Fourier积分算子的局部光滑性及其相关研究

本文综述满足电影型(cinematic)曲率条件的Fourier积分算子的局部光滑性及其相关研究.电影型曲率条件包含非退化条件及曲率条件.作为范例重点讨论如何通过双线性方法建立变系数版本的平方函数不等式,进而改进了Mockenhaupt-Seeger-Sogge局部光滑性的结果.与此同时,本文还分析了解决局部光滑性猜想的困难、可能的途径,以及它与其他数学猜想之间的联系.     

关于复射影空间的常平均曲率的实超曲面

本文研究了复射影空间的常平均曲率的实超曲面关于Ricci曲率、截面曲率、第二基本形式长度平方的量子化现象,得到了关于Ricci曲率、截面曲率、第二基本形式长度平方积分不等式以及相应的Pinching定理,推广并改进了已有结果.     

局部对称流形中具有常平均曲率的完备超曲面

讨论了局部对称黎曼流形中具有常平均曲率的完备超曲面的性质,通过Laplace算子的计算,得到一个关于第二基本形式模长平方S的拼挤定理,推广了已有的结果．     

局部对称黎曼流形中的超曲面

研究了局部对称黎曼流形中具有常平均曲率超曲面的几何性质,得到了关于其第二基本形式模长平方的拼挤定理.     

局部对偶的黎曼空间和引力瞬子解

四维正定黎曼空间R₄能局部地生成两个SU₂规范场和,如果,至少有一个具有自对偶性或反自对偶性,那末空间称为具局部对偶性的.我们证明它们是Einstein空间、数量曲率为0的共形平坦空间以及只R⁺⁺=0(或R^--=0)的空间.文中得出了R⁺⁺=0(R^--≠0)的一类黎曼线素.对曲率张量平方可积的情形,作出了规范场作用量,Euler示性数,Pontrjagin示性数之间的一个不等式,证明它的等号在而且只在R₄具局部对偶性时达到,这结果改进了文献[7]中关于引力瞬子解的研究.并以Hitchin关于4维紧致Einstein形流的一个不等式作为特殊情况.     

余辛流形的半不变子流形的Ricci曲率

本文研究了余辛流形的半不变子流形,得到了这类子流形的Ricci曲率与平均曲率平方之间的—个不等式,并讨论了等式成立的充分必要条件．     

局部对称共形平坦黎曼流形中具有平行平均曲率向量的子流形

本文把[1]的结论推广到了环绕空间是局部对称共形平坦的情形,即获得了:设M~是局部对称共形平坦黎曼流形N~+p(p>1)中具有平行平均曲率向量的紧致子流形,如果则M~位于N~+p的全测地子流形N~+1中。其中S,H分别是M~的第二基本形式长度的平方和M~的平均曲率,T_C、t_c分别是N~+p的Ricci曲率的上、下确界,K是N~+p的数量曲率。     

局部对称黎曼流形中具有平行平均曲率向量的子流形

设Nn+p是截面曲率KN满足的n+p维局部对称完备黎曼流形,p≥2.M是Nn+p的具有平行平均曲率向量的n维紧致子流形.本文讨论了这类子流形关于第二基本形式模长平方的积分不等式及其Pinching问题.     

局部对称共形平坦黎曼流形中紧致极小子流形的一个刚性定理

本文研究局部对称共形平坦黎曼流形中紧致极小子流形,得到了这类子流形第二基本形式模长平方关于外围空间Ricci曲率的—个拼挤定理,推广了文[1]中的结果.     

四元数空间形式中全复子流形的内蕴积分等式

本文给出了四元数空间形式中全复子流形的一个性质．即设Ｍ２ｎ是的全复子流形，ρ，‖Ｒｉｅｍ‖２，‖Ｒｉｃｃｉ‖２分别表示Ｍ２ｎ的纯量曲率和黎曼曲率，Ｒｉｃｃｉ曲率的模长平方，则在Ｍ上处处成立．     

Impossibility of gravitational collapse

In the framework of the relativistic theory of gravity, we show that a universal mechanism for stopping the process of gravitational compression of a body with large mass with its subsequent radial expansion appears because of the gravitational field tensor. This excludes the gravitational collapse and the possibility of black hole formation.     

