On the constraint equation in the relativistic theory of gravitation

Expressions for covariant and contravariant components of the metric tensor for the effective Riemann space-time and for its determinant are constructed on the basis of the relations of tensor algebra.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 508–511, December, 1995.     

The Hyperbolic Geometric Flow on Riemann Surfaces

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ?² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric g _ij keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably “large", then the solution metric g _ij converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature R(t, x) goes to positive infinity as (t, x) tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed.     

Cylindrical wave solutions of the field equations representing zero-rest-mass scalar fields and those of lichnerowicz's « total radiation » in a generalized Einstein-Rosen space-time

Summary Lichnerowicz (1960), by using a property of the Riemann tensor, has given the field equations, R_ij=ϱw_iw_j, w_iwⁱ=0, ϱ being a scalar and has termed them as representing a state of ? total radiation ?. Recently Rao (1970) derived these field equations under a geometrical relation which as stated by him imposes on the Rieman tensor a severe restriction, and he obtained a class of exact solutions of the above field equations corresponding to the cylindrically symmetric space-time with two degrees of freedom. The present authors, in this paper, have obtained exact wave solutions of the field equations representing zero-rest-mass scalar field in the generalized Einstein-Rosen space-time, and it has been shown that by a suitable choice of the scalar field, the field equations considered by Lichnerowicz and Rao can be deduced without imposing any condition on the geometrical nature of the Riemann tensor. Finally a method is given by which most general solutions of these field equations, can be generated from those of the empty space field equations. It is further shown that the solutions of Rao (1960) are only a special case of those found in this paper. Entrata in Redazione il 15 settembre 1976.     

Symmetric energy-momentum tensor of spinor fields

We construct a symmetric energy-momentum tensor for spinor fields with an arbitrary Lagrange function on Riemannian space-time manifolds. We show that this tensor can be considered as the metric energy-momentum tensor.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 306–314, August, 1996.     

一类Riemann张量指标表达式的标准型完全分类及其在微分几何中的应用

本文研究了n维微分几何中Riemann张量指标表达式的标准型完全分类问题, 通过引入指标结构图的概念, 证明了规范类型单项式都是标准型, 并且构成次数不大于5 的Sakai类型单项式的正交基底, 由此得到Sakai类型单项式的标准型完全分类, 这是次数大于3时标准型完全分类问题的第一个结果. 同时给出了相应标准化算法, 通过比较说明了该算法比现有算法更加简便, 最后应用于自动推导和证明微分几何中关于Riemann张量的一些公式.     

Addendum to: Curvature of contact manifolds

If the sectional curvature of any plane containing the characteristic vector field of a contact metric manifold is non-vanishing, then we prove that a vector field leaving the Riemann curvature tensor invariant; is homothetic. In addition, two corollaries have been obtained.Dedicated to the memory of Professor B.B. Sinha     

On Differentiability of Metric Projections onto Moving Convex Sets

We consider properties of the metric projections onto moving convex sets in normed linear spaces. Under certain conditions about the norm, directional differentiability of first and higher order of the metric projections at boundary points is characterized. The conditions are formulated in terms of differentiability of multifunctions and properties of the set-derivatives are shown.     

The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.     

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.     

Special conformal mappings in the general theory of relativity

Some special conformal mappings of relativistic spaces are studied. The amount of arbitrariness with which one can describe a nontrivial special conformai mapping is determined. Necessary and sufficient conditions are found under which a space-timeV admits a special conformal mapping to space-time ¯V in which the metric tensor of bothV and ¯V satisfy the Einstein equations with the energy-momentum tensor of an ideal fluid having a geodesic velocity field.It is proved that such spaces are spaces of standard cosmological models.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 43–50.     

Curvature tensors of singular spaces

A sequence of tensor-valued measures of certain singular spaces (e.g., subanalytic or convex sets) is constructed. The first three terms can be interpreted as scalar curvature, Einstein tensor and (modified) Riemann tensor. It is shown that these measures are independent of the ambient space, i.e., they are intrinsic. In contrast to this, there exists no intrinsic tensor-valued measure corresponding to the Ricci tensor.     

