Weyl homogeneous manifolds modeled on compact Lie groups

TL;DR

This paper proves that Weyl homogeneous manifolds modeled on certain symmetric spaces are conformally equivalent to those spaces, extending understanding of their geometric structure.

Contribution

It establishes that Weyl homogeneous manifolds modeled on irreducible symmetric spaces of types II or IV are conformally equivalent to the model space.

Findings

Weyl homogeneous manifolds modeled on specific symmetric spaces are conformally equivalent to the model.

The result applies to manifolds of dimension at least 4.

It characterizes the geometric structure of these manifolds in terms of conformal equivalence.

Abstract

A Riemannian manifold is called Weyl homogeneous, if its Weyl tensors at any two points are "the same", up to a positive multiple. A Weyl homogeneous manifold is modeled on a homogeneous space $M_0$, if its Weyl tensor at every point is "the same" as the Weyl tensor of $M_0$, up to a positive multiple. We prove that a Weyl homogeneous manifold $M^n, n \ge 4$, modeled on an irreducible symmetric space $M_0$ of types II or IV (compact simple Lie group with a bi-invariant metric or its noncompact dual) is conformally equivalent to $M_0$.