Can the Notion of a Homogeneous Gravitational Field be Transferred from Classical Mechanics to the Relativistic Theory of Gravity?

Bogorodskii generalized the classical mechanical concept of a homogeneous gravitational field to the case of Einstein's general relativity. We seek such a generalization to the case of the relativistic theory of gravity. The corresponding solutions in these two theories differ substantially. The solution obtained in accordance with the relativistic theory of gravity does not satisfy the causality principle in that theory. The problem of constructing a generalization of the classical notion of a homogeneous gravitational field in the framework of the relativistic theory of gravity therefore remains open.     

Is the Causality Principle Violated for Gravitational Waves?

We verify the causality principle for the relativistic theory of gravity in the linear approximation. We show that the contribution of weak gravitational waves is considerably less than the contributions of the static part of the solution and the cosmological background. Therefore, the presence of weak gravitational waves does not violate the causality principle for the relativistic theory of gravity.     

On the possibility of gravitational collapse in the relativistic theory of gravity

In the example of dust matter, it is shown that a gravitational attractive force does not lead to the formation of “black holes” in the relativistic theory of gravity. It is proved that in the absence of matter, the gravitational field is also absent. Therefore, a vacuum is not a source of a gravitational field. The mechanism of energy production in the process of the accumulation of matter into massive objects is discussed.     

The stability of the grioli precession

The motion of a rigid body about a fixed point in a uniform gravitational field is considered. The body is not dynamically symmetric, but its centre of gravity is on the perpendicular, erected from the fixed point, to a circular section of the inertia ellipsoid. Grioli proved that a rigid body with such mass geometry may precess regularly about a non-vertical axis. The problem of the stability of this precession is solved.     

Relativistic theory of gravity and a torsion field

The fundamentals of gravity theory are stated in a Minkowski space with an effective nonzero-torsion Riemann-Cartan space-time, which is more general than the Riemannian space. The theory presented thus includes a torsion field of the Einstein-Cartan type in the general concept of the relativistic theory of gravity. Expressions for the metric and canonical energy-momentum tensors of the gravitational field and nongravitational matter in the Minkowski space are found. Noncoordinate gauge transformations are introduced under which the variation of the density of the gravitational Lagrangian is a divergence expression. Translated from Teoreticheskaya i Matematischeskaya Fizika, Vol. 118, No. 1, pp. 126–132, January, 1999.     

The Gravitational Field of an Electrically Charged Mass Point and the Causality Principle in the RTG

We find the effective Riemannian space–time corresponding to the gravitational field generated by a charged mass point in the framework of the relativistic theory of gravity. The causality principle plays an important role in solving this problem. The analytic form and the domain of definition, i.e., the gravitational radius, of the obtained solution differ from the corresponding results in Einstein's general relativity theory.     

Nonsymmetrical tensor potential in the relativistic theory of gravity

We suggest a theory of nonsymmetrical tensor fields in the Minkowski space. This theory generalizes the relativistic theory of gravity with a symmetrical potential and preserves all its basic postulates. The connection of the effective space-time in which the gravitational field equations can be represented possesses torsion and nonmetricity. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 521–528, June, 2000.     

Absence of gravitational radiation from a nonstatic spherically symmetric body in the relativistic theory of gravity

In the framework of the relativistic theory of gravity, we establish the absence of gravitational radiation from a nonstatic spherically symmetric source.     

Gravitational waves in the relativistic theory of gravity

We demonstrate that because of the causality condition, gravitational waves have no unphysical “ghost” states in the relativistic theory of gravity with the graviton having a rest mass.