On the Exponentiality of Affine Symmetric Spaces

An affine symmetric space $G/H$ is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in $G/H$. For the group space $(G\times G)/G_{\rm diag}$ of a Lie group $G$, these properties are equivalent to the exponentiality and weak exponentiality of $G$, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.