arXiv subjects
Publications and source records attributed to Silas Longwap.
Let $M$ be an $n$-dimensional manifold with a torsion free affine connection $\nabla$ and let $T^*M$ be the cotangent bundle. In this paper, we consider some of the geometrical aspect of a twisted Riemannian extension which provide a link between the affine geometry of $(M,\nabla)$ and the neutral signature pseudo-Riemannian geometry of $T^*M$. We investigate the spectral geometry of the Szabó operator on $M$ and on $T^*M$.
In this paper, we consider the cyclic parallel Ricci tensor condition, which is a necessary condition for an affine manifold to be Szabó. We show that, in dimension $3$, there are affine manifolds which satisfy the cyclic parallel Ricci tensor but are not Szabó. Conversely, it is known that in dimension $2$, the cyclic parallel Ricci tensor forces the affine manifold to be Szabó. Examples of $3$-dimensional affine Szabo manifolds are also given. Finally, we give some properties of Riemannian extensions defined on the cotangent bundle over an affine Szabó manifold.