A characterization of totally geodesic hypersurfaces of 𝑆ⁿ⁺¹ and 𝐶𝑃ⁿ⁺¹

Abe, Kinetsu

doi:10.1090/S0002-9939-1981-0601739-3

A characterization of totally geodesic hypersurfaces of $S^{n+1}$ and $\textbf {C}P^{n+1}$
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Proc. Amer. Math. Soc. 81 (1981), 603-606 Request permission

Abstract:

Let $M$ be a complete hypersurface in ${S^{n + 1}}$ (or ${\mathbf {C}}{P^{n + 1}}$). Assume that through each point $x$ of $M$ a (local) $\mu (x)$-dimensional totally geodesic submanifold ${S_x}$ of ${S^{n + 1}}$ (or ${\mathbf {C}}{P^{n + 1}}$) exists in $M$. A sufficient condition for $M$ itself to be totally geodesic is given in terms of $\mu (x)$.

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