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Nov 28, 2018 at 17:22 comment added hakunamatata @Raziel yes exactly what I thought.
Nov 28, 2018 at 7:14 comment added Raziel @daene, the Hopf fibration has totally geodesic leaves, but the leaves are flat and the total space is not.
Nov 28, 2018 at 2:30 comment added Deane Yang If the leaves are totally geodesic, the sectional curvatures will be the same. If $s\ne 0$, then there are values of $n$ and $k$ for which this is also a necessary condition.
Nov 27, 2018 at 9:20 comment added hakunamatata Of course. By the way, is there any result pointing to the fact when my question has a positive answer? Just curious!
Nov 26, 2018 at 18:44 comment added Raziel If you liked it, then feel free to accept the answer :)
Nov 26, 2018 at 17:13 comment added hakunamatata Thanks. the first example is a nice one.
Nov 26, 2018 at 8:07 history answered Raziel CC BY-SA 4.0