Let $\gamma : S^1 \rightarrow M$ be a smooth map from a circle of length 1 to a closed manifold $M$ with nonpositive curvature. Could we find a constant $C > 0$ depending only on $M$ such that $$\int_{S^1}||\nabla_{\dot{\gamma}}\nabla_{\dot{\gamma}}\dot{\gamma}||^2 \ge C\int_{S^1}||\nabla_{\dot{\gamma}}\dot{\gamma}||^2$$ for arbitrary $\gamma$?
I have constructed counterexamples on the 2-sphere and a noncompact manifold with negative curvature; that's the reason for the assumption on $M$.