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Исследуется многообразие Кенмоцу с нулевым тензором Схоутена. Доказывается, что такое многообразие является многообразием Эйнштейна. На многообразии Кенмоцу определяется N-связность и доказывается, что в случае обращения тензора Схоутена в нуль тензор кривизны N-связности равен нулю тогда и только тогда, когда эндоморфизм N ковариантно постоянен относительно внутренней связности.

А non-holonomic Kenmotsu manifold equipped with a connection analogous to the generalized Tanaka — Webster connection, is consid­ered. The studied connection is obtained from the generalized Tanaka — Webster connection by replacing the first structural endomorphism by the second structural endomorphism. The obtained connection is also called in the work the generalized Tanaka — Webster connection. Unlike a Kenmotsu manifold, the structure form of a non-holonomic Kenmotsu manifold is not closed. The consequence of this single differ­ence is a significant discrepancy in the properties of such manifolds. For example, it is proved in the paper that the alternation of the Ricci-Schouten tensor of a non-holonomic Kenmotsu manifold, which is a transverse analogue of the Ricci tensor, is proportional to the external differential of the structural form. At the same time, in the classical case of a Kenmotsu manifold, the Ricci — Schouten tensor is a symmetric tensor. It is proved that a Tanaka — Webster connection is a metric connec­tion. It is also proved that from the fact that the alternation of the Ricci-Schouten tensor is proportional to the external differential of the structur­al form, the following statement holds: if a non-holonomic Kenmotsu manifold is an Einstein manifold with respect to the generalized Tanaka — Webster connection, then it is Ricci-flat with respect to the same con­nection.

The article discusses the environmental approach as the basis for the developing of the future teachers’ research competencies. Research competencies are among the general competencies provided for by the Federal State Educational Standard of Higher Education. Therefore, the problem of training a teacher-researcher is one of the relevant issues of higher education pedagogy. The purpose of the research is to study the dynamics of the formation of  future teachers’ research competencies under the influence of  the university educational environment. As  a result of  a comparative analysis of  the knowledge and skills of students in the field of research activities, a conclusion was made about the influence of the educational environment on the formation of a future teacher’s research position. Summarizing the experience of three regional universities in building the educational environment, its main components were identified (spatial, institutional, programmatic, organizational, content and competitive) are identified.