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A purely algebraic formulation (i.e., polynomial equations only) of the minimum-cost multi-impulse orbit-transfer problem without time constraints is presented, while keeping all the variables with a precise physical meaning. General algebraic techniques are applied to solve these equations (resultants, Grobner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, a proof of the optimality of the Hohmann transfer for the minimum-fuel two-impulse circular-to-circular orbit-transfer problem is provided. Finally, a general formula is also provided for the optimal two-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which, it is conjectured, can be removed).

We provide an approximate zero $$\widetilde{S}(g,L)$$ for the hyperbolic Kepler’s equation $$S-g\,{{\mathrm{arcsinh}}}(S)-L=0$$ 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 $$|S_n-S|\le 0.5^{2^n-1}|\widetilde{S}-S|$$ . 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.

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.

We prove that every contact metric ( κ , μ ) -space admits a canonical η -Einstein Sasakian or η -Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κ and μ 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 ( κ , μ ) -structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of ( κ , μ ) -spaces related to the existence of Einstein–Weyl and Lorentzian–Sasaki–Einstein structures.

We study the Riemann curvature tensor of (κ, μ, ν)-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 (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.

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

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

We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be Kahler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space $${{\mathbb{C}P^{2n+1}}}$$ (4) with the Fubini-Study metric. On the other hand, it is not possible for normal complex contact metric manifolds to be conformally flat.

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 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

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 $$|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|$$ . 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.

In the last years, a growing number of researchers have studied paracontact metric geometry. We shall present here some of the recently obtained results in this topic.

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