Liviu Popescu

DUAL STRUCTURES ON THE PROLONGATIONS OF A LIE ALGEBROID

Abstract In the present paper we study the properties of dual structures on the prolongations of a Lie algebroid. We introduce the dynamical covariant derivative on Lie algebroids and prove that the nonlinear connection induced by a regular Lagrangian is compatible with the metric and symplectic structures. The notions of mechanical structure and semi-Hamiltonian section are introduced on the prolongation of the Lie algebroid to its dual bundle and their properties are investigated. Finally, we prove the equivalence between the metric nonlinear connection and semi-Hamiltonian section, using the Legendre transformation induced by a regular Hamiltonian.

Metric Non-Linear Connections on the Prolongation of a Lie Algebroid to its Dual Bundle

Metric Non-Linear Connections on the Prolongation of a Lie Algebroid to its Dual Bundle In the present paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied. The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found. We prove that the nonlinear connection induced by a regular Hamiltonian on a Lie algebroid is the unique connection which is compatible with the metric and symplectic structures.

Symmetries of second order differential equations on Lie algebroids

Abstract In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.

Geometrical structures on the cotangent bundle

In this paper we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on

Applications of Adapted Frames to the Geometry of Black Holes

T^{*}M

Geometrical structures on Lie algebroids

fix a nonlinear connection for a given

A Note on Poisson Lie Algebroids

\mathcal{J}

Nonlinear connections on dual Lie algebroids

-regular vector field. Using the Legendre transformation induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector field on

LIE ALGEBROIDS FRAMEWORK FOR DISTRIBUTIONAL SYSTEMS

T^{*}M

Metric nonlinear connections on Lie algebroids

corresponds to a semispray on