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We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct., Comment: 14 pages

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic., Comment: 15 pages

We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gr\"obner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed)., Comment: 17 pages, 6 images in 4 figure environments

We provide an approximate zero $\widetilde{S}(g,L)$ for the hyperbolic Kepler's equation $S-g\, {\rm asinh} (S)-L=0$ for $g\in(0,1)$ and $L\in[0,\infty)$. We prove, by using Smale's $\alpha$-theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution $S(g,L)$ at quadratic speed, i.e. if $S_n$ is the value obtained after $n$ iterations, then $|S_n-S|\leq 0.5^{2^n-1}|\widetilde{S}-S|$. The approximate zero $\widetilde{S}(g,L)$ is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bounded regions of $(0,1) \times [0,\infty)$ that exclude a small neighborhood of $g=1, L=0$, we also provide a method to construct simpler starters involving only constants., Comment: 14 pages, 2 figures

We study a remarkable class of paracontact metric manifolds which have no contact metric counterpart: the paracontact metric $(-1,\widetilde\mu)$-spaces which are not paraSasakian (i.e. have $\widetilde h\neq0$). We present explicit examples with $\widetilde h$ of every possible constant rank and some with non-constant rank, which were not known to exist until recently., Comment: 5 pages, a short review of previous papers in the field

We study paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We will also show explicit examples of paracontact metric $(-1, \mu)$-spaces with tensor $h$ of non-constant rank, which were not known to exist until now., Comment: 9 pages

We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler's equation $E-e\sin(E)=M$ for any $e\in[0,1)$ and $M\in[0,\pi]$. Our solution is guaranteed, via Smale's $\alpha$-theory, to converge to the actual solution $E$ through Newton's method at quadratic speed, i.e. the $n$-th iteration produces a value $E_n$ such that $|E_n-E|\leq (\frac12)^{2^n-1}|\tilde{E}-E|$. The formula provided for $\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$, where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $[0,1)\times[0,\pi]$ if only rational functions are allowed in each branch.

We study non-paraSasakian paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of $h$. We will also give explicit examples of every possible constant rank of $h$., Comment: 12 pages; several corrections have been made in this version

Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,\langle,\rangle)$, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $\mathcal D$-homothetic deformations, and classify paraSasakian and paracontact $(\kappa,\mu)$-spaces inside this class. We also present a way to build paracontact $(\kappa,\mu)$-spaces from corresponding contact metric structures on $T_1 M$., Comment: 21 pages

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (\kappa,\mu,\nu)-spaces. We present examples or obstruction results of these spaces in all possible cases.

We prove that any non-Sasakian contact metric (\kappa,\mu)-space admits a canonical \eta-Einstein Sasakian or \eta-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of \kappa and \mu for which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (\kappa,\mu)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some topological and geometrical properties of (\kappa,\mu)-spaces related to the existence of Eistein-Weyl and Lorentzian Sasakian Einstein structures.

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to be conformally flat.

Generalized (\kappa ,\mu)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case., Comment: 20 pages, several changes have been done in this version