Mediterr. J. Math.
Springer International Publishing AG 2017
Connections on the Total Space
of a Holomorphic Lie Algebroid
Alexandru Ionescu and Gheorghe Munteanu
Abstract. The main purpose of the paper is the study of the total space
of a holomorphic Lie algebroid E. The paper is structured in three parts.
In the ﬁrst section, we brieﬂy introduce basic notions on holomorphic
Lie algebroids. The local expressions are written and the complexiﬁed
holomorphic bundle is introduced. The second section presents two ap-
proaches on the study of the geometry of the complex manifold E.The
ﬁrst part contains the study of the tangent bundle T
E = T
E ⊕ T
and its link, via the tangent anchor map, with the complexiﬁed tangent
M) ⊕ T
M). A holomorphic Lie algebroid
structure is emphasized on T
E. A special study is made for integral
curves of a spray on T
E.Theorem2.8 gives the coeﬃcients of a spray,
called canonical, obtained from a complex Lagrangian on T
second part of section two, we study the holomorphic prolongation T
of the Lie algebroid E. In the third section, we study how a complex
Lagrange (Finsler) structure on T
M induces a Lagrangian structure on
E. Three particular cases are analysed by the rank of the anchor map,
the dimensions of manifold M, and those of the ﬁbres. We obtain the
correspondent on E of the Chern–Lagrange nonlinear connection from
Mathematics Subject Classiﬁcation. Primary 17B66; Secondary 53B40.
Keywords. Holomorphic Lie algebroid, anchor map, spray, nonlinear
connection, prolongation, Lagrangian structures.
Lie algebroids are a generalization of Lie algebras and vector bundles. They
are anchored vector bundles with a Lie bracket deﬁned on the modules of
sections. The Lie bracket is induced on the algebroid from the standard Lie
bracket on the tangent bundle of the basis manifold. Lie algebroids provide a
natural setting in which one can develop the theory of diﬀerential operators
such as the exterior derivative of forms and the Lie derivative with respect
to a vector ﬁeld. This setting is more general than that of the tangent and
cotangent bundles of a smooth manifold and their exterior powers.