Calibrating Einstein field equations using rippling 3-Riemannian structure for gravity with dark matter effects

The paper presents modifications to Einstein field equations (EFEs) based on the model proposed in the working paper, 'Rippling 3-Riemannian structure describing gravity with dark matter effects'. The model proposes matter and energy are separate entities and energy is a property of three dimensional probabilistic structure spanning space. Mass interacts by binding energy density causing variations in length and time scales, mathematically equivalent to spacetime curvature in general relativity. Gravity is thus described as flow and distribution of energy density. Bounded energy density is the additional source of gravity leading to dark matter observations. The results of the model proposes two EFEs for large and largest scales and further predicts dependence of cosmological constant on space and time coordinates.

Towards a Geometrization of Renormalization Group Histories in Asymptotic Safety

Considering the scale-dependent effective spacetimes implied by the functional renormalization group in d-dimensional quantum Einstein gravity, we discuss the representation of entire evolution histories by means of a single, (d+1)-dimensional manifold furnished with a fixed (pseudo-) Riemannian structure. This “scale-spacetime” carries a natural foliation whose leaves are the ordinary spacetimes seen at a given resolution. We propose a universal form of the higher dimensional metric and discuss its properties. We show that, under precise conditions, this metric is always Ricci flat and admits a homothetic Killing vector field; if the evolving spacetimes are maximally symmetric, their (d+1)-dimensional representative has a vanishing Riemann tensor even. The non-degeneracy of the higher dimensional metric that “geometrizes” a given RG trajectory is linked to a monotonicity requirement for the running of the cosmological constant, which we test in the case of asymptotic safety.

Normal forms for the endpoint map near nice singular curves for rank-two distributions

Given a rank-two sub-Riemannian structure $(M,\Delta)$ and a point $x_0\in M$, a singular curve is a critical point of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space of horizontal curves starting at $x_0$. The typical least degenerate singular curves of these structures are called \emph{regular singular curves}; they are \emph{nice} if their endpoint is not conjugate along $\gamma$. The main goal of this paper is to show that locally around a nice singular curve $\gamma$, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which $F$ writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.

The super-Sasaki metric on the antitangent bundle

We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalization of Sasaki’s construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.

Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version of Pontryagin’s maximum principle. More precisely, the local coordinates on a Riemannian manifold play the role of evolution variables (“multitime”), the Riemannian structure, and the corresponding Levi–Civita linear connection become state variables, while the control variables are represented by some objects with the properties of the Riemann curvature tensor field. Moreover, the constraints are provided by the second order partial differential equations describing the dynamics of the Riemannian structure. The shift from formal analysis to optimal Riemannian control takes deeply into account the symmetries (or anti-symmetries) these geometric elements or equations rely on. In addition, various submanifold integral cost functionals are considered as controlled payoffs.