The combination of these features serves to penalize biaxiality less, encouraging localized biaxiality more as a way for equilibrium

tensor fields to avoid the large free energy costs of isotropic cores in defects, and this is the motivation for papers such as [4,13] to analyze this limit rigorously.

Vector fields and Lie series solve systems of ordinary differential equations, and metric

tensor fields model curves and Riemann manifolds.

Sorella, "Perturbation theory for antisymmetric

tensor fields in four dimensions," International Journal of Modern Physics A, vol.

Two new features of the present paper are: i) the use of local expressions for all the involved objects, which leads to a better picture; for example in Section 2, for the structural and virtual

tensor fields of a pair (linear connection, endomorphism), ii) a special attention is given to the mean covariant derivative [[nabla].sup.0] which parallelizes the given [lambda].

Let us consider a sequence of

tensor fields {[[LAMBDA].sub.1], [[LAMBDA].sub.2], ..., [[LAMBDA].sub.n]} generated by a Lame

tensor field [[LAMBDA].sub.1] :[OMEGA] [??] L(V; V) and by a sequence of positive scalar functions ..., [h.sub.n]}; that is, [h.sub.i] : [X.sub.i] [subset or equal to] R [??]]0, + [infinity][, according to the rule:

The "spatial" objects are defined above: the scalar w, the vectors [PHI], P, and the tensor [SIGMA] are simply and rigorously

tensor fields on the manifold [M.sub.F].

Thus, the two (1,1)-type

tensor fields l and h are symmetric and satisfy

Moreover, we observe that there is a close relationship between (1,1)-tensors satisfying certain conditions in terms of

tensor fields defined on generalized manifolds and multiplicative forms.

Weickert, "Tensor median filtering and M-smoothing," in Visualization and Processing of

Tensor Fields, Springer, Berlin, Germany, 2006.

In a previous paper [1], we proposed a natural decomposition of spacetime continuum

tensor fields, based on the continuum mechanical decomposition of tensors in terms of dilatations and distortions.

Hart, "Strategies for direct volume rendering of diffusion

tensor fields," IEEE Transactions on Visualization and Computer Graphics, vol.