Professor Dr. Vasile Crucianu was a distinguished member of the Faculty of Mathematics from the “AI. I. Cuza” University of lasi, Romania. Born in Cäbesti, Bacäu county of Romania on November 1, 1931, he graduated the college “Gh. R. Codreanu” in Bîrlad city and he enrolled as student in Mathematics at the “Al.I.Cuza” University of lasi in 1951. When he graduated the Faculty of Mathematics, in 1955, he received a Diploma of Excellence. In the same year he was appointed as assistant and four years later he moved as lecturer at the Chair of Geometry. In 1966 he has got a position of Associated Professor and then, in 1971, he became Full Professor at the same Chair.

In his long university career, Prof. Dr. V. Crucianu was strongly involved in the life of academic community. Thus he served many years as member of Faculty Council and as chairman of the Chair of Geometry.

Along years, Prof. Dr. V. Crucianu taught many courses at undergraduate, graduate and postgraduate levels in almost all fields of Differential Geometry. No matter of the levels his lectures were always rigorous, clear, well organized and opened to new developments.

Prof. Dr. V. Crucianu has always shown talent and passion for research. The results he had obtained were so many and so important that consecrated him as an outstanding researcher in Differential Geometry. He has got the title of “Doctor in Mathematical Sciences” in 1964. A summary of his Ph.D. Thesis was published in the Proceedings of the ICM, Moscow, 1966. For the important results about geometrical objects on manifolds and bundles, Prof. Dr. V. Crucianu received the award “Gh. Tzitzeica” from the Romanian Academy in 1982.

Professor Vasile Crucianu during a conference in 2007

In the nineties, Prof. Dr. V. Crucianu establishes an agreement with a group of geometers from Spain. Several joint papers stating the paracomplex geometry were published. Prof. Dr. V. Crucianu attended many scientific meetings and organized several ones in Romania and abroad. His papers that have been published in important mathematical journals from Romania or from abroad were continued and used in many Ph. Thesis. He is a member of several mathematical societies and reviewer at some mathematical publications. He gave lectures at important universities from Europe.

### Academic Biography

The following short guide to the research of **Prof. Vasile Cruceanu** was written by **Prof. Mihai Anastasiei** in 2006 and published in: **Vasile Cruceanu, Selected Papers, Editura PIM, 2006, 398 p.**

In [2] Vasile Cruceanu introduces the notion of torse forming directions on an nonholonomic manifold (distribution) V_n^m in a manifold V_n endowed with a metric connection. Using it one studies the curvature lines on V_m^n. Considering in particular the concurrent (in Myller sense) or the parallel (in Levi-Civita sense) directions one investigates the Myller nets and Czebyshev nets on nonolonomic manifold V_m^2.

The paper [4] is devoted to a study of surfaces in a new geometry, that of a 3-dimensional space having as fundamental group a subgroup of the projective group whose elements preserve a plane as well as a straight line situated in that plane. Such a space is called parabolic axial-affine. In four dimensions, these spaces play an important role in constructing a geometrical model for Newtonian Mechanics.

In the papers [6], [12] and [15] one studies the geometry of manifolds endowed with a centro-affine connection. These manifolds were introduced by Prof. Vasile Cruceanu in his Ph.D. Thesis. first, the general theory of centro-affine manifolds is developed. Then the cases when these manifolds are endowed with a metric, symplectic or complex structure are minutely analyzed. A theory of hypersurfaces in manifolds with normal centro-affine connections is also developed. And so many results due to Gh. Tzitzeica, Al. Myller, O. Mayer, V. Vagner and others in classical centro-affine geometry are extended to this new framework.

In the paper [7] one studies the nonholonomic manifolds V_n^(n-1) in classical centro-affine geometry. One introduces, among the other things, the Tzitzeica nonholonomic manifolds and the polarity of Pantazi.

