Left-symmetric bialgebroids and their corresponding Manin triples

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer-Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid.