Related papers: Embedding the Schwarzschild Ideal Fluid Metric

Certain semi-Riemannian metrics may be decomposed into a Riemannian part and an isochronal part. We use this idea and an idea of Kasner to construct a manifold in 6+1 Minkowski space with a well known metric. The full embedding we display…

We suggest a method to search the embeddings of Riemannian spaces with a high enough symmetry in a flat ambient space. It is based on a procedure of construction surfaces with a given symmetry. The method is used to classify the embeddings…

Streamlines of a relativistic perfect isentropic fluid are geodesics of a Riemannian space whose metric is defined by enthalpy of the fluid. This fact simplifies the solution of some problems, as is also of interest from the point of view…

The isometric embedding problem for Riemannian manifolds, which connects intrinsic and extrinsic geometry, is a central question in differential geometry with deep theoretical significance and wide-ranging applications. Despite extensive…

Due to the growing interest in embeddings of space-time in higher-dimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we…

In this work we give a detailed description of Matthias G\"unther's proof of the Isometric Embedding Theorem of Riemannian manifolds. Subsequently we will use this method to show that it is possible to construct an isometric embedding of a…

We study isometric embeddings of some solutions of the Einstein equations with suffciently high symmetries into a flat ambient space. We briefly describe a method for constructing surfaces with a given symmetry. We discuss all minimal…

An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study…

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat…

Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy distribution, and its time rate of change…

When solving the equations of General Relativity in a symmetric sector, it is natural to consider the same symmetry for the geometry and stress-energy. This implies that for static and isotropic spacetimes, the most general natural…

The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as…

This paper examines the inhomogeneous Einstein equation for a static spherically symmetric metric with a source term corresponding to a perfect fluid with p=-rho. By a careful treatment of the equation near the origin we find an analytic…

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit…

In this article, a special static spherically symmetric perfect fluid solution of Einstein's equations is provided. Though pressure and density both diverge at the origin, their ratio remains constant. The solution presented here fails to…

If one assumes a particular form of non-minimal coupling, called the conformal coupling, of a perfect fluid with gravity in the fluid-gravity Lagrangian then one gets modified Einstein field equation. In the modified Einstein equation, the…

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves…

We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in…

In this paper we utilize symmetries in order to exhibit exact solutions to Einstein's equation of a perfect fluid on a static manifold all of whose spatial factor belongs to the conformal class of a Riemannian space of constant curvature.

We study acoustic geometry in fluids with external pressure. In particular, we examine the conditions under which the acoustic metric mimics a Schwarzschild-like metric. We demonstrate that it is possible to mimic a Schwarzschild-like…