In the next paper [9] a theory of submanifolds in manifolds endowed with metrical projective connections is presented. An interesting analogy with the theory of submanifolds in Riemannian spaces of constant curvature is discovered.

The papers [11], [13] were written in collaboration with Radu Miron. Here one studies, on a differentiable manifold, the pairs of connections which are conjugated with respect to a metric, symplectic or a complex structure. This study was extended by Prof. Vasile Cruceanu to Banach manifolds in [21]. Here the results are given in an unitary and elegant manner using the fact that the set of affine connections has a structure of affine module and the operators involved are affine symmetries or projections.

The papers [14], [19], [24] are devoted to an integrated study of projective, affine and centro-affine structures and connections. A subordination relation among these structures and connections as well as a duality principle appear.

A new definition of a connection in a principal bundle using an almost product structure is given in [16].

The papers [25], [31], [39], [41], [48] contain several of the important contributions of Prof. Vasile Cruceanu to the geometry of tangent and cotangent bundles. One defines in a new and very natural way, various lifts of geometrical objects on base to the total space. Also, numerous structures on total space, that are induced by given structures on base, are discovered and their properties are described.

In the short paper [26] Klein‘s viewpoint on geometries with fundamental group is extended by using the theory of categories and functors.

In [29] a very large extension of the notion of affine space is proposed by introducing the notion of G—affine space for any group G. Many notions and properties from classical affine geometry were restated in this general framework.

In the papers [29], [30], [32] the geometry of bundle of (1, 1)-tensors is developed. In is shown that this bundle has a richness of interesting properties especially because of his structure of bundle of Lie algebras.

In the next paper [33] one assumes that a vector bundle endowed with a linear connection splits in a Whitney sum of two subbundles and one establishes the basic equations (of type Gauss and Weingarten) for the pair of these subbundles. The integrability conditions for these equations are established, as well.

In the papers [36], [38] one studies certain classes of curves in Hilbert and Koehler manifolds, that generalize the helices from Euclidian space. These curves are characterized using the torse forming, concurrent or parallel series of directions rigidly associated to them.

The complex centro—affine geometry of hypersurfaces is approached in [40]. One establishes the fundamental equations, the integrability conditions and the associated linear connections. A fundamental theorem for these hypersurfaces is given.

In the paper [42] some properties of the derivations and linear connections which are compatible with the bracket in a Lie algebra bundle are established.

The papers [44], [46] are devoted to the geometry of vector bundles. In the first, [44], one shows that the set of vector fields on the total space, which are invariant by the one parameter group of homotheties becomes a Lie algebra which is isomorphic to the Lie algebra of derivations in the tensor algebra of the bundle. In the second, [46], one modifies, clarfies and extends the definitions of various lifts for vector bundles according to a new and natural viewpoint. Various geometrical characterizations of the lifts are provided.

The papers [43], [47], [48] have been elaborated in a collaboration with the Spanish geometers P. Gadea, J. Masqué, E. Reyes, F. Etayo and P. Fortuny, in the framework of an agreement between the ”Al.I. Cuza” University of Iasi, Romania, and the Consejo Superior de Investigaciones Cientificas from Spain. Here the paracomplex, parahermitian, parakahlerian structures as well as those of electromagnetic type on manifolds and vector bundles are studied. Also, a large and deep account on the researches in this field and on applications of these structures in the theory of field, to symmetric spaces and to homogeneous spaces is provided.

The paper [51], written together with M. Popescu and P. Popescu contains an exhaustive study of the geometrical objects (tensor fields, derivations and linear connections) on the total space E of a fibred manifold \csi, which are projectable and N—projectable, where N is a normalisation of the vertical foliation.

In the papers [52], [54], [55] one studies some geometrical structures on manifolds, called by the author almost hyperproduct, almost complex biproduct and almost product bicomplex. One proves the equivalence of these structures with the other important structures and one determines the metrics and linear connections which are compatible with them. Their compatibility is characterized and examples of manifolds carrying such structures are provided.