INVARIANT VARIATIONAL PRINCIPLES INVOLVING VECTOR AND METRIC FIELDS IN 4-DIMENSIONS

Abstract

Variational principles in which the Lagrangian is a scalar density and a function of a metric tensor and a vector field, together with their first derivatives, are investigated in a 4-dimensional space. Associated with such Lagrangians are two expressions, the metric Euler-Lagrange expression and the vector Euler-Lagrange expression. The most general Lagrangians (of this kind) for which either of these Euler-Lagrange expressions vanishes identically, are obtained.

The most general Lagrangian (of this kind) for which the vector Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although this Lagrangian is more general than the one commonly used, it still has essentially the same energy-momentum tensor.

The most general Lagrangian (of this kind) for which the metric Euler-Lagrange expression is precisely the electromagnetic energy-momentum tensor is derived. Although this Lagrangian is also more general than the one commonly used, the associated vector Euler-Lagrange equations are still Maxwell's equations.

Finally it is shown that, in contrast to the situation which obtains in the case of scalar densities which are functions of up to second derivatives of the metric and first derivatives of the vector field, there does not exist a Lagrangian, of the kind under investigation, for which the metric Euler-Lagrange expression is precisely the Einstein tensor and the vector Euler-Lagrange expression vanishes identically.     

Almost Cosymplectic and Almost Kenmotsu (κ, μ, ν)-Spaces

We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.     

Bergman interpolation on finite Riemann surfaces. Part I: asymptotically flat case

The article considers the Bergman space interpolation problem on open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. Such a surface is equipped with what we call an asymptotically flat conformal metric, i.e., a complete metric with zero curvature outside a compact subset. Sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces are then established. When the weights have curvature that is quasi-isometric to the asymptotically flat boundary metric, these sufficient conditions are shown to be necessary, unless the surface has at least one cylindrical end, in which case, the necessary conditions are slightly weaker than the sufficient conditions.     

Tetrad-Gauge Theory of Gravity

We present a tetrad–gauge theory of gravity based on the local Lorentz group in a four-dimensional Riemann–Cartan space–time. Using the tetrad formalism allows avoiding problems connected with the noncompactness of the group and includes the possibility of choosing the local inertial reference frame arbitrarily at any point in the space–time. The initial quantities of the theory are the tetrad and gauge fields in terms of which we express the metric, connection, torsion, and curvature tensor. The gauge fields of the theory are coupled only to the gravitational field described by the tetrad fields. The equations in the theory can be solved both for a many-body system like the Solar System and in the general case of a static centrally symmetric field. The metric thus found coincides with the metric obtained in general relativity using the same approximations, but the interpretation of gravity is quite different. Here, the space–time torsion is responsible for gravity, and there is no curvature because the curvature tensor is a linear combination of the gauge field tensors, which are absent in the case of pure gravity. The gauge fields of the theory, which (together with the tetrad fields) define the structure of space–time, are not directly coupled to ordinary matter and can be interpreted as the fields describing dark energy and dark matter.     

Dirichlet函数与Riemann函数的性质及应用

探讨Dirichlet函数和Riemann函数的连续性、可导性和可积性,并由此分别构造出在一点连续、多点连续、一点可导、多点可导的函数.     

Analogues of Riemann tensors for the odd metric on supermanifolds

Structure functions constitute the complete set of obstructions to integrability of a G-structure on a manifold. For a Riemannian manifold the structure function is the Riemann tensor. In this work, we compute structure functions for the odd analogue of the metric on supermanifolds and for several related structures. Structure functions take values in Spencer cohomology groups, which we describe by means of the representation theory of Lie algebras and Lie superalgebras.     

（0,2）型张量场诱导的线性变换的显表达式提取方法

对于黎曼流形（M,g）上（0,2）型光滑张量场R诱导出的线性变换R＊：X（M）→X（M）,其隐表达式为g（R＊（X）,Y）=S（X,Y）,X,Y∈X（M）,研究了从局部上提取出R＊（X）的显表达式的方法.提出一种所谓g正定提取方法,并将这种方法进一步应用到度量保形变换下的线性变换Φ＊（X）,L＊（X）.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.