Riemannian Space Vn Research Articles

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

All external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators are found, provided that in the space-time V4 a group of motion G3 acts simply transitively on a non-null subspace of transitivity V3. It is shown that in the case of a Riemannian space Vn, in which the group Gr acts simply transitively, the algebra of symmetry operators of the n-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group Gr.

Proportional nets in Weyl spaces

α-proportional and proportional nets in n-dimensional Weyl space Wn and Riemannian space Vn are introduced. The fundamental forms of the spaces Wn and Vn in the parameters of the α-proportional and proportional nets are found. The theorem - Weyl space \( W_n, n>2 \) which contains a proportional net is Riemannian Vn - is proved.

The role of the internal metric in generalized Kaluza–Klein theories

The dimensional reduction of a Weyl space WN of N=4+n dimensions to a principal fiber bundle P̃(W4,G̃n) over a four-dimensional space–time W4 with structural group G̃n of dimension n arising from the existence of n conformal Killing vector fields of the original N-metric is studied. The framework of a Weyl geometry is adopted in order to investigate conformal rescalings of the metric on the bundle P̃(W4,G̃n) obtained. The Weyl symmetry is then, finally, broken again by choosing a particular Weyl gauge in which the internal, i.e., fiber metric, is of constant Cartan–Killing form. This choice of gauge, yielding a Riemannian theory, forces the internal metric to play no dynamical role in the theory, as is usually assumed to be the case in non-Abelian gauge theories. However, this gauge induces a conformal transformation of the metric in the space–time base of P̃ compared to the space–time metric obtained by the ordinary Kaluza–Klein reduction of a Riemannian space VN. The role of vector torsion in this dimensional reduction by isometries and scale transformations is also investigated.

Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times

Katzin et al. [G. H. Katzin, J. Levine, and W. R. Davis, J. Math. Phys. 10, 617 (1969)] introduced curvature collineations (CC), defined by a vector ξ, satisfying LξRbcda=0, where Rbcda is the Riemann curvature tensor of a Riemannian space Vn and Lξ denotes the Lie derivative. They proved that a CC is related to a special conformal motion which implies the existence of a covariant constant vector field. Unfortunately, recent study indicates that the existence of a covariant constant vector restricts Vn to a very rare special case with limited physical use. In particular, for a fluid space time with special conformal motion, either stiff or unphysical equations of state are singled out. Moreover, perfect fluid space times do not admit special conformal motions. This information was not available, in 1969, when CC symmetry was introduced. In this paper, CC is generalized to another symmetry called ‘‘curvature inheritance’’ (CI) satisfying LξRbcda=2αRbcda, where α is a scalar function. We prove that a proper CI (i.e., α≠0) has direct interplay with the physically significant proper conformal motions. As an application, we show that a proper CI, which is also a conformal Killing vector (CKV), can generate new and physically relevant solutions for a variety of fluid spacetimes. In particular, it is shown, that, for CI with CKV, the known stiff or unphysical equations of state are not singled out.

THE PROBLEM OF THE CORRECTNESS OF SCHUR'S THEOREM

This paper considers the problem of the correctness of Schur's theorem for an n-dimensional Riemannian space Vn. We show that in the general case it is not correct, that is, it may happen that, for an arbitrarily small variation of the curvature of the space due to rotations of two-dimensional elements of area at points of a given domain, the variation of the curvature from point to point of the domain is arbitrarily large. Bibliography: 8 titles.

A function of direction associated with n congruences of an orthogonal ennuple in a Riemannian hypersurface

In a hypersurface Vn belonging to a Riemannian space Vn+1 we choose n congruences of an orthogonal ennuple of unit vectors of contravariant components λ h¦ i (h=1,2,...,n) and denote, by ωkk the normal curvature of the hypersurface in the direction of the unit vector of components λ k¦ i . In the present paper, we have shown that the expression $$\frac{\partial }{{\partial s_k }}\omega _{kk} - 2\mathop \Sigma \limits_{h = 1}^n \omega _{kh} \gamma _{hkk}$$ is a function of direction, where the symbol ∂/∂sk indicates the differentiation in the direction of the vector of components λ k¦ i and that ωkh (h≠k) and γlhk (l,h,k=1,2,...,n) are, respectively, the invariants of the geodesic torsion of the curve of the congruence with unit tangent vector of components λ k¦ i and Ricci's coefficients of rotation of the orthogonal ennuple.

