  A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed
with a homothetic vector field $\xi$. We characterize global selfsimilar
manifolds and describe the structure of local selfsimilar manifolds. We prove
that any selfsimilar manifold with a potential homothetic vector field is a
conical Riemannian manifold or a Eucledean space. A radiant Hessian manifold is
selfsimilar Hessian manifold $\left(M,\nabla,g,\xi\right)$ such that
$\nabla\xi=\lambda \text{Id}$. We prove that any selfsimilar Hessian manifold
with a potential homothetic vector field is locally isomorphic to a product
radiant Hessian manifolds and describe the local structure of radiant
selfsimialar Hessian manifolds.
