The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. For instance, we obtain that the E-Chern form of E 1,0 associated to an almost complex connection ∇ on E can be expressed in terms of the matrix JER, where JE is the almost complex structure of E and R is the curvature of ∇. Also, we consider a metric product connection associated to an almost Hermitian Lie algebroid and we prove that the mean curvature section of E 0,1 vanishes and the second fundamental 2-form section of E 0,1 vanishes iff the Lie algebroid is Hermitian.

In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3,4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.

In this paper we study some problems related to a vertical Liouville distribution (called vertical Liouville-Hamilton distribution) on the cotangent bundle of a Cartan space. We study the existence of some linear connections of Vrȃnceanu type on Cartan spaces related to some foliated structures. Also, we identify a certain (n, 2n − 1)codimensional subfoliation (F V , F C * ) on T * M 0 given by vertical foliation F V and the line foliation F C * spanned by the vertical Liouville-Hamilton vector field C * and we give a triplet of basic connections adapted to this subfoliation. Finally, using the vertical Liouville foliation F V C * and the natural almost complex structure on T * M 0 we study some aspects concerning the cohomology of c-indicatrix cotangent bundle.

Vertical Chern Type Classes on Complex Finsler BundlesIn the present paper, we define vertical Chern type classes on complex Finsler bundles, as an extension of thev-cohomology groups theory on complex Finsler manifolds. These classes are introduced in a classical way by using closed differential forms with respect to the conjugated vertical differential in terms of the vertical curvature form of Chern-Finsler linear connection. Also, some invariance properties of these classes are studied.

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