Geodesic first integrals with explicit path-parameter dependence in Riemannian space–times

In a Riemannian space Vn general formulas are obtained for geodesic first integrals which are mth order polynomials in the tangent vector and which are assumed to depend explicitly on the path parameter s. It is found that such first integrals must also be polynomials in s. Necessary and sufficient conditions are found for the existence of these first integrals. The existence of many well-known symmetries such as homothetic motions (scale change), affine collineations, conformal motions, projective collineations, conformal collineations, or special curvature collineations are shown to be sufficient for the existence of such first integrals with explicit path-parameter dependence. To illustrate the theory, geodesic first integrals of this type have been calculated for four Riemannian space–times of general relativity.

Lie derivatives and deviation equations in Riemannian spaces

The connection between Lie derivatives and the deviation equations has been investigated in Riemannian spacesVn. On this basis the deviation equations of the geodesies have been obtained, in spaces with symmetries, as well as deviation equations of nongeodesic trajectories, through imposing certain conditions on the Lie derivatives with respect to the tangential vector of the basic trajectory.

H-transformations in Riemannian spaces

A Riemannian space Vn (n = mr), equipped with an integrable regular H-structure isomorphic to a hypercomplex algebra h (dim h = r), is considered as a real realization of a hypercomplex manifold\(\mathop V\limits^* _m \) over the algebra h. The geometry of\(\mathop V\limits^* _m \) can be mapped into the geometry of Vn. In particular, with the transformations of\(\mathop V\limits^* _m \) are associated H transformations (preserving the H-structure of the space) in Vn. The H-conformal and the H-projective transformations of Vn are investigated.

Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. II. Partial separation

Sufficient conditions are given for coordinate systems in which the Hamilton–Jacobi equation and the Schrödinger and related equations are partially separable in n dimensions. For the first equation, the solution is assumed to be a sum of ν+τ functions of a single variable, and of μ[0⩽μ⩽ (n−ν−τ)/2] groups of other variables; for the other equations, products of such functions are assumed. These assumptions lead to ν+τ completely separated differential equations, of which ν are linear in the separation constants, and μ partially separated equations depending on the remaining n−ν−τ variables. The general forms of the various metric tensors gkl of the Riemannian spaces Vn as well as of the allowed potentials V corresponding to the different possible types of such equations are determined; they are identical for the Hamilton–Jacobi equation and for the other equations studied, except that for the latter some of the metrics are further restricted by a condition on their determinants. The results are established by methods similar to those used in Paper I of this series for complete separation, and include the results obtained there as special cases. In the course of determining the allowed forms of the gkl it is also established that there exist ν+τ+μ independent first integrals linear and quadratic in the momenta for the dynamical systems descrived by the Hamilton–Jacobi or the Schrödinger equation. The ν linear ones are homogeneous, and the τ+μ quadratic ones correspond to homogeneous quadratic integrals of the geodesics of the Vn. These results imply the existence of ν Killing vectors and of τ+μ Killing tensors of rank two for the Vn. Further polynomial integrals can be constructed; those integrals of degree r which are independent of the original ν+τ+μ integrals each correspond to an independent Killing tensor of rank r.

Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation

It was established by Levi-Civita that in n dimensions there exist n+1 types of coordinate systems in which the Hamilton–Jacobi equation is separable, n of which are in general nonorthogonal; the form of the separated equations was given by Burgatti and Dall’Acqua. In this paper first the general forms of the n+1 types of metric tensors of the corresponding corresponding Riemannian spaces Vn are determined. Then, sufficient conditions are given for coordinate systems in which the Schrödinger, Helmholtz, and Laplace equation are separable. It is shown that there again exist n+1 types of such systems, whose metric tensors are of the same form as those of the Hamilton–Jacobi equation. However, except for the ’’essentially geodesic case’’ of Levi-Civita they are further restricted by a condition on the determinant of the metric; this condition is a generalization of that found by Robertson for orthogonal systems in the case of the Schrödinger equation.

