On the partial Ricci curvature of foliations

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.

arXiv comment20 pages

Cited by in corpus (6)

• The mixed Yamabe problem for harmonic foliations
• The mixed Einstein-Hilbert action and extrinsic geometry of foliated manifolds
• The partial Ricci flow on one-dimensional foliations
• The Einstein-Hilbert type action on foliated pseudo-Riemannian manifolds
• The proof of Gromoll-Walschap conjecture
• The Partial Ricci Flow on $\mathfrak{g}$-foliations