Essential Killing helices of order less than five on a non-flat complex space form.(English)Zbl 1245.53040

Some extrinsic helical properties of curves on submanifolds in a non-flat complex space form characterize these submanifolds. Every helix on a real space form is generated by some Killing vector field, which is not the case for a non-flat complex space form. In a previous paper, conditions for helices on a non-flat complex space form to be generated by some Killing vector field were given. In another paper, it was shown that there are bounded helices of proper order 3 in a complex hyperbolic space. In a Euclidean space and in a real hyperbolic space, all helices of proper order 3 are unbounded.
In this paper, helices of proper order less than 5 on a non-flat complex space form generated by some Killing vector field and lying in some totally geodesic complex plane are studied. These helices are called essential and Killing. The properties of boundedness, closedness and length of these helices are studied. In a previous paper, lamination structures on moduli spaces of helices of proper order less than 4 in a real space form have been given. As a continuation, in the present paper, lamination structures on moduli spaces of Killing helices of proper order less than 5 on non-flat complex space forms are associated with the length spectrum. The structure of these moduli spaces for complex space forms is different from those of the real case.

MSC:

 53C22 Geodesics in global differential geometry 53C12 Foliations (differential geometric aspects)
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References:

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