[U.sup.*kl.sub.jk]--displacement fundamental tensor
of classic elasticity theory ;
of the fundamental tensor
[g.sub.ij], where [psi] is a smooth function on the manifold.
Tavakol in 1994: G[L.sup.n] = (M, g) where g (the fundamental tensor) is a distinguished tensor field on [??], of type (0,2) symmetric, of rank n and having a constant signature on [??].
, [y.sup.(k)])), F : [T.sup.k]M [right arrow] R being positive k--homogeneous on the fibres of [T.sup.k]M and fundamental tensor has a constant signature.
As a submanifold of C2 it is not minimal but its second fundamental tensor field is parallel relative to the induced connection.
If a surface f : M [right arrow] [C.sup.2] is affine Lagrangian and its second fundamental tensor is parallel relative to the induced connection, then the surface is up to a complex affine transformation of [C.sup.2] the Clifford torus.
These theories are based on the geometrical interpretation of gravitation and electromagnetism by using a five dimensional generalized Riemannian spacetime of which non-symmetric fundamental tensor
consists of symmetric part coinciding with the fundamental tensor
of space-time representing the gravitational potential and the skew symmetric part signifying the electromagnetic field.
Since the metric (1.1) plays the part of a fundamental tensor
, we can introduce the contravariant components of the skew-symmetrical tensor field [summation] [V.sub.[alpha][beta]] [cross product] [dx.sub.[alpha][dx.sub.[beta]] with respect to (1.1).
The fundamental tensor fields of [pi] (O'Neill tensors) are given by
since the fundamental tensor A of [pi] satisfies [A.sub.x]Y = -[A.sub.y]X, .
We may therefore define an asymmetric fundamental tensor
of the gravoelectromagnetic manifold [S.sub.2] by
As it is known, the interval [ds.sup.2] = [g.sub.[alpha][beta]][dx.sup.[alpha]][dx.sup.[beta]] can not be fully degenerated in a Riemannian space ([double dagger]): the condition is that the determinant of the metric fundamental tensor
[g.sub.[alpha][beta]] must be strictly negative g = det ||[g.sub.[alpha][beta]]|| < 0 by definition of Riemannian spaces.