{
  "id": "1912.12986",
  "version": "v1",
  "published": "2019-12-30T16:26:43.000Z",
  "updated": "2019-12-30T16:26:43.000Z",
  "title": "On the $R_{f_B}$ condition for the (2+m)-Einstein warped product manifolds and some almost-complex structure cases",
  "authors": [
    "Alexander Pigazzini",
    "Cenap Ozel",
    "Saeid Jafari"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "For the studied cases in [14], the author showed that having the $f$-curvature-Base ($R_{f_B}$) is equal to requiring a flat metric on the base-manifold. In [15] the authors used the condition $R_{f_B}$ on (2+m)-Einstein warped product manifold, in the case $\\lambda=\\mu=0$, to built a new kind of manifolds, composed by positive-dimensional manifold and negative-dimensional manifold, the so called POLJ-manifolds. This have opened up the possibility of seeking the existence of others $(n, m)$-POLJ manifolds. The aim of this paper is to extend the work done in [14] to $m$-dimensional fiber, and show if the value of $m$ can influence the result, i.e., finding base-manifolds with non-flat metric for $dimF \\neq 2$ and, at the same time, check if in these cases can exists other kinds of (2, m)-POLJ manifolds. In the second part of the paper, we put in relation a particular type of almost-Hermitian manifolds with POLJ-manifolds and we proceeded to give the definitions of: almost Hermitian-POLJ manifold (aH-POLJs) and pseudo almost Hermitian-POLJ manifolds (paH-POLJs). As a result, we find out that the dimension of fiber-manifold does not change the result of [14]. Moreover, we show that the (2,m)-almost hyperbolic Hermitian-POLJ manifolds with flat fiber and quasi-constant curvature, and (2, m)-quasi-Einstein almost hyperbolic Hermitian-POLJ manifolds with flat fiber and quasi-constant curvature can not exist. Finally we add a Special Remark about the possible use of the $(n,-n)$-POLJs in superconductor graphene theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-30T16:26:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "warped product manifold",
      "almost-complex structure cases",
      "hyperbolic hermitian-polj manifolds",
      "flat fiber",
      "quasi-constant curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}