Abstract

The study of circles is one of the interesting objects in differential geometry. A curve γ(s) on a Riemannian manifold M parametrized by its arc length 5 is called a circle, if there exists a field of unit vectors Ys along the curve which satisfies, together with the unit tangent vectors Xs — Ϋ(s)9 the differential equations : FSXS = kYs and FsYs= — kXs, where k is a positive constant, which is called the curvature of the circle γ(s) and Vs denotes the covariant differentiation along γ(s) with respect to the Aiemannian connection V of M. For given a point lEJIί, orthonormal pair of vectors u, v^ TXM and for any given positive constant k, we have a unique circle γ(s) such that γ(0)=x, γ(0) = u and (Psγ(s))s=o=kv. It is known that in a complete Riemannian manifold every circle can be defined for o o < 5 < o o (Cf. [6]). The study of global behaviours of circles is very interesting. However there are few results in this direction except for the global existence theorem. In general, a circle in a Riemannian manifold is not closed. Here we call a circle γ(s) closed if there exists So with 7(so) = /(0), XSo Xo and YSo— Yo. Of course, any circles in Euclidean m-space E are closed. And also any circles in Euclidean m-sphere S(c) are closed. But in the case of a real hyperbolic space H(c), there exist many open circles. It is well-known that in H(c) circles with curvature not exceeding v |c | are open and circles with curvature greater than v \c are closed (cf. [3]). That is, in H{c) the answer to the question Is a circle γ(s) closed? depends on its curvature. In this paper, we are concerned with circles in an n-dimensional complex projective space CP(c) of constant holomorphic sectional curvature c. In Section 1, by using Submanifold Theory we give an interesting family of open circles and closed circles with the same curvature /c/2^ in CP(c). In his paper [5], Naitoh

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