Differential geometry of Fermat quartic surface
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We examine the differential geometry of the Fermat Quartic surface X4/0+X4/3-X4/1-X4/2 = 0 in CP^3 with induced Fubini-Study metric. We show that the differential equations of geodesics, when restricted to the real Fermat quartic surface inside the full complex quartic, can be reduced to two non-linear differential equations with rational coefficients along especially chosen geodesics. This simplification opens up the possibility of parametrizing these geodesics in terms of genus three Abelian integrals and their inversions. Furthermore the identity component of the differential Galois group of normal variational equation, derived from the geodesic equation along one of these selected curves, is SL(2,C). By Morales-Ramis theory the Hamiltonian system defining the geodesic equations is not integrable in a neighborhood of this solution by meromorphic integrals.
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