# Maximally-warped metrics with harmonic curvature

## TL;DR

This paper characterizes Riemannian manifolds with harmonic curvature that admit many warped-product decompositions and have Ricci tensors with simple eigenvalues, revealing a finite-dimensional moduli space of such geometries.

## Contribution

It provides a detailed description of the local structure of these manifolds and establishes the existence of a finite-dimensional moduli space of their local-isometry types.

## Key findings

- Manifolds with harmonic curvature and maximal warped-product decompositions are classified.
- A finite-dimensional moduli space of such manifolds exists in each dimension > 2.
- Examples include irreducible metrics that are neither Ricci-parallel nor conformally flat in higher dimensions.

## Abstract

We describe the local structure of Riemannian manifolds with harmonic curvature which admit a maximum number, in a well-defined sense, of local warped-product decompositions, and at the same time their Ricci tensor has, at some point, only simple eigenvalues. We also prove that, in every given dimension greater than two, the local-isometry types of such manifolds form a finite-dimensional moduli space, and a nonempty open subset of this moduli space is realized by locally irreducible complete metrics which are neither Ricci-parallel, nor -- for dimensions greater than three -- conformally flat.