Affine connections and Gauss-Bonnet theorems in the Heisenberg group
aa r X i v : . [ m a t h . DG ] F e b AFFINE CONNECTIONS AND GAUSS-BONNET THEOREMS IN THEHEISENBERG GROUP
YONG WANG
Abstract.
In this paper, we compute sub-Riemannian limits of Gaussian curvatureassociated to two kinds of Schouten-Van Kampen affine connections and the adaptedconnection for a Euclidean C -smooth surface in the Heisenberg group away from char-acteristic points and signed geodesic curvature associated to two kinds of Schouten-VanKampen affine connections and the adapted connection for Euclidean C -smooth curveson surfaces. We get Gauss-Bonnet theorems associated to two kinds of Schouten-VanKampen affine connections in the Heisenberg group. Introduction
In [5], Gaussian curvature for non-horizontal surfaces in sub-Riemannian Heisenbergspace H was defined and a Gauss-Bonnet theorem was proved. In [1],[2], Balogh-Tyson-Vecchi used a Riemannnian approximation scheme to define a notion of intrinsic Gauss-ian curvature for a Euclidean C -smooth surface in the Heisenberg group H away fromcharacteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean C -smooth curves on surfaces. These results were then used to prove a Heisenberg ver-sion of the Gauss-Bonnet theorem. In [8], Veloso verified that Gausssian curvature ofsurfaces and normal curvature of curves in surfaces introduced by [5] and by [1] to proveGauss-Bonnet theorems in Heisenberg space H were unequal and he applied the sameformalism of [5] to get the curvatures of [1]. With the obtained formulas, it is possibleto prove the Gauss-Bonnet theorem in [1] as a straightforward application of the Stokestheorem. In [9], we proved Gauss-Bonnet theorems for the affine group and the group ofrigid motions of the Minkowski plane. In [10], we obtained Gauss-Bonnet theorems forBCV spaces and the twisted Heisenberg group.In [6], Klatt proved a Gauss-Bonnet theorem associated to a metric connection (seeProposition 5.2 in [6]). When a Riemannian manifold has a splitting tangent bundle, wecan define a Schouten-Van Kampen affine connection which is a metric connection. In[3], [7], Schouten-Van Kampen affine connections on foliations and almost (para) contact Key words and phrases:
Schouten-van Kampen affine connections; the adapted connection, Gauss-Bonnet theo-rem; sub-Riemannian limit, Heisenberg group
Corresponding author:
Yong Wang . manifolds were studied. In [5], in order to prove a Gauss-Bonnet theorem in the Heisen-berg group, the adapted connection was introduced. The adapted connection is a metricconnection. Motivated by above works, it is interesting to study Gauss-Bonnet theoremsassociated to Schouten-Van Kampen affine connections and the adapted connection in theHeisenberg group. Let K Σ , ∇ , ∞ and k ∞ , ∇ ,sγ i , Σ be the intrinsic Gauss curvature associatedto the second kind of Schouten-Van Kampen affine connections and the intrinsic signedgeodesic curvature associated to the second kind of Schouten-van Kampen affine connec-tions. Our main theorem (Theorem 3.9) in this paper is as following: (see Section 3 forrelated definitions) Theorem 1.1.
