REPRESENTATION THEOREMS IN A 4-DIMENSIONAL EUCLIDEAN SPACE. THE CASE WITH ONLY SKEW-SYMMETRIC TENSORS

Abstract

Listen

Translate article

In a 4-dimensional Euclidean space, representation theorems have been recently obtained for isotropic functions depending on an arbitrary number of scalars, skew-symmetric second order tensors and symmetric second order tensors; the cases has been treated where at least one of these last ones has an eigenvalue with multiplicity 1 or two distinct eigenvalues with multiplicity 2. The case with at least a non null vector, among the independent variables, was already treated in literature. There remain the case where every symmetric tensor has an eigenvalue with multiplicity 4; but, in this case, it plays a role only through its trace. Consequently, it remains the case where the independent variables, besides scalars, are skew-symmetric tensors. This case is treated in the present paper. As in the other cases, the result is a finite set of scalar valued isotropic functions such that every other scalar function of the same variables can be expressed as a function of the elements of this set. Similarly, a set of tensor valued isotropic functions is found such that every other tensor valued function of the same variables can be expressed as a linear combination, trough scalar coefficients, of the elements of this set. This result i s achieved both for symmetric functions , and for skew-symmetric functions.