Following the study on volume of indicatrices in a real Finsler space, in this paper we are investigating some volume properties of the indicatrix considered in an arbitrary fixed point of a complex Finsler manifold. Since for each point of a complex Finsler space the indicatrix is an embedded CR-hypersurface of the punctured holomorphic tangent bundle, by means of its normal vector, the volume element of the indicatrix is determined. | Turk J Math (2017) 41: 267 – 281 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On the volume of the indicatrix of a complex Finsler space Elena POPOVICI∗ Department of Mathematics and Informatics, Faculty of Mathematics and Informatics, Transilvania University of Bra¸sov, Romania Received: • Accepted/Published Online: • Final Version: Abstract: Following the study on volume of indicatrices in a real Finsler space, in this paper we are investigating some volume properties of the indicatrix considered in an arbitrary fixed point of a complex Finsler manifold. Since for each point of a complex Finsler space the indicatrix is an embedded CR-hypersurface of the punctured holomorphic tangent bundle, by means of its normal vector, the volume element of the indicatrix is determined. Thus, the volume function is pointed out and its variation is studied. Conditions under which the volume is constant are also determined and some classes of complex Finsler spaces with constant indicatrix volume are given. Moreover, the length of the complex indicatrix of Riemann surfaces is found to be constant. In addition, considering submersions from the complex indicatrix onto almost Hermitian surfaces, we obtain that the volume of the submersed manifold is constant. Key words: Complex Finsler space, complex indicatrix, volume element, volume variation, indicatrix length, submersion volume 1. Introduction The study of the unit tangent sphere, or indicatrix, in real Finsler spaces is one of interest ([15, 19, 21, 22], etc.), mainly because it is a compact and strictly convex set surrounding the origin. For example, the indicatrix plays a special role in the volume definition of a Finsler space. However, in the present paper, based on some ideas from the real case, the volume element and volume function of the indicatrix in a complex Finsler manifold (M, F ) are .
