Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Einstein multiply warped products and generalized Kasner manifolds with multidimensional base

Fernando Dobarro, Carolina Rey

TL;DR

The paper develops a comprehensive framework for Einstein multiply warped products, with emphasis on generalized Kasner manifolds, by deriving explicit Ricci and scalar-curvature formulas and translating Einstein conditions into tractable systems for base, fibers, and warping functions. It separates the analysis into single- and multi-fiber cases, and further treats compact and noncompact bases, obtaining sharp relations between base geometry, warping Hessians, and fiber curvatures; it also provides an eigenvalue interpretation for the compact-base Kasner-type setting. A key outcome is a set of necessary and (in favorable cases) sufficient conditions that exclude nontrivial Einstein mwp in several regimes, along with precise bounds on the Einstein parameter $\lambda$ and its relation to total scalar curvature. The results illuminate when nontrivial gK Einstein metrics can exist and connect the problem to spectral properties of the base, offering directions for future work in higher-fiber configurations and nonlinear elliptic analyses related to Besse’s conjecture.

Abstract

The purpose of this paper is to provide conditions for the existence or non existence of non trivial Einstein multiply warped products, specially of generalised Kasner type; as well as to show estimates of the Einstein parameter that condition the existence of such metrics.

Einstein multiply warped products and generalized Kasner manifolds with multidimensional base

TL;DR

The paper develops a comprehensive framework for Einstein multiply warped products, with emphasis on generalized Kasner manifolds, by deriving explicit Ricci and scalar-curvature formulas and translating Einstein conditions into tractable systems for base, fibers, and warping functions. It separates the analysis into single- and multi-fiber cases, and further treats compact and noncompact bases, obtaining sharp relations between base geometry, warping Hessians, and fiber curvatures; it also provides an eigenvalue interpretation for the compact-base Kasner-type setting. A key outcome is a set of necessary and (in favorable cases) sufficient conditions that exclude nontrivial Einstein mwp in several regimes, along with precise bounds on the Einstein parameter and its relation to total scalar curvature. The results illuminate when nontrivial gK Einstein metrics can exist and connect the problem to spectral properties of the base, offering directions for future work in higher-fiber configurations and nonlinear elliptic analyses related to Besse’s conjecture.

Abstract

The purpose of this paper is to provide conditions for the existence or non existence of non trivial Einstein multiply warped products, specially of generalised Kasner type; as well as to show estimates of the Einstein parameter that condition the existence of such metrics.
Paper Structure (12 sections, 16 theorems, 120 equations)

This paper contains 12 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries about Multiply Warped Products
  3. Generalized Kasner Semi-Riemannian Manifolds
  4. Einstein multiple warped manifolds
  5. Case of one single fiber
  6. Case of multiple fibers
  7. Einstein gK manifolds with compact Riemannian base
  8. Conclusions and future developments
  9. Case Lg, i.e. singly warped product
  10. mwp with compact Riemannian base
  11. Generalized Kasner manifold with compact Riemannian base
  12. What happens if Lg and Lg?

Key Result

Theorem 1.1

Let $\left(B^{n}, g_{B}\right)$ be a compact Riemannian manifold of dimension $n \geqq 2$. Suppose that $b_i$ are non-constant smooth functions on $B$ satisfying (Einst-mwp-a) for a constant $\lambda \in \mathbb{R}$ and some positive integers $s_i$, $i=1,\dots,m$. Then $b_1, \dots, b_m$ satisfy Note

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Proposition 3.6
  • ...and 27 more