Let Σ ⊂ ( H , g L ) be a regular surface with finitely many boundary com-ponents ( ∂ Σ) i , i ∈ { , · · · , n } , given by Euclidean C -smooth regular and closed curves γ i : [0 , π ] → ( ∂ Σ) i . Suppose that the characteristic set C (Σ) satisfies H ( C (Σ)) = 0 andthat ||∇ H u || − H is locally summable with respect to the Euclidean -dimensional Hausdorffmeasure near the characteristic set C (Σ) , then (1.1) Z Σ K Σ , ∇ , ∞ dσ Σ + n X i =1 Z γ i k ∞ , ∇ ,sγ i , Σ ds = 0 . In Section 2, we compute sub-Riemannian limits of Gaussian curvature associated to thefirst kind of Schouten-Van Kampen affine connections for a Euclidean C -smooth surfacein the Heisenberg group away from characteristic points and signed geodesic curvatureassociated to the first kind of Schouten-Van Kampen affine connections for Euclidean C -smooth curves on surfaces. We get the Gauss-Bonnet theorem associated to the firstkind of Schouten-Van Kampen affine connections in the Heisenberg group. In Section3, we compute sub-Riemannian limits of Gaussian curvature associated to the secondkind of Schouten-Van Kampen affine connections for a Euclidean C -smooth surface inthe Heisenberg group away from characteristic points and signed geodesic curvature as-sociated to the second kind of Schouten-Van Kampen affine connections for Euclidean C -smooth curves on surfaces. We get the Gauss-Bonnet theorem associated to the sec-ond kind of Schouten-Van Kampen affine connections in the Heisenberg group. In Section4, we compute sub-Riemannian limits of Gaussian curvature associated to the adaptedconnection for a Euclidean C -smooth surface in the Heisenberg group away from char-acteristic points and signed geodesic curvature associated to the adapted connection forEuclidean C -smooth curves on surfaces.2. The Gauss-Bonnet theorem associated to the first kind ofSchouten-Van Kampen affine connections in the Heisenberg group
Firstly we introduce some notations on the Heisenberg group. Let H be the Heisenberggroup R where the non-commutative group law is given by( a, b, c ) ⋆ ( x, y, z ) = ( a + x, b + y, c + z −
12 ( xb − ya )) . auss Bonnet theorems 3 Let(2.1) X = ∂ x − x ∂ x , X = ∂ x + x ∂ x , X = ∂ x , and span { X , X , X } = T H . Let H = span { X , X } be the horizontal distribution on H Let ω = dx , ω = dx , ω = ω = dx + ( x dx − x dx ) . For the constant
L >
0, let g L = ω ⊗ ω + ω ⊗ ω + Lω ⊗ ω be the Riemannian metric on H . Then X , X , f X := L − X are orthonormal basis on T H with respect to g L . We have(2.2) [ X , X ] = X , [ X , X ] = 0 , [ X , X ] = 0 . Let ∇ L be the Levi-Civita connection on H with respect to g L . By Lemma 2.8 in [1], wehave Lemma 2.1.
Let H be the Heisenberg group, then ∇ LX j X j = 0 , ≤ j ≤ , ∇ LX X = 12 X , ∇ LX X = − X , (2.3) ∇ LX X = − L X , ∇ LX X = − L X , ∇ LX X = ∇ LX X = L X . Let H ⊥ = span { X } and P : T H → H and P ⊥ : T H → H ⊥ be the projections. Wedefine the first kind of Schouten-Van Kampen affine connections in the Heisenberg group:(2.4) ∇ X Y = P ∇ LX P Y + P ⊥ ∇ LX P ⊥ Y. By Definition 3.1 in [1], we have
Definition 2.2.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth curve. We say that γ is regular if ˙ γ = 0 for every t ∈ [ a, b ] . Moreover we say that γ ( t ) is a horizontal point of γ if ω ( ˙ γ ( t )) = ˙ γ ( t ) γ ( t ) − ˙ γ ( t ) = 0 . Definition 2.3.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . The curvature k L, ∇ γ associated to ∇ of γ at γ ( t ) is definedas (2.5) k L, ∇ γ := s ||∇ ˙ γ ˙ γ || L || ˙ γ || L − h∇ ˙ γ ˙ γ, ˙ γ i L || ˙ γ || L . By (2.3) and (2.4), we have
Lemma 2.4.
