## Abstract

The mixed scalar curvature is the simplest curvature invariant of a foliated Riemannian manifold. We explore the problem of prescribing the leafwise constant mixed scalar curvature of a foliated Riemann–Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibered manifold, we reduce the problem to solution of a nonlinear leafwise elliptic equation for the conformal factor. We are looking for its solutions that are stable stationary solutions of the associated parabolic equation. Our main tool is using of majorizing and minorizing nonlinear heat equations with constant coefficients and application of comparison theorems for solutions of Cauchy's problem for parabolic equations.

Original language | English |
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Pages (from-to) | 42-67 |

Number of pages | 26 |

Journal | Journal of Geometry and Physics |

Volume | 126 |

DOIs | |

State | Published - Mar 2018 |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier B.V.

## Keywords

- Conformal
- Foliation
- Mixed scalar curvature
- Nonlinear heat equation
- Riemann–Cartan manifold
- Schrödinger operator

## ASJC Scopus subject areas

- Mathematical Physics
- General Physics and Astronomy
- Geometry and Topology