Develops a model for the gravitational interaction in which the gravitational field is identified with the functions describing the embedding of spacetime in a ten-dimensional flat space. The choice of ten dimensions is dictated by the local isometric embedding theorem. No physical meaning is given to the extra dimensions; the embedding is used just as a mathematical tool in order to write the metric in a particular way. The Lagrangian for the model is the Hilbert Lagrangian, but is considered this time as a functional of the embedding functions rather than of the metric. The field equations are the Einstein equations contracted with the ten Gauss tensors of the embedding. For a static spherically symmetric field the correct asymptotic behaviour, the Schwarzschild solution, is obtained. Only six of the ten field equations are linearly independent, therefore a fixing of the gauge is necessary; the author fixes it in such a way that the dynamical fields measure the departure from flatness of spacetime. The ungauged energy-momentum tensor is identically zero; after fixing the gauge it turns out to be proportional to the Einstein tensor.

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We consider the conformal properties of geometries described by higher-rank line elements. A crucial role is played by the conformal Killing equation (CKE). We introduce the concept of null-flat spaces in which the line element can be written as ds r = r!dζ 1 · · · dζ r . We then show that, for null-flat spaces, the critical dimension, for which the CKE has infinitely many solutions, is equal to the rank of the metric. Therefore, in order to construct an integrable conformal field theory in 4 dimensions we need to rely on fourth-rank geometry. We consider the simple model L = 1 4 G µνλρ ∂ µ φ∂ ν φ∂ λ φ∂ ρ φ and show that it is an integrable conformal model in 4 dimensions. Furthermore, the associated symmetry group is V ir 4 . ICP. 04.50.+h Unified field theories and other theories of gravitation. *