On a type of Sasakian space

A Sasakian space [1]Mn (n = 2m + 1) is a Riemannian n-space with a positive definite metric tensor gij and a unit Killing vector field η which satisfies where the comma denotes covariant differentiation with respect to the metic tensor. In a recent paper [2] M. C. Chaki and A. N. Roy Chowdhury studied conformally recurrent spaces of second order, or briefly conformally 2-recurrent spaces, that is, non-flat Riemannian spaces Vn (n > 3) defined by where is the conformal curvature tensor: and alm is a tensor not identically zero.

Integral formulae on hypersurfaces in Riemannian manifolds

The coefficients of the complete set of n fundamental forms of a hypersuface Vn−1 imbedded in an n-dimensional Riemannian space Vn, as recently introduced[(5)], are used to construct certain tensor fields over Vn−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of Vn, the coefficients of the n fundamental forms, and the rth curvatures of Vn−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on Vn−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of Vn−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever Vn is a space of constant curvature.

Polynomial conformal tensors

AbstractThe conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.

Groups of Curvature Collineations in Riemannian Space-Times Which Admit Fields of Parallel Vectors

By definition, a Riemannian space Vn admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξi of the Riemann curvature tensor vanishes. It is shown that if a Vn admits a parallel vector field, then it will admit groups of CC's. It follows that every space-time with an expansion-free, shear-free, rotation-free, geodesic congruence admits groups of CC's, and hence gravitational pp waves admit such groups of symmetries.

Curvature Collineations: A Fundamental Symmetry Property of the Space-Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space-time is strongly motivated by the all-important role of the Riemannian curvature tensor in the theory of general relativity. For space-times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null-gravitational space-times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space-times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel-Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space-time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space-time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space-times is analyzed in detail to obtain the explicit CC vectors which they admit.

Projective-symmetric spaces

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

The Riemannian geometry of physical systems of curves

The properties of a physical system Sk where k ≠−1, of ∞2n−1 trajectories C. in a Riemannian space Vn are developed. The intrinsic differential equations and the equations of Lagrange, of a physical system Sk, are derived. The Lagrangian function L and the Hamiltonian function H, are studied in the conservative case. Also included are systems of the type (G), curvature trajectories, and natural families. The Appell transformation T of a dynamical system S0 in a Riemannian space Vn, is obtained. Finally, contact transformations and the transformation theory of a physical system Sk where k ≠−1, are considered in detail.

Subspaces Of Riemannian Spaces

In this paper, results obtained by the author for Riemannian Spaces Vn imbedded in Euclidean Spaces EN (3; 4) are extended to Vn imbedded in VN.The first section is introductory. In §2 the general result is obtained. This is the establishment of a certain dependency among the three basic sets of equations of the Vn with respect to the VN, namely the equations of Gauss, Codazzi and Kuehne.

XXI.—Relative Co-ordinates

If C is a given curve in a Riemannian space Vn, a system of co-ordinates (z1, z2, …, zn) can be set up at each point P of C, thus generalising moving axes along a twisted curve. If in these co-ordinates a point Q is defined relative to the point P, then Q traces some curve as P moves along C; any curve C' can be defined in this way by setting up a (1, 1) correspondence between points of C and C′. We shall take the relative co-ordinates at P to be normal co-ordinates with origin at P, the parametric directions at P being the directions of an orthogonal ennuple defined at points of C. In general, this work is too heavy except for the consideration of points within a certain distance from the curve C, and we shall therefore consider only points the cube of whose distance from C may be neglected. This is sufficient for the application to such problems as the motion of a small rigid body in space-time.