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A local convergence analysis of Inexact Newton's method with relativeresidual error tolerance for finding a singularity of a differentiable vectorfield defined on a complete Riemannian manifold, based on majorant principle,is presented in this paper. We prove that under local assumptions, the inexactNewton method with a fixed relative residual error tolerance converges Q-linearly to a singularity of the vector field under consideration. Using thisresult we show that the inexact Newton method to find a zero of an analyticvector field can be implemented with a fixed relative residual error tolerance.In the absence of errors, our analysis retrieve the classical local theorem onthe Newton method in Riemannian context.

Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian Manifolds