On the Relationships Between Affine Manifolds and Complex Manifolds

In this work, we study the relationships between affine manifolds and complex manifolds. We prove that a linear connection and a Riemannian metric on an affine manifold M of dimension n induce a complex structure and a Hermitian metric on both the product M×Rn and the tangent bundle TM. We also discuss some geometric relations among the affine manifold M and the Hermitian manifolds M×Rn and TM. As an application in analysis, we obtain the generalized maximum principle on complete affine Riemannian manifolds, which can be used to study partial differential equations. It is worth noting that Hessian manifolds, as a special case of affine manifolds, have broad application potential in statistics and information geometry.