Hyperbolic Ricci-Bourguignon flow

The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy–momentum tensor of the thin shell is proportional to Diracs delta-function. We constructed the theory of the spherically symmetric thin shells for three types of gravitational theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl) gravity where the gravitational part of the total Lagrangian is just the square of the Weyl tensor; (3) Weyl–Einstein gravity. The results are compared with these in General Relativity (Israel equations). We considered in detail the shells immersed in the vacuum. Some peculiar properties of such shells are found. In particular, for the traceless ([Formula: see text] massless) shell, it is shown that their dynamics cannot be derived from the matching conditions and, thus, is completely arbitrary. On the contrary, in the case of the Weyl–Einstein gravity, the trajectory of the same type of shell is completely restored even without knowledge of the outside solution.

While every manifold with parallel Ricci tensor has harmonic curvature (i.e. satisfies 6R = 0), there are examples ([3], Theorem 5.2) of open Riemannian manifolds with 6R = 0 and VS ~ O. In [1] J.P. Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this note is to describe an easy example answering this question in the negative. More precisely, metrics with 6R = 0 and VS ~ 0 are exhibited on S 1 × N 3, N 3 being e.g. the 3-sphere or a lens space. By taking products of these manifolds with themselves or with arbitrary compact Einstein manifolds, one gets similar examples in all dimensions greater {han three.

The paper undertakes certain special forms of the quarter symmetric metric and non-metric connections on an ε-anti-Kähler manifold. Firstly, we deduce the relation between the Riemannian connection and the special forms of the quarter symmetric metric and non-metric connections. Then, we present some results concerning the torsion tensors of these connections. In addition, we find the forms of the curvature tensor, the Ricci curvature tensor and scalar curvature of such connections and we search the conditions for the ε-anti-Kähler manifold to be an Einstein space with respect to these connections. Finally, we study U(Ric)-vector fields with respect to these connections and give some results related to them.

Purpose With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively. Design/methodology/approach Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively. Findings This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement. Originality/value In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.

On recurrent Riemannian and Ricci curvatures of Finsler metrics

We prove that if $$M^n(n\ge 4)$$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $$R_0>\sigma _{n}K_{\max }$$ , where $$\sigma _n\in (\frac{1}{4},1)$$ is an explicit positive constant depending only on $$n$$ , then $$M$$ must be isometric to a spherical space form. Moreover, we prove that if an $$n(\ge {\!\!4})$$ -dimensional compact Einstein manifold satisfies $$K_{\min }\ge \eta _n R_0,$$ where $$\eta _n\in (\frac{1}{4},1)$$ is an explicit positive constant, then $$M$$ is locally symmetric. It should be emphasized that the pinching constant $$\eta _n$$ is optimal when $$n$$ is even. We then obtain some rigidity theorems for Einstein manifolds under $$(n-2)$$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $$M$$ is an $$n(\ge {\!\!4})$$ -dimensional compact Einstein submanifold in the simply connected space form $$F^{N}(c)$$ with constant curvature $$c\ge 0$$ , and the normalized scalar curvature $$R_0$$ of $$M$$ satisfies $$R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$$ where $$A_n=n^3-5n^2+8n$$ , and $$H$$ is the mean curvature of $$M$$ , then $$M$$ is isometric to a standard $$n$$ -sphere.

The metric is an important tensorial object that introduces more structure into a (differential) manifold. The metric coefficients allow for example to determine the length of parameter curves in the manifold and make it possible to relate corresponding co- and contravariant objects defined on the manifold. Thus the inner product of either co- or contravariant quantities and in particular the angle between two vectors or (covectors) may be computed. Adopting the Ricci postulate of vanishing covariant derivative of the metric coefficients results in a decomposition of the fully covariant connection into its Riemann part, that depends exclusively on the metric, and the contortion, that depends exclusively on the torsion. Moreover, in this case the length of vectors and their angle with respect to geodesics are preserved upon parallel transport. Based on the metric coefficients the fully covariant curvature tensor, displaying left and right skew symmetry, is introduced and the corresponding modifications of the Bianchi identities are highlighted. Likewise the metric allows to introduce the mixed-variant Ricci tensor together with corresponding identities. Finally the metric enables to compute the Ricci scalar from the previous curvature tensors. The particular case of a symmetric, metrically connected manifold represents a Riemann geometry with associated Riemann curvature tensor as fundamental for example in the Einstein Theory of General Relativity/Gravitation. A generalization is obtained by allowing for metric manifolds with non-vanishing covariant derivative of its metric coefficients. The presence of the non-metricity is then subsequently reflected by extra terms in the decomposition of the connection and in the explicit representation of the curvature tensor, the Ricci tensor, the Ricci scalar, and the corresponding Bianchi identities for the various curvature quantities.

