Abstract.
In our previous paper, for a generic rank-2 vector distributions on an n-dimensional manifold (n ≥ 5) we constructed a special differential invariant, the fundamental form. In the case n = 5, this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E. Cartan, using his “reduction-prolongation” procedure (we call this form the Cartan tensor). In the present paper, we prove that our fundamental form coincides (up to the constant factor −35) with the Cartan tensor. This result explains geometrically the reason for the existence of the Cartan tensor (originally, this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in the Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for the Cartan tensor in terms of the structural functions of any frame naturally adapted to the distribution.
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2000 Mathematics Subject Classification. 58A30, 53A55.
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Zelenko, I. Fundamental Form and the Cartan Tensor of (2,5)-Distributions Coincide. J Dyn Control Syst 12, 247–276 (2006). https://doi.org/10.1007/s10450-006-0383-1
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DOI: https://doi.org/10.1007/s10450-006-0383-1