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Geodesic spheres in a nonflat complex space form and their integral curves of characteristic vector fields. (English) Zbl 1139.53007
Let \((\widetilde{M}, < , >)\) be a Kähler manifold with complex structure \(J\). A smooth curve \(\nu\) on \(\widetilde{M}\) parametrized by its arc length is a Kähler circle if it satisfies either \(\widetilde{\nabla}_{\dot{\nu}}{\dot{\nu}} = kJ{\dot{\nu}}\) or \(\widetilde{\nabla}_{\dot{\nu}}{\dot{\nu}} = -kJ{\dot{\nu}}\).
The authors study the geodesic sphere in a nonflat complex space form \(M^n(c; C)\) and on the geodesic sphere a classification of smooth curves whose extrinsic shape are Kähler circles in \(M^n(c; C)\), \(c\neq 0\) is obtained. By using the extrinsic shape of the integral curves of the characteristic vector field on their geodesic sphere a characterisation of complex space forms among Kähler manifolds is given.
53B25 Local submanifolds
53C40 Global submanifolds
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