## 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

### MSC:

 53C65 Integral geometry

### Keywords:

curvature; density; Hessian; semi-Riemannian space