The Darboux Tensor on Hypersurfaces in Spaces with Metric and Torsion: A Complete Construction

We present the complete construction of the generalized Darboux tensor on hypersurfaces embedded in a spacetime equipped with a metric and a torsionful connection. By working exclusively with the symmetric part of the second fundamental form π (αβ) , we demonstrate that the classical formula, featuring a free constant C, remains valid when torsion is present. The torsion does not introduce new explicit terms into the Darboux tensor itself; rather, its effects are entirely absorbed into the definition of the symmetric second fundamental form and into the induced connection used for covariant differentiation. We prove that this tensor satisfies the apolarity condition π (αβ) θ αβγ = 0 for any C, derive its associated cubic form, and analyze Darboux directions in the two-dimensional case. An explicit example in ℝ 4 with torsion illustrates the construction. The Cesa`ro polynomial remains projectively invariant, offering a natural separation between gravitational and electromagnetic degrees of freedom when the torsion trace is identified with the electromagnetic potential.