Abstract
Spacelike intrinsic rotational surfaces with constant mean curvature in the Lorentz-Minkowski space $\E_1^3$ have been recently investigated by Brander et al., extending the known Smyth's surfaces in Euclidean space. In this paper, we give an approach to the analogue Smyth's surfaces of $\E_1^3$ . Assuming that the surface is intrinsic rotational with coordinates $(u,v)$ and conformal factor $\rho(u)^2$, we replace the constancy of the mean curvature by the property that the Weingarten endomorphism $A$ can be expressed as $\Phi_{-\alpha(v)}\left(\begin{array}{ll}\lambda_1(u)&0\\ 0&\lambda_2(u)\end{array}\right)\Phi_{\alpha(v)}$, where $\Phi_{\alpha(v)}$ is the (Euclidean or hyperbolic) rotation of angle $\alpha(v)$ at each tangent plane and $\lambda_i$ are the principal curvatures. Under these conditions, it is proved that the mean curvature is constant and $\alpha$ is a linear function. This result also covers the case that the surface is timelike. If the mean curvature is zero, we determine all spacelike and timelike intrinsic rotational surfaces with rotational angle $\alpha$. This family of surfaces includes the spacelike and timelike Enneper surfaces.
References
- [1] S. Akamine, J. Cho and Y. Ogata, Analysis of timelike Thomsen surfaces, J. Geom. Anal. 30, 731–761, 2020
Abstract
References
- [1] S. Akamine, J. Cho and Y. Ogata, Analysis of timelike Thomsen surfaces, J. Geom. Anal. 30, 731–761, 2020
Details
| Primary Language | English |
|---|---|
| Subjects | Algebraic and Differential Geometry |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | April 11, 2025 |
| Publication Date | |
| Submission Date | June 16, 2024 |
| Acceptance Date | November 13, 2024 |
| Published in Issue | Year 2025 Early Access |