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The ohmic decay of magnetic fields confined within the crust of neutron starsis considered by incorporating both the effect of neutron star cooling and theeffect of space-time curvature produced by the intense gravitational field ofthe star. For this purpose a stationary and static gravitational field has beenconsidered with the standard as well as the accelerated cooling models ofneutron stars. It is shown that general relativistic effect reduces themagnetic field decay rate substantially. At the late stage of evolution whenthe field decay is mainly determined by the impurity-electron scattering, theeffect of space-time curvature suppresses the role of the impurity contentsignificantly and reduces the decay rate by more than an order of magnitude.Even with a high impurity content the decay rate is too low to be ofobservational interest if the accelerated cooling model along with the effectof space-time curvature is taken into account. It is, therefore, pointed outthat if a decrease in the magnetic field strength by more than two orders ofmagnitude from its initial value is detected by observation then the existenceof quark in the core of the neutron star would possibly be ruled out.Comment: 15 pages, AAS LATEX macros v4.0, 5 postscript figures, Accepted for  publication in the Astrophysical Journal (Part I

Evolution of Crustal Magnetic Fields in Isolated Neutron Stars: Combined Effects of Cooling and Curvature of Spacetime