The resolution of the acceleration and jerk vectors of a particle moving on a space curve in the Euclidean 3-space is considered. By applying this resolution and Siacci’s theorem, alternative resolutions of acceleration and jerk vectors are derived based on the quasi-frame. In the osculating plane, the acceleration vector is resolved as the sum of its tangential and radial components. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions. The maximum permissible speed on a space curve at all trajectory points is established via the jerk vector formula. Finally, some examples are presented to illustrate how the results work.

This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space ${\mathbb{E}}^3$. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors and offer an alternative resolution of the jerk and snap vectors along the tangential direction and two special radial directions that lie in the osculating and rectifying planes. This alternative resolution for a quasi plane curve in Euclidean 3-space ${\mathbb{E}}^3$ is given as corollary. Moreover, our results are illustrated via some examples.

This study investigates the pseudo-projective curvature tensor within the framework of doubly and twisted warped product manifolds. It offers significant insights into the interaction between the pseudo-projective curvature tensor and both the base and fiber manifolds. The research highlights key geometric characteristics of the base and fiber manifolds as influenced by the pseudo-projective curvature tensor in these structures. Additionally, the paper extends its analysis to examine the behavior of the pseudo-projective curvature tensor in the context of generalized doubly and twisted generalized Robertson-Walker space-times.

This work aims at investigating the geometry of surfaces corresponding to the geometry of solutions of the vortex filament equation in Euclidean 3-space ${\mathbb{E}}^3$ using the quasi-frame. In particular, we discuss some geometric properties and some characterizations of parameter curves of these surfaces in ${\mathbb{E}}^3$.