Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group

Abstract

We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean C 2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in H is provided.

Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane

“…Proof. We know that ||γ(t)|| L = γ 1 γ 1 2 +γ 2 3 + Lω(γ(t)) 2 , similar to the proof of Lemma 6.1 in [1], we can prove (4.2). When ω(γ(t)) = 0, we have…”

“…Proof. By (3.33) and Lemma 6.5, we have By (6.20),(6.23) and Lemma 6.2, similar to the proof of Theorem 1 in [1], we have Theorem 6.7. Let Σ 1 ⊂ (E(1, 1), g L ) be a regular surface with finitely many boundary components (∂Σ 1 ) i , i ∈ {1, · · · , n}, given by Euclidean C 2 -smooth regular and closed curves γ i : [0, 2π] → (∂Σ 1 ) i .…”

“…Remark. Using the similar discussions of the page 27 in [1], we can relax the condition that the characteristic set C(Σ) is the empty set and only suppose that the characteristic set C(Σ) satisfies H 1 (C(Σ)) = 0 where H 1 (C(Σ)) denotes the Euclidean 1-dimensional Hausdorff measure of C(Σ) and that ||∇ H u|| −1 H is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set C(Σ).…”

“…The image of Gauss map was in the cylinder of radius one. In [1], Balogh-Tyson-Vecchi used a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C 2 -smooth surface in the Heisenberg group H 1 away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. These results were then used to prove a Heisenberg version of the Gauss-Bonnet theorem.…”

“…Such cancellation stems from the specific choice of the adapted frame bundle on the surface, and on symmetries of the underlying left-invariant group structure on the Heisenberg group. In [1], they proposed an interesting question to understand to what extent similar phenomena hold in other sub-Riemannian geometric structures. In this paper, we solve this problem for the affine group and the group of rigid motions of the Minkowski plane.…”