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Let $(M^n,g)$ be a closed simply-connected positively curved manifold. Show that if $M$ contains a totally geodesic closed hypersurface (i.e., the second fndamental form or shape operator is zero), then $M$ is homeomorpic to a sphere.

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If $H$ is a hypersurface, then we cut $M$ into two regions $D_i$ where $\partial D_i=H$ Then define $$ f : D_1\rightarrow {\bf R},\ f(p)=d(H,p)$$ where $d$ is a distance function.

If $c$ is a normal geodesic with $c(0)=p$, then $f\circ c(t)$ is concave. Since $M$ is positively curved so $f$ has a unique maximum so that by Morse theory $D_1$ is homeomorphic to a $n$-dimensional disk.

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