scholarly journals Harmonic and Killing tensor fields in Riemannian spaces with boundary.

1958 ◽  
Vol 10 (4) ◽  
pp. 430-437 ◽  
Author(s):  
Kentaro YANO
Keyword(s):  
Killing Tensor ◽  
Tensor Fields ◽  
Riemannian Spaces

scholarly journals Killing tensor fields on the 2-torus

2016 ◽  
Vol 57 (1) ◽  
pp. 155-173 ◽  
Author(s):  
V. A. Sharafutdinov
Keyword(s):  
Killing Tensor ◽  
Tensor Fields

scholarly journals Tensor fields associated with integrable systems of chiral

2019 ◽  
Vol 21 (4) ◽  
pp. 405-412
Author(s):  
Alexander V. Balandin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Necessary Conditions ◽  
Differential Forms ◽  
Type Systems ◽  
Lax Representation ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Linear Differential ◽  
Special Case

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.

EINSTEIN'S THEORY OF GRAVITATION AS A LAGRANGIAN THEORY FOR TENSOR FIELDS

1998 ◽  
Vol 13 (12) ◽  
pp. 1941-1967 ◽  
Author(s):  
S. MANOFF
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein’S Equations ◽  
Tensor Fields ◽  
Einstein's Equations ◽  
Lagrange’S Equations ◽  
Theory Of Gravitation ◽  
Lagrange's Equations ◽  
Riemannian Spaces

Einstein's theory of gravitation (ETG) is considered as a Lagrangian theory of tensor fields over (pseudo) Riemannian spaces without torsion (Vn spaces, n=4) by means of the method of Lagrangians with covariant derivarives (MLCD). In a trivial manner Euler–Lagrange's equations as Einstein's equations are obtained. The corresponding energy–momentum tensors (EMT's) are found for the standard for the ETG Lagrangian invariant on the basis of the covariant Noether identities. The symmetric energy–momentum tensor of Hilbert appears as an element irrelevant to the whole scheme of the considered Lagrangian thoery of tensor fields over Vn spaces despite of the fact that it has some elements of the structure of the variational EMT of Euler–Lagrange. The notion of the active gravitational rest mast density is related to the variational EMT of Euler–Lagrange and on this basis to a certain extent to the EMT of Hilbert.

scholarly journals Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

2021 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Vladimir A. Sharafutdinov
Keyword(s):  
Tensor Field ◽  
Killing Vector ◽  
First Integral ◽  
Symmetric Tensor ◽  
Two Dimensional ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Killing Field ◽  
Quadratic Equations ◽  
Isothermal Coordinates

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero. Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= λ(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function λ satisfies the equation R(∂/∂z（λ(c∆^-1λ_zz+a))= 0 with some complex constants a and c≠0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function λ. If the functions λ and λ + λ_0 satisfy the equation for a real constant λ0, 0, then there exists a non-zero Killing vector field on the torus.

On Horizontal and Complete Lifts of $(1,1)\;$ Tensor Fields F Satisfying ${f(2K+S,S)=0}$

2016 ◽  
Vol 6 (1) ◽  
pp. 143
Author(s):  
Abhishek Singh ◽  
Ramesh Kumar Pandey ◽  
Sachin Khare
Keyword(s):  
Tensor Fields
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