Pushforwards of Riemmanian metrics and geodesics

Question

Let $(M_1,g_1)$ and $(M_2, g_2)$ be two Riemannian manifolds. Assume that $T : M_1 \to M_2$ is a diffeomorphism and $g_2$ is the pushforward of $g_1$ by $T$. Assume that $\gamma$ is a geodesic on $M_1$. Then, is $T \circ \gamma$ a geodesic on $M_2$?