Nonholonomic Jet Deformations and Exact Solutions for Modified Ricci Soliton and Einstein Equations

TL;DR

This paper develops methods to generate exact solutions for modified Einstein and Ricci soliton equations using nonholonomic jet deformations, enabling new insights into off-diagonal metrics and their symmetries.

Contribution

It introduces a framework for constructing explicit off-diagonal solutions with jet variables and nonholonomic structures, extending classical solutions like Kerr metrics.

Findings

Derived classes of exact solutions with nonholonomic jet structures.

Demonstrated how to impose zero torsion to recover Levi-Civita configurations.

Showed relations between solutions via half-conformal and jet deformations.

Abstract

Let $\mathbf{g}$ be a pseudo--Riemanian metric of arbitrary signature on a manifold $\mathbf{V}$ with conventional $n+n$ dimensional splitting, $\ n\geq 2,$ determined by a nonholonomic (non--integrable) distribution $\mathcal{N}$ defining a generalized (nonlinear) connection and associated nonholonomic frame structures. We shall work with a correspondingly adapted linear metric compatible connection $\hat{\mathbf{D}}$ and its nonzero torsion $\hat{\mathcal{T}}$, both completely determined by $\mathbf{g}$. Our first goal is to prove that there are certain generalized frame and/or jet transforms and prolongations with $(\mathbf{g},\mathbf{V})\rightarrow (\hat{\mathbf{g}},\hat{\mathbf{V}})$ into explicit classes of solutions of some generalized Einstein equations $\hat{\mathbf{R}}\mathit{ic}=\Lambda \hat{\mathbf{g}}$, $\Lambda =const$, encoding various types of nonholonomic) Ricci soliton configurations and/or jet variables and symmetries. The second goal is to solve additional constraint equations for zero torsion, $\hat{\mathcal{T}}=0$, on generalized solutions constructed in explicit forms with jet variables and extract Levi-Civita configurations. This allows us to find generic off-diagonal exact solutions depending on all space time coordinates on $\mathbf{V}$ via generating and integration functions and various classes of constant jet parameters and associated symmetries. We shall study (third goal) how such generalized metrics and connections can be related by so-called "half-conformal" and/ or jet deformations of certain sub-classes of solutions with one, or two, Killing symmetries. Finally, there are considered some examples when exact solutions are constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by nonholonomic jet structures and generalized connections.