 1-type Gauss map of the second kind if and only if the profile curve is $z(s) = \bar{b}_0(x(s))^{\pm b}$ for some $\bar{b}_0 \neq 0$.
• Rotational surfaces in $\mathbb{E}^4_2$ with zero mean curvature cannot have pointwise 1-type Gauss map of the first kind, as shown by Corollary 4.3.
• The Gauss map of $M_1(1)$ satisfies $\Delta\nu = f(\nu + C)$ with $f = -8\varepsilon\kappa^2$ and $C = -\frac{1}{2}e_1\wedge e_2 - \frac{1}{2}\varepsilon\varepsilon^*e_3\wedge e_4$, confirming the second-kind condition.