hsjg Hagia Sophia Journal of Geometry 2687-4261 Salim YÜCE Algebraic and Differential Geometry Cebirsel ve Diferansiyel Geometri Directional Tube Surface in Euclidean 4-Space https://orcid.org/0000-0003-4067-3034 Yağbasan Başak https://orcid.org/0000-0002-3247-5727 Ekici Cumali ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ Tozak Hatice 12 30 2023 5 2 18 30 08 29 2023 09 20 2023 Copyright © 2019, Hagia Sophia Journal of Geometry 2019 Hagia Sophia Journal of Geometry

The aim of this paper is to study characterization of tube surfaces (called directional tube surfaces) with respect to the q-frame in Euclidean $4$-space $\mathbb{E}^{4}$. First, a parametrization of these directional tube surfaces in $\mathbb{E}^{4}$ is established. Then, the normals of the directional tube surfaces, denoted as $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$, are determined respectively. Furthermore, the Gaussian curvature $K$ and the mean curvature $H$ of the directional tube surfaces are investigated. Subsequently, an example of a directional tube surface is given in $\mathbb{E}^{4}$, together with visual representations of this tube surfaces in projection space.

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