Let H be the Heisenberg group, then ∇ X X = − L X , ∇ X X = L X , ∇ X j X k = 0 , for other X j , X k . (2.6) Wang
Let γ ( t ) = ( γ ( t ) , γ ( t ) , γ ( t )), then(2.7) ˙ γ ( t ) = ˙ γ X + ˙ γ X + ω ( ˙ γ ( t )) X . By (2.6) and (2.7), we have ∇ ˙ γ ˙ γ = (cid:20) ¨ γ + Lω ( ˙ γ ( t )) ˙ γ (cid:21) X + (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) X + ddt ( ω ( ˙ γ ( t ))) X . (2.8)Similar to Lemma 2.4 in [9], we have Lemma 2.5.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . Then k L, ∇ γ = (((cid:20) ¨ γ + Lω ( ˙ γ ( t )) ˙ γ (cid:21) + (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) + L (cid:20) ddt ( ω ( ˙ γ ( t ))) (cid:21) ) (2.9) · (cid:2) ˙ γ + ˙ γ + L ( ω ( ˙ γ ( t ))) (cid:3) − − (cid:26) ˙ γ (cid:20) ¨ γ + Lω ( ˙ γ ( t )) ˙ γ (cid:21) + ˙ γ (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) + Lω ( ˙ γ ( t )) ddt ( ω ( ˙ γ ( t ))) (cid:27) · (cid:2) ˙ γ + ˙ γ + L ( ω ( ˙ γ ( t ))) (cid:3) − o . In particular, if γ ( t ) is a horizontal point of γ , k L, ∇ γ = (( ¨ γ + ¨ γ + L (cid:20) ddt ( ω ( ˙ γ ( t ))) (cid:21) ) · (cid:2) ˙ γ + ˙ γ (cid:3) − (2.10) − { ˙ γ ¨ γ + ˙ γ ¨ γ } · (cid:2) ˙ γ + ˙ γ (cid:3) − o . Definition 2.6.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve inthe Riemannian manifold ( H , g L ) . We define the intrinsic curvature associated to theconnection ∇ , k ∞ , ∇ γ of γ at γ ( t ) to be k ∞ , ∇ γ := lim L → + ∞ k L, ∇ γ , if the limit exists. We introduce the following notation: for continuous functions f , f : (0 , + ∞ ) → R ,(2.11) f ( L ) ∼ f ( L ) , as L → + ∞ ⇔ lim L → + ∞ f ( L ) f ( L ) = 1 . Similar to Lemma 2.6 in [9], we have auss Bonnet theorems 5
Lemma 2.7.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . Then (2.12) k ∞ , ∇ γ = p ˙ γ + ˙ γ | ω ( ˙ γ ( t )) | , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ γ = | ¨ γ ˙ γ − ¨ γ ˙ γ | ( ˙ γ + ˙ γ ) , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 , (2.13)(2.14) lim L → + ∞ k L, ∇ γ √ L = | ddt ( ω ( ˙ γ ( t ))) | ˙ γ + ˙ γ , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 . We will say that a surface Σ ⊂ ( H , g L ) is regular if Σ is a Euclidean C -smooth compactand oriented surface. In particular we will assume that there exists a Euclidean C -smoothfunction u : H → R such thatΣ = { ( x , x , x ) ∈ G : u ( x , x , x ) = 0 } and u x ∂ x + u x ∂ x + u x ∂ x = 0 . Let ∇ H u = X ( u ) X + X ( u ) X . A point x ∈ Σ is called characteristic if ∇ H u ( x ) = 0. We define the characteristic set C (Σ) := { x ∈ Σ |∇ H u ( x ) =0 } . Our computations will be local and away from characteristic points of Σ. Let us definefirst p := X u, q := X u, and r := e X u. We then define l := p p + q , l L := p p + q + r , p := pl , (2.15) q := ql , p L := pl L , q L := ql L , r L := rl L . In particular, p + q = 1. These functions are well defined at every non-characteristicpoint. Let v L = p L X + q L X + r L f X , e = qX − pX , e = r L pX + r L qX − ll L f X , (2.16)then v L is the Riemannian unit normal vector to Σ and e , e are the orthonormal basisof Σ. On T Σ we define a linear transformation J L : T Σ → T Σ such that(2.17) J L ( e ) := e ; J L ( e ) := − e . For every
