On the partial Ricci curvature of foliations

TL;DR

This paper studies how to prescribe partial Ricci curvature on conformally flat manifolds with orthogonal distributions, providing explicit solutions and exploring geometric optimization related to distributions.

Contribution

It offers explicit solutions for the partial Ricci curvature equations on conformally flat manifolds with orthogonal distributions, and investigates geometric optimization problems.

Findings

Explicit solutions for partial Ricci curvature equations.

Conditions for symmetric tensors to admit conformal metrics solving the equations.

Examples of prescribing mixed scalar curvature and optimizing distribution placement.

Abstract

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold $(M^n, g)$ endowed with the complementary orthogonal distributions $D_1$ and $D_2$. We provide conditions for symmetric $(0,2)$-tensors $T$ of a simple form (defined on $M$) to admit metrics $\tilde g$, conformal to $g$, that solve the partial Ricci equations. The solutions are given explicitly. Using above solutions, we also give examples to the problem of prescribing the mixed scalar curvature related to $D_i$. In aim to find "optimally placed" distributions, we calculate the variations of the total mixed scalar curvature (where again the partial Ricci curvature plays a key role), and give examples concerning minimization of a total energy and bending of a distribution.