Smarandache Ruled Surfaces According to Bishop Frame in E 3 S¨ uleyman S¸ENYURT 1 , Davut CANLI 2* , Kebire Hilal AYVACI 3 1,2,3 Ordu University / Faculty of Arts and Science Department of Mathematics * : corresponding author Abstract This paper introduces new ruled surfaces according to Bishop frame by referring the main idea of Smarandache geometry. The fundamental forms and the corresponding curvatures are provided to put forth some characteristics of each surface. Finally, an example is given to illustrate the constructed surfaces. Keywords: Smarandache ruled surfaces; Bishop frame; fundamental forms; mean and Gaus- sian curvatures; developable surfaces; minimal surfaces. 1 Introduction Ruled surfaces are special kind of surfaces that are easy to handle and have the potential use of related fields such as engineering, computational constructions, architectural structures, computer graphics, works of art, geometric designs, textile, automobile industry, etc. The basic theory re- lated to ruled surfaces can be found in many differential geometry textbooks such as [4, 5, 7, 15]. Generalization of ruled surfaces was introduced by Juza in the 1960s [6]. In addition, some char- acteristic properties of the ruled surface with Frenet frame of a non-cylindrical ruled surface were investigated in 2020 by Ouarab and Chahdi in [10]. Apart from Frenet frame, in [8] and in [17], Tunc¸er, (2015) and Masal and Azak, (2019), separately, studied some characteristics of the ruled surfaces according to Bishop frame (introduced by Bishop, 1975 in [2]), whereas Ouarab et al. (2018) provided the main properties of ruled surfaces according to alternative frame in [9]. Recently, Ouarab, (2021a) put forth a method to generate new ruled surfaces by taking the ad- vantage of the idea of Smarandache geometry [1, 18]. By assigning the base curve as one of the Smarandache curves and taking the generator as the another vector element of Frenet frame, she introduced these ruled surfaces as Smarandache ruled surfaces according to Frenet frame in [11]. The same method of generating such ruled surfaces is applied to the Darboux frame by Ouarab, (2021b) in [12] and according to the alternative frame by Ouarab, (2021c) in [13]. There are other 1 arXiv:2112.05530v1 [math.GM] 19 Nov 2021