U, V ∈ T Σ, we define ∇ Σ U V = π ∇ U V where π : T H → T Σ is the projection.Then ∇ Σ is the metric connection on Σ with respect to the metric g L . By (2.8),(2.16),we have(2.18) ∇ Σ˙ γ ˙ γ = h∇ ˙ γ ˙ γ, e i L e + h∇ ˙ γ ˙ γ, e i L e , Wang we have ∇ Σ˙ γ ˙ γ = (cid:26) q (cid:20) ¨ γ + Lω ( ˙ γ ( t )) ˙ γ (cid:21) − p (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21)(cid:27) e (2.19) + (cid:26) r L p (cid:20) ¨ γ + Lω ( ˙ γ ( t )) ˙ γ (cid:21) + r L q (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) − ll L L ddt ( ω ( ˙ γ ( t ))) (cid:21)(cid:27) e . Definition 2.8.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. The geodesic curvature associated to ∇ , k L, ∇ γ, Σ of γ at γ ( t ) isdefined as (2.20) k L, ∇ γ, Σ := s ||∇ Σ˙ γ ˙ γ || ,L || ˙ γ || ,L − h∇ Σ˙ γ ˙ γ, ˙ γ i ,L || ˙ γ || ,L . Definition 2.9.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. We define the intrinsic geodesic curvature associated to ∇ , k ∞ , ∇ γ, Σ of γ at γ ( t ) to be k ∞ , ∇ γ, Σ := lim L → + ∞ k L, ∇ γ, Σ , if the limit exists. Similar to Lemma 3.3 in [9], we have
Lemma 2.10.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. Then (2.21) k ∞ , ∇ γ, Σ = | p ˙ γ + q ˙ γ | | ω ( ˙ γ ( t )) | , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ γ, Σ = 0 if ω ( ˙ γ ( t )) = 0 , and ddt ( ω ( ˙ γ ( t ))) = 0 , (2.22) lim L → + ∞ k L, ∇ γ, Σ √ L = | ddt ( ω ( ˙ γ ( t ))) | ( q ˙ γ − p ˙ γ ) , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 . Definition 2.11.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. The signed geodesic curvature associated to ∇ , k L, ∇ ,sγ, Σ of γ at γ ( t ) is defined as (2.23) k L, ∇ ,sγ, Σ := h∇ Σ˙ γ ˙ γ, J L ( ˙ γ ) i Σ ,L || ˙ γ || ,L . auss Bonnet theorems 7 Definition 2.12.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. We define the intrinsic geodesic curvature associated to ∇ , k ∞ , ∇ ,sγ, Σ of γ at the non-characteristic point γ ( t ) to be k ∞ , ∇ ,sγ, Σ := lim L → + ∞ k L, ∇ ,sγ, Σ , if the limit exists. Similar to Lemma 3.6 in [9], we have
Lemma 2.13.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. Then (2.24) k ∞ , ∇ ,sγ, Σ = p ˙ γ + q ˙ γ | ω ( ˙ γ ( t )) | , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ ,sγ, Σ = 0 if ω ( ˙ γ ( t )) = 0 , and ddt ( ω ( ˙ γ ( t ))) = 0 , (2.25) lim L → + ∞ k L, ∇ ,sγ, Σ √ L = ( − q ˙ γ + p ˙ γ ) ddt ( ω ( ˙ γ ( t ))) | q ˙ γ − p ˙ γ | , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 . In the following, we compute the sub-Riemannian limit of the Gaussian curvature asso-ciated to ∇ of surfaces in the Heisenberg group. We define the second fundamental formassociated to ∇ , II ∇ ,L of the embedding of Σ into ( H , g L ):(2.26) II ∇ ,L = (cid:18) h∇ e v L , e ) i L , h∇ e v L , e ) i L h∇ e v L , e ) i L , h∇ e v L , e ) i L (cid:19) . Similarly to Theorem 4.3 in [4], we have
Theorem 2.14.
The second fundamental form II ∇ ,L of the embedding of Σ into ( H , g L ) is given by (2.27) II ∇ ,L = (cid:18) h , h h , h (cid:19) , where h = ll L [ X ( p ) + X ( q )] , h = − l L l h e , ∇ H ( r L ) i L ,h = − l L l h e , ∇ H ( r L ) i L − √ L − √ L r L ,h = − l l L h e , ∇ H ( rl ) i L + f X ( r L ) . Wang
Proof.
By Theorem 4.3 in [4] and Lemma 2.4, we have h∇ e v L , e i L = h∇ Le v L , e i L , h∇ e v L , e i L = h∇ Le v L , e i L + √ L , (2.28) h∇ e v L , e i L = h∇ Le v L , e i L − √ L r L , h∇ e v L , e i L = h∇ Le v L , e i L , By Theorem 4.3 in [4] and (2.28), we get this theorem. (cid:3)
The mean curvature associated to ∇ , H ∇ ,L of Σ is defined by H ∇ ,L := tr( II ∇ ,L ) . Define the curvature of a connection ∇ by(2.29) R ( X, Y ) Z = ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X,Y ] . Let(2.30) K Σ , ∇ ( e , e ) = −h R Σ ( e , e ) e , e i Σ ,L , K ∇ ( e , e ) = −h R ( e , e ) e , e i L . By the Gauss equation (in fact the Gauss equation holds for any metric connections), wehave(2.31) K Σ , ∇ ( e , e ) = K ∇ ( e , e ) + det( II ∇ ,L ) . Similar to Proposition 3.8 in [9], we have
Proposition 2.15.
Away from characteristic points, the horizontal mean curvature as-sociated to ∇ , H ∇ , ∞ of Σ ⊂ H is given by (2.32) H ∇ , ∞ = lim L → + ∞ H ∇ ,L = X ( p ) + X ( q ) . By Lemma 2.4 and (2.29), we have
Lemma 2.16.
