Geodesic submersion on the orthogonal complement of kernel

Question

I found this assertion while reading a paper:

Since $F$ is a Riemannian submersion, we have $(\nabla F_*)(Z_1,Z_2)=0$ (F is geodesic map) for $Z_1, Z_2\in \Gamma((KerF_*)^\perp).$

Here $\nabla$ is the Levi-Civita connection. What would be the proof of this, I cannot understand?