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About the densities for straight lines in semi-Riemannian spaces. (English) Zbl 1190.53072
The author considers some families of all linear \(r\)-subspaces of a given \(n\)-dimensional space. The aim is to find the densities of such families, which are \((n-r)(r+1)\)-differential forms.
First, for \(n=3\), the cases \(r=2\) and \(r=1\) are considered, and the corresponding densities are computed in the Euclidean space.
For \(n=2\) and \(r=1\) the density is determined in the case of a semi-Riemannian space. From this, new formulas can be derived for the particular cases of Euclidean and Lorentzian plane.
Further, the density for the family of lines which are tangent to a differentiable curve in the Euclidean plane is obtained as a function of the curvature of the curve.
Also, the family of planes which are tangent to a differential surface is considered, and its density is computed in terms of the Hessian of the surface itself
53C65 Integral geometry