Let H be the Heisenberg group, then R ( X , X ) X = L X , R ( X , X ) X = − L X , R ( X i , X j ) X k = 0 , for other i, j, k. Proposition 2.17.
Away from characteristic points, we have (2.33) K Σ , ∇ ( e , e ) → K Σ , ∇ , ∞ + O ( 1 √ L ) , as L → + ∞ , where (2.34) K Σ , ∇ , ∞ := − h e , ∇ H ( X u |∇ H u | ) i − ( X u ) p + q ) . auss Bonnet theorems 9 Proof.
By Lemma 2.16 and similar to (3.33) and (3.34) in [9], we have K ∇ ( e , e ) = − L r L . (2.35)By Theorem 2.14, (2.31) and (2.35), similar to Proposition 3.10 in [9], we can obtain thisproposition. (cid:3) Let us first consider the case of a regular curve γ : [ a, b ] → ( H , g L ). We define theRiemannian length measure ds L = || ˙ γ || L dt. By [1], we have(2.36) 1 √ L ds L → ds := | ω ( ˙ γ ( t )) | dt as L → + ∞ . (2.37) 1 √ L e ∗ ∧ e ∗ → dσ Σ := pω ∧ ω − qω ∧ ω as L → + ∞ , where e ∗ , e ∗ are the dual basis of e , e . We recall the local Gauss-Bonnet theorem for themetric connection(see Proposition 5.2 in [6]). Theorem 2.18.
Let Σ be an oriented compact two-dimensional manifold with manyboundary components ( ∂ Σ) i , i ∈ { , · · · , n } , given by Euclidean C -smooth regular andclosed curves γ i : [0 , π ] → ( ∂ Σ) i . Let ∇ be a metric connection and K ∇ be the Gausscurvature associated to ∇ and k s, ∇ γ i be the signed geodesic curvature associated to ∇ , then (2.38) Z Σ K ∇ dσ Σ + n X i =1 Z γ i k s, ∇ γ i ds = 2 πχ ( M ) . By Lemma 2.13 and Proposition 2.17 and Theorem 2.18, similar to the proof of Theorem1.1 in [1], we have
Theorem 2.19.
Let Σ ⊂ ( H , g L ) be a regular surface with finitely many boundary com-ponents ( ∂ Σ) i , i ∈ { , · · · , n } , given by Euclidean C -smooth regular and closed curves γ i : [0 , π ] → ( ∂ Σ) i . Suppose that the characteristic set C (Σ) satisfies H ( C (Σ)) = 0 andthat ||∇ H u || − H is locally summable with respect to the Euclidean -dimensional Hausdorffmeasure near the characteristic set C (Σ) , then (2.39) Z Σ K Σ , ∇ , ∞ dσ Σ + n X i =1 Z γ i k ∞ , ∇ ,sγ i , Σ ds = 0 . By Lemma 2.13 and (2.34), we note that Theorem 2.19 is the same as Theorem 1.1 in[1] up to the scaler .3. The Gauss-Bonnet theorem associated to the second kind ofSchouten-van Kampen affine connections in the Heisenberg group
Let H = span { X , X } and H ⊥ = span { X } and P : T H → H and P ⊥ : T H → H ⊥ bethe projections. We define the second kind of Schouten-van Kampen affine connectionsin the Heisenberg group:(3.1) ∇ X Y = P ∇ LX P Y + P ⊥ ∇ LX P ⊥ Y. By Lemma 2.1 and (3.1), we have
Lemma 3.1.
