In this paper, we study the extended Ricci flow on a complete noncompact Riemannian manifold of dimension n introduced by List in List (2008), and prove the short-time existence with bounded Lp norm ...of Riemann curvature. In the critical case p=n2, we replace the bounded Lp norm of Riemann curvature by the bounded Lp norm of Ricci curvature in the short-time existence.
In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian ...manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated M− and V−fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space Rn that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of Rn endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.
In this paper we demonstrate the generation of gravitational caustics that appear due to the geodesic focusing in a self-gravitating N-body system. The gravitational caustics are space regions where ...the density of particles is higher than the average density in the surrounding space. It is suggested that the intrinsic mechanism of caustics generation is responsible for the formation of the cosmological Large Scale Structure that consists of matter concentrations in the form of galaxies, galactic clusters, filaments, and vast regions devoid of galaxies.
In our approach the dynamics of a self-gravitating N-body system is formulated in terms of a geodesic flow on a curved Riemannian manifold of dimension 3N equipped by the Maupertuis’s metric. We investigate the sign of the sectional curvatures that defines the stability of geodesic trajectories in different parts of the phase space. The regions of negative sectional curvatures are responsible for the exponential instability of geodesic trajectories, deterministic chaos and relaxation phenomena of globular clusters and galaxies, while the regions of positive sectional curvatures are responsible for the gravitational geodesic focusing and generation of caustics. By solving the Jacobi and the Raychaudhuri equations we estimated the characteristic time scale of generation of gravitational caustics, calculated the density contrast on the caustics and compared it with the density contrasts generated by the Jeans–Bonnor–Lifshitz–Khalatnikov gravitational instability and that of the spherical top-hat model of Gunn and Gott.
•The gravitational caustics are generated in self-gravitating N-body systems.•The gravitational caustics are regions in space where the matter density is high.•In cosmology caustics represent regions where the density of galaxies is high.•Caustics are responsible for the formation of the cosmological Large Scale Structure.•Solution of Jacobi and Raychaudhuri equations provides a characteristic time scale.
On projective Riemann curvature Zhu, Hongmei; Li, Ranran
Differential geometry and its applications,
June 2022, 2022-06-00, Letnik:
82
Journal Article
Recenzirano
With a volume form on a manifold, every spray can be deformed to a projective spray. The Riemann curvature of a projective spray is called the projective Riemmann curvature. In this paper, we ...introduce the notion of projectively R-flat sprays/Finsler metrics. As an application, we investigate and characterize projectively R-flat Randers metrics. Moreover, we obtained a rigidity theorem on a closed Finsler manifold.
In this work, we consider the effects of repulsive gravity in an invariant way for four static 3D regular black holes, using the eigenvalues of the Riemann curvature tensor, the Ricci scalar, and the ...strong energy conditions. The eigenvalues of the solutions are non-vanishing asymptotically (in asymptotically AdS) and increase as the source of gravity is approached, providing a radius at which the passage from attractive to repulsive gravity might occur. We compute the onsets and the regions of repulsive gravity and conclude that the regular behavior of the solutions at the origin of coordinates can be interpreted as due to the presence of repulsive gravity, which also turns out to be related with the violation of the strong energy condition. We showed that in all of the solutions for the allowed region of parameters, gravity changes its sign, but the repulsive regions only for the non-logarithmic solution are affected by the mass that generates the regular black hole. The repulsive regions for the logarithmic solutions are dependent on electric charge and the AdS
3
length. The implications and physical consequences of these results are discussed in detail.
In this paper, we study the left invariant spray geometry on a connected Lie group. Using the technique of invariant frames, we find the ordinary differential equations on the Lie algebra describing ...for a left invariant spray structure the linearly parallel translations along a geodesic and the nonlinearly parallel translations along a smooth curve. In these equations, the connection operator plays an important role. Using parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. In particular, the Riemannian curvture appears in both a double Lie derivative along the spray vector and brackets between smooth vector fields induced by the connection operator. We propose two questions in left invariant spray geometry. One question generalizes Landsberg Problem in Finsler geometry, and the other concerns the restricted holonomy group.
A theory for the evolution of a metric g driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving ...three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash–Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving C1 manifold. The theory is applied to the incompressible Euler and Navier–Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier–Stokes equations.
Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We ...generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature tensor of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersections, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.
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