Abstract
In this paper, we prove some results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, a part of which generalizes classical results about loxodromes on rotational surfaces in \({{\mathbb {R}}}^3\). In particular, we show how to parametrize a loxodrome on an invariant surface of \({\mathbb {H}}^2\times {{\mathbb {R}}}\) and \({\mathbb {H}}_3\), and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.
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Notes
The elliptic integral of the second kind is defined by
$$\begin{aligned} E(\phi ,m)=\int _{0}^{\phi }\sqrt{1-m\sin ^2\theta }\,\text {d}\theta . \end{aligned}$$The Gaussian hypergeometric function \(F(\alpha _1,\alpha _2,\beta _1,z)\) is defined for \(|z|<1\) by the Pochhammer power series:
$$\begin{aligned} F(\alpha _1,\alpha _2,\beta _1,z) \overset{not}{=} _2F_1[\alpha _1,\alpha _2,\beta _1,z]=\sum _{n=0}^\infty \frac{(\alpha _1)_n\,(\alpha _2)_n}{(\beta _1)_n}\frac{z^n}{n!}. \end{aligned}$$Here, \(\alpha _1,\alpha _2,\beta _1\in {\mathbb {R}}\), \(c\notin {\mathbb {Z}}_{\le 0}\), and \((\lambda )_n =\lambda (\lambda +1)\cdots (\lambda +n-1)\).
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The second author was supported by grant 2016/24707-4, São Paulo Research Foundation (Fapesp), and by CNPq productivity grant 312700/2017-2. The other two authors were supported by GNSAGA-INdAM, Italy.
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Caddeo, R., Onnis, I.I. & Piu, P. Loxodromes on Invariant Surfaces in Three-Manifolds. Mediterr. J. Math. 17, 6 (2020). https://doi.org/10.1007/s00009-019-1439-2
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DOI: https://doi.org/10.1007/s00009-019-1439-2