Differential geometry of circles in a complex projective space (Q2715742)

A smooth curve \(\gamma(s)\) in a Riemannian manifold parametrized by a length parameter \(s\) is called a circle of curvature \(\kappa\) if there exists a vector field \(Y\) of unit length along the curve satisfying the conditions \(\nabla_XX=\kappa Y\) and \(\nabla_XY=-\kappa X\), where \(X\) is the unit tangent vector of the curve. It is called a closed circle if it is periodic: \(\gamma(s+p)=\gamma(s)\) for all \(s\). The periods \(p\) form the length spectrum. A length spectrum is said to be simple if there exists a closed circle unique up to the isometry group. This paper gives a detailed study of closed circles and length spectra. In particular, it gives an answer to the question which length spectrum is simple.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].