On Riemann submersion of submanifold

Let π∶N~(n+r)→N′~n be a Riemann submersion.If f∶M~(m+r)→N~(n+r) and f′∶M′~m→N′~n are isometric immersions,and M respects the Riemann submersion π,then we have the following reults:1) If M is a totally umbilical submanifold of N,then M′ is a totally umbilical submanifold of N′;2) If(f(M)) is an isotropic submanifold of N,then f′(M′) is an isotropic submanifold of N′;3) If M lies in a totally geodesic submanifold N~(m+r+1) of N,then M′ lies in a totally geodesic submanifold N′~(m+1) of N′.In particular,we obtain a pinching theorem of Simons type of submanifold and a character of totally umbilical submanifold in complex projective space and quatermionic projective space.