Let $M^{n}$ a Riemannian manifold orientable with positive curvature and even dimension. Let $\gamma$ a closed geodesic. Prove that $\gamma$ is homotopic to a closed curve whose lenght is strictly less than $\gamma$. I would any tips for solve this problem, because , i honestly i do know which tool use.
Thanks.
Either I didn't understand you, or the statement is false. There exist non- simply connected positively curved manifolds of even dimension (e.g. product of two lens spaces). So choose any non-trivial class in fundamental group and consider the shortest (smooth) loop in this class by minimizing the length functional. It will be geodesic.