Having as a model the metric contact case of V. Brınzanescu; R. Slobodeanu, we study two similar subjects in the paracontact (metric) geometry: Distributions that are invariant with respect to the structure endomorphism φ, the class of vector fields of holomorphic type. As examples we consider both the 3-dimensional case and the general dimensional case through a Heisenberg-type structure inspired also by contact geometry. | Turk J Math (2015) 39: 467 – 476 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Invariant distributions and holomorphic vector fields in paracontact geometry Mircea CRASMAREANU1 , Laurian-Ioan PIS ¸ CORAN2,∗ Faculty of Mathematics, University “Al. I. Cuza”, Ia¸si, Romania 2 North University Center of Baia Mare, Technical University of Cluj, Baia Mare, Romania 1 Received: • Accepted/Published Online: • Printed: Abstract: Having as a model the metric contact case of V. Brˆınz˘ anescu; R. Slobodeanu, we study two similar subjects in the paracontact (metric) geometry: a) distributions that are invariant with respect to the structure endomorphism φ ; b) the class of vector fields of holomorphic type. As examples we consider both the 3 -dimensional case and the general dimensional case through a Heisenberg-type structure inspired also by contact geometry. Key words: Paracontact metric manifold, invariant distribution, paracontact-holomorphic vector field 1. Introduction Paracontact geometry [7, 13] appears as a natural counterpart of the contact geometry in [9]. Compared with the huge literature in (metric) contact geometry, it seems that new studies are necessary in almost paracontact geometry; a very interesting paper connecting these fields is [5]. The present work is another step in this direction, more precisely from the point of view of some subjects of [4]. The first section deals with the distributions V , which are invariant with respect to the structure endomorphism φ, one trivial example being the canonical distribution D provided by the annihilator of the paracontact 1 -form η . As in the contact case, the characteristic vector field ξ must belong to V or V ⊥ . Two important tools in this study are the second fundamental form and the integrability tensor field, both satisfying important (skew)-commutation formulas in the paracontact .
