Classification of locally homogeneous affine connections with arbitrary torsion on 2-dimensional manifolds

Four-dimensional neutral signature self-dual gradient Ricci solitons

“…Further, if a connection is of Type B, all Christoffel symbols are given by (27) in some local coordinates (x 1 , x 2 ) and one has that {x = ∂ x 2 , y = x 1 ∂ x 1 + x 2 ∂ x 2 } are linearly independent affine Killing vector fields satisfying [x, y] = x. The converse statement holds true, and types A and B above are characterized by the existence of linearly independent affine Killing vector fields x, y satisfying [x, y] = 0 and [x, y] = x, respectively [2]. It is worth emphasizing here that types A and B are not disjoint.…”

“…Let f ∈ C ∞ (M ), we say that the triple (M, g, f ) is a gradient Ricci soliton if the following equation is satisfied: (1) Hes f +ρ = λ g , for some λ ∈ R, where ρ is the Ricci tensor, and Hes f is the Hessian tensor acting on f defined by (2) Hes f (x, y) = (∇ x df )(y) = xy(f ) − (∇ x y)(f ) .…”

“…[3,12,14]). We emphasize here that the geometry of (anti)-self-dual conformal structures is much richer in neutral signature (2,2) than the corresponding Riemannian one (see, for example [17] and the references therein).…”

“…Affine (2). In order to state our results, we briefly recall some facts of the geometry of cotangent bundles.…”

The moduli space of Type A surfaces with torsion and non-singular symmetric Ricci tensor

“…The following result was first proved in the torsion free setting by Opozda [26] and subsequently extended to surfaces with torsion by Arias-Marco and Kowalski [4], see also [4,11,16,21,22,27] for related work. Theorem 1.1.…”