We study Riemannian submersions with pointwise k-planar horizontal sections. We provide examples, obtain characterizations and give a geometric interpretation of such property.

In this study, we consider bi-f-harmonic Legendre curves in Sasakian space forms. We investigate necessary and sufficient conditions for a Legendre curve to be bi-f-harmonic in various cases.

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In this paper, we handle set-theoretical solutions of Yang-Baxter equation and Lyubashenko set theoretical solutions in Lie algebras. We present a new commutative binary operation on these structures, and we obtain new set theoretical solutions including this operation by using property of commutativity of it. Also, we show that some set theoretical solutions of Yang-Baxter equation corresponds to the Lyubashenko set theoretical solutions on these structures. Additionally, we give some relations to verify set theoretical solution of Yang-Baxter equation. Moreover, we put an interpretation for these solutions from the point of geometrical view in Euclidean space, Minkowski space and differentiable manifolds by using Lie algebras.

In this paper, we determine necessary and sufficient conditions for a non-Frenet Legendre curve to be f -harmonic, f -biharmonic, bi-f -harmonic, biminimal and f -biminimal in three-dimensional normal almost paracontact metric manifold. Besides, we obtain some nonexistence theorems.

This paper is devoted to examine necessary and sufficient conditions for a Frenet curve to be f-harmonic, f-biharmonic, bi-f-harmonic and f-biminimal in three-dimensional $$\beta $$ β -Kenmotsu manifolds. In addition, such conditions are investigated for slant curves.

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