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f-left-invariant Riemannian metrics on Lie groups

Hamid Reza Salimi Moghaddam

Abstract

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Riemannian metric to be a Ricci soliton. Using this result, for any expansion constant $λ$, we obtain a flat gradient Ricci soliton on some two and three-dimensional non-abelian Lie groups. We give an example of a non-flat steady gradient Ricci soliton and construct some examples of non-flat shrinking, steady, and expanding non-gradient Ricci solitons on the non-abelian Lie group $\Bbb{R}\rtimes\Bbb{R}^{+}$. Finally, we study f-left-invariant Riemannian metrics on the Heisenberg group.

f-left-invariant Riemannian metrics on Lie groups

Abstract

With a f-left-invariant Riemannian metric on a Lie group , we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor . In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Riemannian metric to be a Ricci soliton. Using this result, for any expansion constant , we obtain a flat gradient Ricci soliton on some two and three-dimensional non-abelian Lie groups. We give an example of a non-flat steady gradient Ricci soliton and construct some examples of non-flat shrinking, steady, and expanding non-gradient Ricci solitons on the non-abelian Lie group . Finally, we study f-left-invariant Riemannian metrics on the Heisenberg group.

Paper Structure

This paper contains 8 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Riemannian metrics which are conformally equivalent to the left-invariant metrics
  3. f-left-invariant Ricci solitons on two-dimensional Lie groups
  4. Lie group $G=\Bbb{R}^2$
  5. Lie group $G=\Bbb{R}\rtimes\Bbb{R}^{+}$
  6. f-left-invariant Ricci solitons on $\Bbb{R}^2\rtimes\Bbb{R}^{+}$ and the Heisenberg group
  7. Lie group $G=\Bbb{R}^2\rtimes\Bbb{R}^{+}$
  8. The Heisenberg group $H_3$

Key Result

Lemma 2.4

Suppose that $f$ is a smooth real positive function on a Lie group $G$ such that $f(e)=1$. A Riemannian metric $\tilde{g}$ on $G$ is $f$-left-invariant if and only if, for any $a,b\in G$, Similarly, the Riemannian metric $\tilde{g}$ is $f$-right-invariant if and only if for all $a,b\in G$.

Theorems & Definitions (26)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Lemma 2.7
  • proof
  • ...and 16 more