# Minimal representative of the elements of the fundamental group of a negatively curved manifold

Let (M,g) be a negatively curved manifold , let p be any point of M and denote by G=π1(M,p) . the minimal representative (by minimal i mean the smallest length representative ) of every α in G is a simple closed geodesic loop at p . my question is why it should be simple ?

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If by "simple" you mean "has no self-intersections", that statement is false. –  Igor Rivin Nov 25 '11 at 12:45
yeah that's what i mean . what is true or special if you want for negatively curved manifold concerning this subject ( representative of based point class ,length of a representative ,...) a reference would be most welcomed also thank you –  student Nov 25 '11 at 12:54