Diagonalizing the Ricci Tensor

TL;DR

This paper characterizes 'nice' bases in semisimple Lie algebras of compact type where diagonal left-invariant metrics have diagonal Ricci tensors, extending previous nilpotent Lie algebra results and analyzing Ricci flow on certain manifolds.

Contribution

It provides a Lie algebraic condition for diagonal Ricci tensors in compact type semisimple Lie algebras, extending prior work on nilpotent cases and exploring Ricci flow dynamics.

Findings

Characterization of 'nice' bases in compact Lie algebras

Extension of Ricci tensor diagonalization results to homogeneous spaces

Analysis of Ricci flow behavior on cohomogeneity one manifolds

Abstract

We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being "nice". Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will on nilpotent Lie algebras. The result follows from a more general characterization for diagonalizing the Ricci tensor for homogeneous spaces. Finally, we also study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.