Bulletin of the
Korean Mathematical Society BKMS

Let $K$, $H$, $K_{II}$ and $H_{II}$ be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface $T_\gamma(\alpha)$ with the radius $\gamma$ along a timelike curve $\alpha(s)$ in Minkowski 3-space $E_{1}^3$. We prove that $T_\gamma(\alpha)$ must be a $(K,H)$-Weingarten surface and a $(K,H)$-linear Weingarten surface. We also show that $T_{\gamma}(\alpha)$ is $(X,Y)$-Weingarten type if and only if its central curve is a circle or a helix, where $(X,Y)$ $\in$ $\{(K,K_{II})$, $(K,H_{II})$, $(H,K_{II})$, $(H,H_{II})$, $(K_{II}$, $H_{II}) \}$. Furthermore, we prove that there exist no timelike tubular surfaces of $(X,Y)$-linear Weingarten type, $(X,Y,Z)$-linear Weingarten type and $(K,H,K_{II},H_{II})$-linear Weingarten type along a timelike curve in $E_{1}^3$, where $(X,Y,Z)\in\{(K,H,K_{II})$, $(K,H,H_{II})$, $(K,K_{II},H_{II})$, $(H$, $K_{II},H_{II})\}$.

Keywords: Tubular surface, Minkowski 3-space, Weingarten surface, the second Gaussian curvature, the second mean curvature

MSC numbers: Primary 53B30, 53A35

Supported by: This work was supported by the National Natural Science Foundation of China (Grant No. 11001130) and the Fundamental Research Funds for the Central Universities (Grant No. 30917011335).