  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.
