Are part of great circles geodesics on spherically symmetric manifolds?

Question

While doing some study on geodesics on Riemannian manifolds, I learned that any geodesics on $n$-spheres are part of great circles. I then started wondering if that is true for any spherically symmetric manifold. My thought is, for a manifold $M=N \times S^2$, since this manifold is spherically symmetric I can always choose my coordinate system such that the polar angle is constant for the initial and ending points connected by the geodesic. But how can I prove that along the geodesic the polar angle doesn't change? I'm I wrong in my hypotheses?