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Metrics with equatorial singularities on the sphere. (English) Zbl 1305.53040

On the two-sphere of revolution we consider the coordinates \((\varphi , \theta )\) where \(\varphi \) is the angle along the meridian and \(\theta \) the angle of revolution. The present paper is devoted to metrics of the form \(XR(X)d\theta ^2+d\varphi ^2\) where \(R\) is a rational fraction with a single pole at \(X=1\). Two examples motivating this study are presented; one stems from quantum mechanics, the second from space mechanics. Both are limit cases of optimal control problems, not linear, but affine in the control.
The main aspects of this study concern with relating the structure of the cut and conjugate loci to the convexity of the quasi-period of the \(\theta \)-coordinate. Under appropriate assumptions, the cut locus of a point is reduced to a single segment, and the conjugate locus has at most four cusps. Also, a detailed analysis using a parametrization of geodesics by elliptic curves is presented.

MSC:

53C17 Sub-Riemannian geometry
49K15 Optimality conditions for problems involving ordinary differential equations
53C80 Applications of global differential geometry to the sciences
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