# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 36 (2007) pp.353-363

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### Abstract

For Hopf hypersurfaces in a nonflat complex space form $M^n(c; \Bbb{C})$, integral curves of their characteristic vector fields are ''nice'' curves in the sense that their extrinsic shapes in $M^n(c; \Bbb{C})$ are K\"ahler circles. In this paper we mainly study geodesic spheres in a nonflat complex space form $M^n(c; \Bbb{C})$. On these geodesic spheres we classify smooth curves whose extrinsic shapes are K\"ahler circles in $M^n(c; \Bbb{C}),c\not=0$. We also give a characterization of complex space forms among K\"ahler manifolds by extrinsic shapes of integral curves of characteristic vector fields on their geodesic spheres.

MSC(Primary) 53C40 53B25 complex space forms, geodesic spheres, integral curves of characteristic vector fields, K\"ahler Frenet curves, K\"ahler circles, structure torsion, Hopf hypersurfaces