Tensor Fields on C1 Fuzzy Compact Manifolds

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

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Local well-posedness for the quadratic Schrödinger equation in two-dimensional compact manifolds with boundary

São Paulo Journal of Mathematical Sciences ◽

10.1007/s40863-021-00249-w ◽

2021 ◽

Author(s):

Marcelo Nogueira ◽

Mahendra Panthee

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Compact Manifolds ◽

Well Posedness ◽

Two Dimensional ◽

Manifolds With Boundary

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Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Symmetry ◽

10.3390/sym13050830 ◽

2021 ◽

Vol 13 (5) ◽

pp. 830

Author(s):

Evgeniya V. Goloveshkina ◽

Leonid M. Zubov

Keyword(s):

Nonlinear Theory ◽

Hollow Sphere ◽

Tensor Field ◽

Exact Analytical Solution ◽

Symmetric Tensor ◽

Symmetric Distribution ◽

Spherically Symmetric ◽

Tensor Fields ◽

Spherically Symmetric Distribution ◽

Nonlinear Elastic

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

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Gauge × gauge = gravity on homogeneous spaces using tensor convolutions

Journal of High Energy Physics ◽

10.1007/jhep06(2021)117 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

L. Borsten ◽

I. Jubb ◽

V. Makwana ◽

S. Nagy

Keyword(s):

Homogeneous Spaces ◽

Gauge Fields ◽

Convolution Product ◽

Tensor Fields ◽

Einstein Universe ◽

Gauge Transformations ◽

Yang Mills ◽

Static Einstein Universe ◽

Definition Of ◽

Gauge Gravity

Abstract A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

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