Let H be the Heisenberg group, then ∇ X X = 12 X , ∇ X X = − L X , ∇ X j X k = 0 , for other X j , X k . (3.2)Similar to the definition 2,3, we can define the curvature k L, ∇ γ associated to ∇ of γ at γ ( t ). By Lemma 3.1 and (2.7), we have ∇ γ ˙ γ = ¨ γ X + (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) X + (cid:20) ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ (cid:21) X . (3.3)Similar to Lemma 2.5, we have Lemma 3.2.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . Then k L, ∇ γ = (( ¨ γ + (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) + L (cid:20) ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ (cid:21) ) (3.4) · (cid:2) ˙ γ + ˙ γ + L ( ω ( ˙ γ ( t ))) (cid:3) − − (cid:26) ˙ γ ¨ γ + ˙ γ (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) + Lω ( ˙ γ ( t )) (cid:20) ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ (cid:21)(cid:27) · (cid:2) ˙ γ + ˙ γ + L ( ω ( ˙ γ ( t ))) (cid:3) − o . In particular, if γ ( t ) is a horizontal point of γ , k L, ∇ γ = (( ¨ γ + ¨ γ + L (cid:20) ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ (cid:21) ) · (cid:2) ˙ γ + ˙ γ (cid:3) − (3.5) − { ˙ γ ¨ γ + ˙ γ ¨ γ } · (cid:2) ˙ γ + ˙ γ (cid:3) − o . Similar to the definition 2.6, we can define the intrinsic curvature associated to theconnection ∇ , k ∞ , ∇ γ of γ at γ ( t ). Similar to the lemma 2.7, we have Lemma 3.3.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . Then (3.6) k ∞ , ∇ γ = | ˙ γ | | ω ( ˙ γ ( t )) | , if ω ( ˙ γ ( t )) = 0 , auss Bonnet theorems 11 k ∞ , ∇ γ = | ¨ γ ˙ γ − ¨ γ ˙ γ | ( ˙ γ + ˙ γ ) , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ = 0 , (3.7)(3.8)lim L → + ∞ k L, ∇ γ √ L = | ddt ( ω ( ˙ γ ( t ))) + ˙ γ ˙ γ | ˙ γ + ˙ γ , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ = 0 . For every
U, V ∈ T Σ, we define ∇ , Σ U V = π ∇ U V where π : T H → T Σ is the projection.Similar to (2.18), we have(3.9) ∇ , Σ˙ γ ˙ γ = h∇ γ ˙ γ, e i L e + h∇ γ ˙ γ, e i L e , and ∇ , Σ˙ γ ˙ γ = (cid:26) q ¨ γ − p (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21)(cid:27) e (3.10) + (cid:26) r L p ¨ γ + r L q (cid:20) ¨ γ − Lω ( ˙ γ ( t )) ˙ γ (cid:21) − ll L L (cid:20) ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ (cid:21)(cid:27) e . Similar to Definitions 2.8 and 2.9, we can define the geodesic curvature associated to ∇ , k L, ∇ γ, Σ of γ at γ ( t ) and the intrinsic geodesic curvature associated to ∇ , k ∞ , ∇ γ, Σ of γ at γ ( t ). Similar to Lemma 2.10, we have Lemma 3.4.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. Then (3.11) k ∞ , ∇ γ, Σ = | p ˙ γ | | ω ( ˙ γ ( t )) | , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ γ, Σ = 0 if ω ( ˙ γ ( t )) = 0 , and ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ = 0 , (3.12)lim L → + ∞ k L, ∇ γ, Σ √ L = | ddt ( ω ( ˙ γ ( t ))) + ˙ γ ˙ γ | ( q ˙ γ − p ˙ γ ) , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ = 0 . Similar to the definitions 2.11 and 2.12, we can define the signed geodesic curvatureassociated to ∇ , k L, ∇ ,sγ, Σ of γ at γ ( t ) and the intrinsic geodesic curvature associated to ∇ , k ∞ , ∇ ,sγ, Σ of γ . Similar to Lemma 2.13, we have Lemma 3.5.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. Then (3.13) k ∞ , ∇ ,sγ, Σ = p ˙ γ | ω ( ˙ γ ( t )) | , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ ,sγ, Σ = 0 if ω ( ˙ γ ( t )) = 0 , and ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ = 0 , (3.14) lim L → + ∞ k L, ∇ ,sγ, Σ √ L = ( − q ˙ γ + p ˙ γ ) (cid:2) ddt ( ω ( ˙ γ ( t ))) + ˙ γ ˙ γ (cid:3) | q ˙ γ − p ˙ γ | ,if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) + 12 ˙ γ ˙ γ = 0 . Similar to (2.26), we can define the second fundamental form associated to ∇ , II ∇ ,L of the embedding of Σ into ( H , g L ). Similarly to Theorem 2.14, we have Theorem 3.6.
