Luis C. de Andrés

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle T G of a 2n-dimensional Lie group G, which are left invariant with respect to the Lie group structure on T G induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on G. Using this correspondence and results of (8) and (10), it turns out that when G is nilpotent and four or six dimensional, the cotangent bundle T G always has a hermitian structure. However, we prove that if G is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then T G has no hermitian structure or, equivalently, G has no left invariant generalized complex structure.

Connections on tangent bundles of higher order associated to regular Lagrangians

Given a regular Lagrangian L of order k on M we show that there exists a canonical connection ГL on T2k−1M whose paths are the solutions of the Euler-Lagrange equations for L. If L is the kinetic energy defined by a Riemannian metric then ГL is the Riemannian connection.

CONNECTIONS ON TANGENT BUNDLES OF HIGHER ORDER

Introduction As it is well known the tangent bundle TM of any manifold M carries a canonical integrable almost tangent structure J (see [Go], [Gr], [ YI ] ). By means of J, Grifone [Gr] gave a new definition of (non-homogeneous) connection on M. In fact, a (non-homogeneous) connection on M is a vector 1-form I* on TM such that Jr = J and TJ = -J. The connection T is said to be homogeneous if T is homogeneous as a vector 1-form. Obviously, if r is C°° on all TM, then it is a linear connection (see [V]); so, we must suppose that r