Mathematics
On the principal Ricci curvatures of a Riemannian 3-manifold
Document Type
Article
Abstract
We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: In particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (-μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.
Publication Title
Advances in Geometry
Publication Date
4-24-2019
Volume
19
Issue
2
First Page
251
Last Page
262
ISSN
1615-715X
DOI
10.1515/advgeom-2018-0020
Keywords
Ricci curvature, Riemannian 3-manifold
Repository Citation
Aazami, Amir Babak and Melby-Thompson, Charles M., "On the principal Ricci curvatures of a Riemannian 3-manifold" (2019). Mathematics. 27.
https://commons.clarku.edu/mathematics/27
