On the classification of some totally umbilical submanifolds of C -manifolds Yavuz Selim Balkan and Cenap ¨ Ozel Abstract. In this paper, we give some classifications of submanifolds of C -manifolds, particularly totally umbilical slant submanifolds and to- tally umbilical CR-submanifolds of globally framed type. Firstly, we show that a totally umbilical slant submanifold M of a C -manifold M is either an anti-invariant submanifold or an s-dimensional submanifold. Then we prove that every totally umbilical proper slant submanifold of a C - manifold is totally geodesic. Last, we give the characterization of a totally umbilical globally framed CR-submanifold M of a C -manifold M . M.S.C. 2010: 53D10, 53C15, 53C25, 53C35. Key words: Slant submanifold; totally umbilical; totally geodesic; minimal subman- ifold; globally framed CR-submanifold; C -manifold. 1 Introduction The submanifold theory is an important research topic in Differential Geometry since the famous Nash’s embedding theorem. In time, several authors introduced some different classes of submanifolds. In this work, we particularly focus on the slant submanifolds and CR-submanifolds. The notion of slant submanifolds in the complex spaces was initiated by Chen ([7], [8]). A slant submanifold is a natural generalization of both holomorphic and totally real submanifolds. Since then, many researchers focused on this area and proved the existence of these submanifolds in different known spaces ([5], [14], [19], [17]). Recently, the first author defined the slant submanifolds of globally framed metric f -manifolds in his Ph.D. thesis [1]. On the other hand, in 1978, Bejancu introduced the notion of CR -submanifolds of a K¨ahler manifold as a natural generalization of both the holomorphic and totally real submanifolds of a K¨ahler manifold. [2]. Then, a lot of researchers studied on the CR-submanifolds using different structures ([3], [9], [10], [18]). In the present paper, firstly, we consider slant submanifolds of C -submanifolds and we give some classifications of these type submanifolds under some certain conditions. Secondly, we prove that a globally CR-submanifold of a C -submanifold is totally geodesic or the anti-invariant distribution D ⊥ is s-dimensional or a mean curvature Differential Geometry - Dynamical Systems, Vol.19, 2017, pp. 1-14. c ⃝ Balkan Society of Geometers, Geometry Balkan Press 2017.