Assume that $\pi : X\rightarrow B$ is a Riemannian submersion where $X$ is a closed manifold. If each fiber is a totally geodesic submanifold, then fibers are isometric : Here if $c$ is a curve in $B$, then a lift of $c$ gives an isometry.
Similarly, we can consider the following : If fibers with an induced Riemannian metrics from $X$ are isometric, then there is a Riemannian submersion $\pi' :X\rightarrow B$ s.t. each fiber of $\pi'$ is totally geodesic ?
The assertion is true ? If it is true, then how can we prove this ?