About the densities for straight lines in semi-Riemannian spaces.

*(English)*Zbl 1190.53072The 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

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

Reviewer: Paolo Dulio (Milano)

##### MSC:

53C65 | Integral geometry |