The homeomorphism of Minkowski space and the separable complex Hilbert space: the physical, mathematical and philosophical interpretations

A homeomorphism is built between the separable complex Hilbert space and Minkowski space by meditation of quantum information. That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering. Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy.

Geometry of semi-Riemannian group manifold and its applications in spacetime admitting Ricci solitons

In this work, we show that semi-Riemannian group manifold admits Ricci solitons and satisfies the dynamical cosmology equation of spacetime. In Sec. 2, we introduce and provide some geometric properties of semisymmetric nonmetric connection in semi-Riemannian space. In Sec. 3, we define and show some geometric properties of group manifold endowed with semisymmetric nonmetric connection in semi-Riemannian space. In the section that follows, we give a condition of a group manifold to be Ricci solitons and gradient Ricci soliton. In Sec. 5, we provide the applications of group manifold admitting Ricci solitons in the theory of general relativity.

Geometric theory of non-regular separation of variables and the bi-Helmholtz equation

The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic–hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation.

Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

The recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.

A New Algebraic Inequality and Some Applications in Submanifold Theory

We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.

LAWS OF MOTION IN THE FRAME THEORIES OF GRAVITY

One of the most striking features of the general relativity (GR) is the fact that the matter that generates gravitational field cannot move arbitrarily, but must obey certain equations that follow from equations of the gravitational field as a condition of their compatibility. This fact was first noted in the fundamental Hilbert's work, in which equations of GR saw the world for the first time as variational Lagrange equations. Hilbert showed that in the case when fulfilling equations of the gravitational field which were born by an electromagnetic field, four linear combinations of equations of the electromagnetic field and their derivatives are zero due to the general covariance of the theory. It is known that this is what stimulated E. Noether to invent her famous theorem. As for "solid matter", for the compatibility of equations of the gravitational field, it is necessary that particles of dust matter move along geodesics of Riemannian space, which describes the gravitational field. This fact was pointed out in the work of A. Einstein and J. Grommer and according to V. Fock it is one of the main justifications of GR (although even before the creation of GR it was known that the motion along geodesics is a consequence of the condition of covariant conservation of energy-momentum of matter). This remarkable feature of GR all his life inspired Einstein to search on the basis of GR such theory from which it would be possible to derive all fundamental physics, including quantum mechanics. Interest in this problem (following Einstein, we name it the problem of motion) has resumed in our time in connection with the registration of gravitational waves and analysis of the conditions of their radiation, i.e. the need for its direct application in gravitational-wave astronomy. In this article we consider the problem to what extent the motion of matter that generates the gravitational field can be arbitrary. Considered problem is analyzed from the point of view symmetry of the theory with respect to the generalized gauge deformed groups without specification of Lagrangians. In particular it is shown, that the motion of particles along geodesics of Riemannian space is inherent in an extremely wide range of theories of gravity and is a consequence of the gauge translational invariance of these theories under the condition of fulfilling equations of gravitational field. In addition, we found relationships of equations for some fields that follow from the assumption about fulfilling of equations for other fields, for example, relationships of equations of the gravitational field which follow from the assumption about fulfilling of equations of matter fields.

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.