An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

Chiu, Hung-Lin; Huang, Yen‐Chang; Lai, Sin-Hua

doi:10.3842/sigma.2017.097

Cited by 10 publications

(27 citation statements)

References 24 publications

(32 reference statements)

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“…The results in [4] and [5] have been used by Ferrari and Valdinoci [16, § 2] to obtain geometric inequalities in H 1 . Several recent works use the distance function and techniques of Integral Geometry to obtain geometric inequalities in sub-Riemannian manifolds (e.g., [9], [13], [27]). From the Brunn-Minkowski type inequality obtained by Leonardi and Masnou [30], lower estimates of the volume of a tubular neighborhood of a given set can be obtained.…”

Section: Introductionmentioning

confidence: 99%

Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Ritoré

2017

Advances in Calculus of Variations

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We consider the Carnot-Carath\'eodory distance $\delta_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\mathbb{H}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of $\delta_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\ge 2$, we prove that $\delta_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\partial E$ and of the function $\langle N,T\rangle/|N_h|$ and its tangent derivatives.Comment: 44 pages. Accepted version to appear in Adv. Calc. Va

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Section: Introductionmentioning

confidence: 99%

Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Ritoré

2017

Advances in Calculus of Variations

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“…Along the curve γ, as in [1], we define the p-curvatures κ j (s), 1 ≤ j ≤ n and the contact normality τ (s) as We point out that all quantities above are invariant under the group actions of P SH(n). Our main theorem shows that those invariants completely determine the non-degenerate horizontally regular curve up to a Heisenberg rigid motion, which is analogues to the fundamental theorem of curves in R n .…”

Section: Introductionmentioning

confidence: 99%

The fundamental theorem for hypersurfaces in Heisenberg groups

Chiu

Lai

2015

Calc. Var.

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We study the horizontally regular curves in the Heisenberg groups Hn. We show the fundamental theorem of curves in Hn (n ≥ 2) and define the orders of horizontally regular curves. We also show that the curve γ is of order k if and only if, up to a Heisenberg rigid motion, γ lies in H k but not in H k−1 ; moreover, two curves with the same order differ from a rigid motion if and only if they have the same invariants: p-curvatures and contact normality. Thus, combining with our previous work [1] we have completed the classification of horizontally regular curves in Hn for n ≥ 1.

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“…In Proposition 4.1 in [15], the authors showed that any horizontally regular curve can be parametrized by the horizontal arc-length s such that |γ ξ (s)| = . Throughout the article, we always assume that the curve (or line) is parametrized under this condition.…”

Section: Invariants For Sets Of Horizontal Linesmentioning

confidence: 99%

“…Similarly to the group of rigid motions in R , in the previous work [15] we showed that any pseudohermitian transformation Φ Q,α ∈ PSH( ) can be represented by a left-invariant translation L Q for Q = (a, b, c) ∈ R and a rotation Rα ∈ SO( ). Actually there exists the following one-to-one correspondence between the group actions and the matrix multiplications…”

Section: Introductionmentioning

confidence: 99%

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

Huang

2017

Analysis and Geometry in Metric Spaces

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By studying the group of rigid motions, PSH( ), in the 3D-Heisenberg group H , we de ne a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also de ne the kinematic density for PSH( ) and show that the measure of all segments with length intersecting a convex domain D ⊂ H can be represented by the p-area of the boundary ∂D, the volume of D, and . 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.

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An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

Cited by 10 publications

References 24 publications

Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Tubular neighborhoods in the sub-Riemannian Heisenberg groups

The fundamental theorem for hypersurfaces in Heisenberg groups

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

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