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Special normal form of a hyperbolic CR-manifold in ℂ⁴

100%

Ežov V., Schmalz G.

Annales Polonici Mathematici

1998

tom 70

nr 1

99-107

We give a special normal form for a non-semiquadratic hyperbolic CR-manifold M of codimension 2 in ℂ⁴, i.e., a construction of coordinates where the equation of M satisfies certain conditions. The coordinates are determined up to a linear coordinate change.

Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group

88%

Huang Y.-C.

Analysis and Geometry in Metric Spaces

2016

tom 4

nr 1

By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments with length l intersecting a convex domain D ⊂ H1 can be represented by the p-area of the boundary ∂D, the volume of D, and 2l. Both results show the relationship between geometric probability and the natural geometric quantity in [10] derived by using variational methods. The probability that a line segment be contained in a convex domain is obtained as an application of our results.

Ograniczanie wyników

1 Analysis and Geometry in Metric Spaces

1 Annales Polonici Mathematici

1 Ežov V.

1 Huang Y.-C.

1 Schmalz G.

1 2016

1 1998

Wyniki wyszukiwania

Special normal form of a hyperbolic CR-manifold in ℂ⁴

Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group