Geometry on Metric Manifolds

This chapter contains a 56-page-long list of formulae from the calculus on manifolds, and tables concerning different topics: Lie groups, Lie algebras and symmetric spaces, a list of Poincaré polynomials of compact simple Lie groups, an overview of real forms of classical complex simple Lie algebras and their corresponding simple Lie groups, a table of irreducible Riemannian symmetric spaces of type I and III, a table of Riemannian symmetric spaces of classical type with noncompact isotropy group, etc. One can find the formulae for Christoffel symbols, the curvature tensor, Bianchi identities, Ricci tensor, the basic differential operators, the expression for conformal changes of Riemannian metrics, Cartan structure equations for pseudo-Riemannian manifolds, and many more. Several of these formulae are used throughout the book; others are not, but they have been included since such a collection might prove useful as an aide-mémoire, also to lecturers and researchers.KeywordsRelevant TheoryChristoffel SymbolRiemannian Symmetric SpaceMoving CoframeCartan StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction.Let M be a connected »-dimensional C°°-manifold.If S is any tensor (field) on M, then (i) S(u) denotes the value of 5 at the point w£Af;(ii) 5 = 0 means that 5 is everywhere zero, i.e.S(u) =0 for every i*£M;(iii) S féO means that 5 is not everywhere zero, i.e.S(u)?¿0 for some u£M;(iv) SVO means that S is nowhere zero, i.e.S(u) = 0 for no w£.M.Throughout this paper, a tensor of type (1,0) will be called a vector, and a tensor of type (0, 1) a covector.Summation over a repeated index, Latin or Greek, is always implied.Let 7 be a linear connexion on M. If T,/ (1 5¡a, h, i, j, • ■ ■ , tin) are the components of y in the local coordinate system ( U, uh) in M, then the components in (U, uh) of the torsion tensor T and the curvature tensor R on M are respectively T..h -T>..h _ r>..* Rk»k = d*r** -»¡Tu* + r*.*iy -rjahrki', where dk -d/duk.If y has zero torsion, i.e., if F = 0, then the curvature tensor R satisfies the two Bianchi identities (0.1) Ria" + Ri*" + R*jh = 0, (0.2) v-tRwi» + VkRjU» + VjRuf = 0,where V denotes covariant differentiation with respect to y.A linear connexion 7 on M is said to be with recurrent curvature if its curvature tensor R is not everywhere zero and if the covariant derivative of R is equal to the tensor product of a covector and R itself.We express these conditions in symbol by Rf^O and VZ?= W®R, and call W the recurrence covector.By Theorem 1.2, the conditions Rf^O and VR=W®R imply that R^O.Hence for a linear connexion with recurrent curvature, the curvature tensor is nowhere zero, i.e.Ry^O.In what follows, whenever the condition Vi?= W®R is assumed, the condition RféO and consequently also the condition Rr¿0 are always understood.Recently, generalizing a result of K. Nomizu concerning linear connexions

We consider the Bianchi identity as a field equation for the distortion of the shapes of images produced by weak gravitational lensing. Using the spin coefficient formalism of Newman and Penrose (1962 J. Math. Phys. 3 566–78), we show that certain complex components of the Weyl and Ricci curvature tensors are directly related to fundamental observables in weak gravitational lensing. In the case of weak gravitational fields, we then show that the Bianchi identity provides a field equation for the Ricci tensor assuming a known Weyl tensor. From the Bianchi identity, we derive the integral equation for weak lensing presented by Miralda-Escude (1996 IAU Symp. vol 173 p 131), thus making the Bianchi identity a first principles equation of weak gravitational lensing. This equation is integrated in the important case of an axially symmetric lens and explicitly demonstrated in the case of a point lens and a singular isothermal sphere (SIS) model.

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

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.

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.

In this paper, we concentrate on a new Finsler space that is a combination of two types of Cartan derivatives of third order for Berwald curvature tensor H_jkh^i. We find the condition that Berwald curvature tensor H_jkh^i is special generalized of generalized mixed trirecurrent. Furthermore, we show that the normal projective curvature tensor N_jkh^i is generalized hv- mixed trirecurrent if and only if the tensor ∂ ̇_j (H_hk-H_kh) behaves as mixed trirecurrent. Also, the Ricci tensor N_jk of the normal projective curvature tensor N_jkh^i is non – vanishing if and only if the tensor ((1-n) ∂ ̇_j ∂ ̇_k H+H_jk+H_kj) is mixed trirecurrent.

The first chapter in Part II contains basic information on general relativity and all necessary notations and formulas. The consideration is concise but rather brief; it is not meant to replace a textbook on general relativity. Covariant derivatives, curvature tensors, Bianchi identities, covariant equations, the classical limit and Einstein equations are covered. The fact that the Schwarzschild solution and the cosmological solution contain singularities, which can be interpreted as indicating a need for quantum gravitational theory, is addressed. In addition, the applicability of general relativity and Planck units are discussed.

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

In this paper, we consider the hyperbolic Ricci-Bourguignon flow on a compact manifold M and show that this flow has a unique solution on short-time with imposing on initial conditions. After then, we find evolution equations for Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of M under this flow. In the final section, we give some examples of this flow on some compact manifolds.

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.

This note provides a short guide to dimensional analysis in Lorentzian and general relativity and in differential geometry. It tries to revive Dorgelo and Schouten’s notion of ‘intrinsic’ or ‘absolute’ dimension of a tensorial quantity. The intrinsic dimension is independent of the dimensions of the coordinates and expresses the physical and operational meaning of a tensor. The dimensional analysis of several important tensors and tensor operations is summarized. In particular it is shown that the components of a tensor need not have all the same dimension, and that the Riemann (once contravariant and thrice covariant) and Ricci (fully covariant) curvature tensors are dimensionless. The relation between dimension and operational meaning for the metric and stress–energy–momentum tensors is discussed; and the main conventions for the dimensions of these two tensors and of Einstein's constant are reviewed. A more thorough and updated analysis is available as a preprint.

Differentiation of tensor components in a curved space must be handled with extra care. By adding another term (related to Christoffel symbols) to the ordinary derivative operator, we can form a “covariant derivative”; such a differentiation operation does not spoil the tensor property. The relation between Christoffel symbols and first derivatives of metric functions is re-established. Using the concept of parallel transport, the geometric meaning of covariant differentiation is further clarified. The curvature tensor for an n-dimensional space is derived by the parallel transport of a vector around a closed path. Symmetry and contraction properties of the Riemann curvature tensor are considered. We find just the desired tensor needed for GR field equation.

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.

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.