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preprint, final version 25 pagesInternational audienceIn a complete Riemannian manifold, an embedded geodesic γ with finite length and negative Jacobi operator admits an r-neighborhood Nr(γ) with radius r > 0 small enough such that each couple of points of Nr(γ) can be joined by a unique geodesic contained in Nr(γ) where it minimizes length among the piecewise C1 paths joining its end points

Thinr-neighborhoods of embedded geodesics with finite length and negative Jacobi operator are strongly convex