Search Results for « Affine sub-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.

We are interested in studying the affine and the affine polarized k-symplectic manifolds exploiting the action of the polarized k-symplectic group Sp(k,n;R), especially when it acts properly and discontinuously without fixed point in order to construct a particular class of affine polarized k-symplectic manifolds of odd dimension 2k’+1 with k’∈N∗ which will be a generalization of the case of dimension 3 studied in [8].

In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than n(n+1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto symmetric manifolds.