Study of bi-f-harmonic curve along immersions

In this paper, we characterize the bi-f-harmonic curve on surfaces and then we study the submanifold of a Riemannian manifold using the bi-f-harmonic curve. The conditions for curvature and torsion of bi-f-harmonic curve on surface, ruled surface and 3-dimensional space are derived. In addition, the geometry of the submanifold is studied by taking a bi-f-harmonic curve with immersion from the submanifold to the ambient space. Moreover, the conditions are given for isotropic submanifolds, totally geodesic submanifolds and umbilical submanifolds, so that the immersed curve is a bi-f-harmonic curve in ambient space. Finally, we investigate some important results for a particular case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>f</mml:mi> <mml:mo class="MathClass-rel" stretchy="false">=</mml:mo> <mml:mn>1</mml:mn></mml:mrow></mml:math> of the bi-f-harmonic curve.