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In this paper, we study how undergraduate students define 3D geometrical solids. With this aim, we have identified the routines that are present in the discourse of the students when describing and defining these solids. These routines are one of the properties that characterise the mathematical discourse in the theory of commognition (Sfard 2008). Our results show three different types of routines. The first type is related to the process of describing the solids, the second one to the process of defining the solids and the rest of the routines have a transversal nature. All of them together give us a global vision of the mathematical practice of defining of these undergraduate students. For instance, it seems that some of these students do not have a clear idea of what a definition is. Moreover, there are also differences between the discourse of students when defining 2D figures and the discourse of students when defining 3D solids.

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 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.

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.