Four-dimensional neutral signature self-dual gradient Ricci solitons

Abstract

Abstract. We describe the local structure of self-dual gradient Ricci solitons in neutral signature. If the Ricci soliton is non-isotropic then it is locally conformally flat and locally isometric to a warped product of the form I ×ϕ N (c), where N (c) is a space of constant curvature. If the Ricci soliton is isotropic, then it is locally isometric to the cotangent bundle of an affine surface equipped with the Riemannian extension of the connection, and the Ricci soliton is described by the underlying affine structure. This provides examples of self-dual gradient Ricci solitons which are not locally conformally flat.

Special Structures on General Rotational Surfaces in Pseudo-Euclidean 4-Space With Neutral Metric

“…As usual, (E 4 2 , , ) denotes the pseudo-Euclidean 4-space of signature (+, +, −, −) endowed with the canonical pseudo-Euclidean (neutral) metric given in a rectangular coordinate system (x 1 , x 2 , x 3 , x 4 ) by , = dx 2 1 + dx 2 2 − dx 2 3 − dx 2 4 . A vector v ∈ E 4 2 can have one of the following three casual characters: spacelike, if v, v > 0 or v = 0, timelike if v, v < 0, and lightlike if v, v = 0 and v = 0. This terminology is inspired by General Relativity and the Minkowski 4-space E 4 1 .…”

“…A vector v ∈ E 4 2 can have one of the following three casual characters: spacelike, if v, v > 0 or v = 0, timelike if v, v < 0, and lightlike if v, v = 0 and v = 0. This terminology is inspired by General Relativity and the Minkowski 4-space E 4 1 . A surface M in E 4 2 is called spacelike (resp.…”

“…This terminology is inspired by General Relativity and the Minkowski 4-space E 4 1 . A surface M in E 4 2 is called spacelike (resp. timelike or Lorentz ) if , induces a Riemannian (resp.…”

“…A classification of the spacelike and timelike Weingarten rotational surfaces in S 3 1 was obtained in [20]. The class of Chen spacelike rotational surfaces of hyperbolic or elliptic type in E 4 1 was described in [14]. In the case a = 0 and b = 0, the surfaces defined by (3.2) are analogous to the general rotational surfaces in the Euclidean space E 4 .…”

“…In the case a = 0 and b = 0, the surfaces defined by (3.2) are analogous to the general rotational surfaces in the Euclidean space E 4 . In [15], G. Ganchev and the second author studied spacelike general rotational surfaces with plane meridian curves in E 4 1 and described analytically the following main subclasses: minimal, flat, with flat normal connection. Spacelike general rotational surfaces in E 4 1 with meridian curves lying in 2-dimensional planes and having pointwise 1-type Gauss map were studied in [10].…”