# IJPAM: Volume 93, No. 6 (2014)

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

M.C. Carrisi , S. Montisci , S. Pennisi Dipartimento di Matematica ed Informatica
Università di Cagliari
Via Ospedale 72, 09124, Cagliari, ITALY

Abstract. 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 is achieved both for symmetric functions , and for skew-symmetric functions.

Received: June 1, 2014

AMS Subject Classification: 15A72

Key Words and Phrases: representations theorems, scalar valued isotropic functions, skew-symmetric second order tensors, symmetric second order tensors

DOI: 10.12732/ijpam.v93i6.14 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 93
Issue: 6
Pages: 929 - This work is licensed under the Creative Commons Attribution International License (CC BY).