### Real Hypersurfaces of Complex Projective Spaces.

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We characterize Clifford hypersurfaces and Cartan minimal hypersurfaces in a sphere by some properties of extrinsic shapes of their geodesics.

We characterize totally η-umbilic real hypersurfaces in a nonflat complex space form M̃ₙ(c) (= ℂPⁿ(c) or ℂHⁿ(c)) and a real hypersurface of type (A₂) of radius π/(2√c) in ℂPⁿ(c) by observing the shape of some geodesics on those real hypersurfaces as curves in the ambient manifolds (Theorems 1 and 2).

We characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms in terms of their shape operators and structure tensors.

In this paper we characterize totally umbilic hypersurfaces in a space form by a property of the extrinsic shape of circles on hypersurfaces. This characterization corresponds to characterizations of isoparametric hypersurfaces in a space form by properties of the extrinsic shape of geodesics due to Kimura-Maeda.

We characterize homogeneous real hypersurfaces of types (A₀), (A₁) and (B) in a complex projective space or a complex hyperbolic space.

We characterize homogeneous real hypersurfaces $M$’s of type $\left({A}_{1}\right)$, $\left({A}_{2}\right)$ and $\left(B\right)$ of a complex projective space in the class of real hypersurfaces by studying the holomorphic distribution ${T}^{0}M$ of $M$.

In the class of real hypersurfaces ${M}^{2n-1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_{n}\left(c\right)$ of constant holomorphic sectional curvature $c$ $(\ne 0)$ which is either a complex projective space $\u2102{P}^{n}\left(c\right)$ or a complex hyperbolic space $\u2102{H}^{n}\left(c\right)$ according as $c>0$ or $c<0$, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this...

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