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.

The quasi frame is more efficient than the Frenet frame in investigating surfaces, and it is regarded a generalization frame of both the Frenet and Bishop frames. The geometry of quasi-Hasimoto surfaces in Minkowski 3-space ${\mathbb{E}}_1^3$ is investigated in this paper. For the three situations of non-lightlike curves, the geometric features of the quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ are examined and the Gaussian and mean curvatures for each case are determined. The quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ must satisfy a necessary and sufficient condition to be developable surfaces. As a result, the parameter curves of quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ is described. Thus, the $s$-parameter and $t$-parameter curves of quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ are said to be geodesics, asymptotic, and curvature lines under necessary and sufficient circumstances are proved. Finally, quasi curves and associated quasi-Hasimoto surface correspondences are discussed.

This study investigated the concepts of (0, 2)-involute and (1, 3)-evolute curves associated with quaternionic Bertrand curves within the context of four-dimensional Euclidean space. Using a type-2 quaternionic frame, we derived mathematical expressions that define these interacting and evolute curves. The (0, 2)-involute curve is characterized by tangents orthogonal to points on the original quaternionic Bertrand curve, while the (1, 3)-evolute curve is constructed using specific normal vectors related to curvature properties. We presented a comprehensive framework that clarifies the interrelationships between the curvature functions of involute and evolute pairs and their connections to the Frenet frame. This framework provides a geometric basis for analyzing curves in higher-dimensional spaces. The findings enhance the understanding of quaternionic curves and their geometric properties, contributing to the broader field of differential geometry.

This paper presents a geometric study of three types of ruled surfaces generated from the tangent, normal, and binormal unit vectors of unit speed space curves. Using the T-pedal curve construction as a foundation, we analyze these surfaces through their fundamental geometric forms, including curvature properties, the striction curve geometry, and the distribution parameter. The theoretical framework is used to analyze problems in computational geometry and shape modeling, with results relevant to both mathematical research and engineering applications. The work establishes fundamental geometric insights while providing tools for applied shape modeling and analysis.