[Paper Review] Conformal de Rham decomposition of Riemannian manifolds

Ruy Tojeiro|ArXiv.org|Apr 23, 2004

Geometric Analysis and Curvature Flows7 references19 citations

TL;DR

This paper establishes conformal analogues of de Rham's and Hiepko's decomposition theorems for Riemannian manifolds, providing necessary and sufficient conditions for a manifold to be locally conformal to a Riemannian or warped product. The key contribution is a characterization of Riemannian manifolds admitting a Codazzi tensor with two distinct eigenvalues, showing they are locally conformal to twisted products, with explicit classification when eigenvalues satisfy functional relations and are constant along their eigendistributions.

ABSTRACT

We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We also obtain other related de Rham-type decomposition theorems. As an application, we study Riemannian manifolds that admit a Codazzi tensor with two distinct eigenvalues everywhere.

Motivation & Objective

• To extend de Rham's and Hiepko's local decomposition theorems to the conformal setting, characterizing when a Riemannian manifold is locally conformal to a Riemannian or warped product.
• To study orthogonal nets on Riemannian manifolds and determine conditions under which they admit local product representations conformal to Riemannian or warped product metrics.
• To apply the conformal decomposition framework to classify Riemannian manifolds that carry a Codazzi tensor with exactly two distinct eigenvalues.
• To determine the structure of such manifolds when the eigenvalues are constant along their respective eigendistributions and related by a smooth function.

Proposed method

• Introduce the concept of a twisted product metric via a smooth twist-function ρ, generalizing warped and Riemannian products.
• Define a net morphism and net isomorphism to formalize local product structures on manifolds with orthogonal distributions.
• Use the Levi-Civita connection and curvature properties of twisted products to derive conditions for conformal decomposition.
• Apply the Codazzi tensor condition to derive differential equations involving eigenvalues and their gradients, particularly focusing on the case of two distinct eigenvalues.
• Utilize the mean curvature normal and the condition that eigenvalues are constant along eigendistributions to classify the resulting geometric structures.
• Derive the key relation ∫ dμ̃ / (h(μ̃) − μ̃) = log σ to link the warping function σ and the eigenvalue function h(μ̃) in the warped product case.

Experimental results

Research questions

• RQ1 Under what conditions is a Riemannian manifold locally conformal to a Riemannian product of Riemannian manifolds?
• RQ2 When is a Riemannian manifold locally conformal to a warped product of Riemannian manifolds?
• RQ3 What geometric structure arises when a Codazzi tensor with two distinct eigenvalues exists and one eigenvalue is constant along its eigendistribution?
• RQ4 How do functional relations between eigenvalues (e.g., λ = h(μ)) constrain the local geometry of the manifold?
• RQ5 What is the complete classification of Riemannian manifolds admitting a Codazzi tensor with two distinct eigenvalues, both constant along their eigendistributions?

Key findings

• A Riemannian manifold with a Codazzi tensor having two distinct eigenvalues λ and μ, where μ is constant along Eμ and λ = h(μ) for a smooth function h, is locally conformal to a twisted product metric.
• If both λ and μ are constant along their respective eigendistributions, the manifold is locally isometric to a Riemannian product metric.
• When Eλ has rank one and μ is constant along Eμ, the manifold is locally isometric to a warped product metric with warping function σ satisfying ∫ dμ̃ / (h(μ̃) − μ̃) = log σ.
• The eigenvalue function h(μ̃) and the warping function σ are related through a first-order ODE, fully determining the warped product structure.
• The results generalize classical characterizations of isothermic surfaces, with the conformal de Rham theorem extending the classical criterion for isothermic nets in terms of geodesic curvatures.
• The classification includes all such manifolds where the Codazzi tensor's eigenvalues satisfy the functional and constancy conditions, yielding a complete local geometric description.