In this study, some special curves defined on the alternative frame are investigated by means of the conformable derivative. The study consists of three chapters. The first chapter is devoted to the introduction. The second chapter is divided into three subsections. In the first subsection, basic definitions and fundamental concepts related to the alternative frame are presented. The second subsection is concerned with the basic definitions and concepts of the conformable derivative and integral. Finally, the third subsection includes known fundamental definitions and theorems concerning some special curves defined by means of the conformable derivative and integral. In the third chapter, these two structures are combined, forming the original part of the study. In the three-dimensional Euclidean space, some special curves are defined on the alternative frame constructed by means of the conformable derivative and integral, and their characterizations are obtained.
We complete the results from [14] concerning tangent sphere bundles of constant radius $r$, previously valid for nonflat base manifolds of dimension $n>2$, by taking as base manifold an arbitrary $2$-dimensional space form$(M,g)$. We derive expressions for the metric ${}^rG$ induced on $T_rM$ by a natural diagonal metric from $TM$, such that $(T_rM,{}^rG)$ is an Einstein manifold, i.e. its Ricci tensor field is $Ric=\rho\cdot{}^rG$. In particular, if the metric is induced by the Sasaki metric, $T_rM$ is an Einstein manifold if and only if $(M,g)$ has positive constant sectional curvature $c$, $r=\frac{\sqrt{c}}{c}$, $\rho=\frac{c}{2}$, or $(M,g)$ is flat, and then $T_rM$ is Ricci-flat. Moreover, we show that for a flat $n$-dimensional Riemannian manifold $(M,g)$, $T_rM$ is Einstein with respect to the metric induced by the natural diagonal metric from $TM$ if and only if $n=2$, $(T_rM,{}^rG)$ is Ricci flat, and a certain function coefficient of the metric vanishes. Finally, we consider the Sasakian manifold $(T_rM,\varphi,\xi,\eta,G)$ from \cite{DrutaOpr} and provide a simpler characterization of the $\eta$--Einstein Sasakian tangent sphere bundles in terms of rational expressions involving the coefficients of the metric, the dimension of the base manifold, and the radius $r$.
In this paper, we present a sufficient conditions under which a linear relation (multivalued linear operator) admits an invariant subspace on a Banach space $X$. Also, we give some Fredholm properties of the linear relation $T$ using properties of its restriction to an invariant subspace $M$ and the linear relation induced by $T$ over the quotient subspace $X/M$.
We analyze the evolution of certain parabolic-type energy expressions associated with linear heat-type equations in the setting of compact Riemannian manifolds undergoing the Ricci–Bourguignon and Yamabe flows. By formulating appropriate frequency-type quantities tailored to the underlying geometric dynamics, we demonstrate their monotonic nature under specific curvature assumptions and flow-adapted conditions. This study contributes to the understanding of the interaction between geometric evolution and analytic invariants.
We introduce a unified framework for Golden Poisson Geometry, built around an operator $\varphi$ satisfying the Golden identity $\varphi^2=\varphi+I$. By developing the notions of Golden algebras, Golden Poisson algebras, and Golden Lie algebras, we describe how $\varphi$ governs new compatibility patterns between algebraic operations, Lie brackets, and Poisson tensors. Linear Golden Poisson structures naturally arise on the duals of Golden Lie algebras, revealing Poisson manifolds endowed with intrinsic Fibonacci-type symmetries. Conceptual parallels with the Poisson–Nijenhuis theory of Kosmann-Schwarzbach (1990) show how Golden operators extend classical recursion phenomena. This framework opens new directions in deformation theory and in the geometric study of polynomial identities. Our main results include the construction of an infinite Golden-Fibonacci hierarchy of compatible Poisson tensors . We establish Golden recurrence relations for hamiltonian vector fields and prove functorial properties showing that the Golden Poisson morphisms transport the entire hierarchical structure. We completely characterize the compatibility conditions between Golden structures and linear Poisson brackets, analyze the associated symplectic foliations, and develop the theory of Golden Poisson morphisms.
