ar X iv : 0 70 5 . 21 12 v 1 [ nl in . S I ] 1 5 M ay 2 00 7 Density of real closed geodesics on ellipsoids related to
نویسنده
چکیده
We prove that any real doubly periodic geodesic on an n dimensional ellipsoid with distinct semiaxes and caustic parameters is uniquely associated to a real hyperelliptic tangential cover and that the following density property holds: given a real closed geodesic on the ellipsoid Q = {X2 1/a1 + · · ·+X2 n+1/an+1 = 1} with caustic parameters cj, j = 1, . . . , n − 1, for any ǫ > 0 there exist (a, c) ∈ R satisfying n−1 ∑ j=1 (cj − cǫj)2 + n+1 ∑ i=1 (ai − aǫi)2 < ǫ and such that the geodesics on Q = {X2 1/aǫ1 + · · · + X n+1/aǫn+1 = 1} with caustic parameters cǫj , j = 1, . . . , n − 1, are real doubly–periodic. In particular, if n = 2 (triaxial ellipsoid), the closed geodesics associated to (Q, c) and to (Q, c) have the same length T and the same value of the period mapping. Moreover, for real doubly–periodic geodesics on triaxial ellipsoids, we show how to evaluate the period mapping using the topological classification of the second covering associated to the genus 2 hyperelliptic curve. Finally we prove the existence of real doubly–periodic geodesics associated to a family of degree 2 coverings with extra automorphisms and show their relation to hyperelliptic tangential covers.
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تاریخ انتشار 2008