Abstract. In the present paper, we introduce a type of Riemann-ian manifolds (namely, W 4 -birecurrent manifold) and study the sev-eral properties of such a manifold on which some geometric condi-tions are imposed. 1. IntroductionIn [7], Lichnerowicz introduced a recurrent manifold of second or-der (called birecurrent manifold) and obtained some interesting results.Later the weak notion of this manifold was studied by many authors in-cluding Thompson [9], Chaki and Roy Chowdhury [2], Ewert-Krzemieni-ewski [3,4], Garai [5]. In [8], Pokhariyal de ned some curvature tensorswith the help of Weyl’s projective curvature tensor and studied theirphysical and geometrical properties. One of the curvature tensors men-tioned in [8] was the W 4 -curvature tensor de ned by W 4 (X;Y;Z;T) = R(X;Y;Z;T) +1(n 1)[g(X;Z)r(Y;T) g(X;Y)r(Z;T)] or in local coordinates,(1.1) W 4ijkl = R ijkl +1(n 1)[g ik r jl g ij r kl ];where Rand rare the Riemannian curvature tensor and the Ricci tensor,respectively. The W 4

Let (M, g) be an n-dimensional Riemannian manifold. We say M has k-positive Ricci curvature if at each point p ∈ M the sum of the k smallest eigenvalues of the Ricci curvature at p is positive. We say that the k-positive Ricci curvature is bounded below by α if the sum of the k smallest eigenvalues is greater than

In this paper the Feynman Green function for Maxwell’s theory in curved space-time is studied by using the Fock–Schwinger–DeWitt asymptotic expansion; the point-splitting method is then applied, since it is a valuable tool for regularizing divergent observables. Among these, the stress-energy tensor is expressed in terms of second covariant derivatives of the Hadamard Green function, which is also closely linked to the effective action; therefore one obtains a series expansion for the stress-energy tensor. Its divergent part can be isolated, and a concise formula is here obtained: by dimensional analysis and combinatorics, there are two kinds of terms: quadratic in curvature tensors (Riemann, Ricci tensors and scalar curvature) and linear in their second covariant derivatives. This formula holds for every space-time metric; it is made even more explicit in the physically relevant particular cases of Ricci-flat and maximally symmetric spaces, and fully evaluated for some examples of physical interest: Kerr and Schwarzschild metrics and de Sitter space-time.

For compact, connected Riemannian manifolds with Ricci curvature bounded below by a constant, what is the linear approximation of the first eigenvalue of Laplacian? The answer is presented with computer assisted proof and the result is optimal in certain sense.

We study the construction and classification of weakly Bochnerflat (WBF) metrics (i.e., Kähler metrics with coclosed Bochner tensor) on compact complex manifolds.A Kähler metric is WBF if and only if its 'normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series.It follows that WBF Kähler metrics are extremal.We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kähler 6-manifolds, extending work by the first three authors on weakly selfdual Kähler 4-manifolds.The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kähler manifolds with a hamiltonian 2-form [3].

We classify almost C(α)-manifolds, which satisfy the curvature conditions (Z ) (ξ,X)R=0, (Z ) (ξ,X) (Z ) =0, (Z ) (ξ,X)S=0 and (Z ) (ξ,X)P=0, where (Z ) is the concircular curvature tensor, P is the Weyl projective curvature tensor, S is the Ricci tensor and R is Riemannian curvature tensor of manifold.

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis, we construct explicit examples of Calabi–Yau with torsion Hermitian structures and prove a uniqueness result for them.

When a conformai structure on a manifold is defined by a Riemannian metric g , how to detect conformai flatness on g ? The answer, due to Weyl and Schouten, is given in § C, and some applications are derived in § D. It turns out that the three dimensional case, i.e. the case where the curvature tensor is determined by the Ricci tensor, needs a special treatment. An example of that situation is given in § E. We also give some global properties of compact conformally flat manifolds: the nullity of their Pontryagin numbers, (Chern-Simons), a vanishing theorem for middle-dimensional cohomology when the scalar curvature is positive (Bourguignon)and a structure’ theorem when the scalar curvature is zero.