The second fundamental form II ∇ ,L of the embedding of Σ into ( H , g L ) is given by (3.15) II ∇ ,L = (cid:18) h , h h , h (cid:19) , where h = ll L [ X ( p ) + X ( q )] + √ Lp q r L ,h = − l L l h e , ∇ H ( r L ) i L − r L q √ L − l l L q q L √ L,h = − l L l h e , ∇ H ( r L ) i L − √ L √ L l l L − √ L r L q ,h = − l l L h e , ∇ H ( rl ) i L + f X ( r L ) − √ L ll L p q L r L − √ L p q r L . Similar to (2.29) and (2.30), we can define R ( X, Y ) Z , K Σ , ∇ ( e , e ) and K ∇ ( e , e )(2.31) is correct for ∇ . Similar to Proposition 2.15, we have Proposition 3.7.
Away from characteristic points, the horizontal mean curvature asso-ciated to ∇ , H ∇ , ∞ of Σ ⊂ H is given by (3.16) H ∇ , ∞ = lim L → + ∞ H ∇ ,L = X ( p ) + X ( q ) . By Lemma 3.1, we have R ( X i , X j ) X k = 0 for any i, j, k . Similar to Proposition 2.17,we have Proposition 3.8.
Away from characteristic points, we have (3.17) K Σ , ∇ ( e , e ) → K Σ , ∇ , ∞ + O ( 1 √ L ) , as L → + ∞ , where (3.18) K Σ , ∇ , ∞ := − p q ( X ( u ))2 p p + q [ X ( p ) + X ( q )] − q (cid:20) h e , ∇ H ( X u |∇ H u | ) i + ( X u ) p + q (cid:21) . By Lemma 3.5 and Proposition 3.8, similar to Theorem 2.19, we have auss Bonnet theorems 13
Theorem 3.9.
Let Σ ⊂ ( H , g L ) be a regular surface with finitely many boundary com-ponents ( ∂ Σ) i , i ∈ { , · · · , n } , given by Euclidean C -smooth regular and closed curves γ i : [0 , π ] → ( ∂ Σ) i . Suppose that the characteristic set C (Σ) satisfies H ( C (Σ)) = 0 andthat ||∇ H u || − H is locally summable with respect to the Euclidean -dimensional Hausdorffmeasure near the characteristic set C (Σ) , then (3.19) Z Σ K Σ , ∇ , ∞ dσ Σ + n X i =1 Z γ i k ∞ , ∇ ,sγ i , Σ ds = 0 . We note that Theorem 3.9 is different from Theorem 1.1 in [1].4.
The sub-Riemannian limit and the adapted connection
We define the adapted connection ∇ on the Heisenberg group by ∇ X j X k = 0 for any j, k . Then ∇ is a metric connection. Similar to the definition 2.3, we can define thecurvature k L, ∇ γ associated to ∇ of γ at γ ( t ). We have ∇ γ ˙ γ = ¨ γ X + ¨ γ X + ddt ( ω ( ˙ γ ( t ))) X . (4.1)Similar to Lemma 2.5, we have Lemma 4.1.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . Then k L, ∇ γ = (( ¨ γ + ¨ γ + L (cid:20) ddt ( ω ( ˙ γ ( t ))) (cid:21) ) (4.2) · (cid:2) ˙ γ + ˙ γ + L ( ω ( ˙ γ ( t ))) (cid:3) − − (cid:26) ˙ γ ¨ γ + ˙ γ ¨ γ + Lω ( ˙ γ ( t )) ddt ( ω ( ˙ γ ( t ))) (cid:27) · (cid:2) ˙ γ + ˙ γ + L ( ω ( ˙ γ ( t ))) (cid:3) − o . In particular, if γ ( t ) is a horizontal point of γ , k L, ∇ γ = (( ¨ γ + ¨ γ + L (cid:20) ddt ( ω ( ˙ γ ( t ))) (cid:21) ) · (cid:2) ˙ γ + ˙ γ (cid:3) − (4.3) − { ˙ γ ¨ γ + ˙ γ ¨ γ } · (cid:2) ˙ γ + ˙ γ (cid:3) − o . Similar to the definition 2.6, we can define the intrinsic curvature associated to theconnection ∇ , k ∞ , ∇ γ of γ at γ ( t ). Similar to the lemma 2.7, we have Lemma 4.2.
Let γ : [ a, b ] → ( H , g L ) be a Euclidean C -smooth regular curve in theRiemannian manifold ( H , g L ) . Then (4.4) k ∞ , ∇ γ = 0 , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ γ = | ¨ γ ˙ γ − ¨ γ ˙ γ | ( ˙ γ + ˙ γ ) , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 , (4.5)(4.6) lim L → + ∞ k L, ∇ γ √ L = | ddt ( ω ( ˙ γ ( t ))) | ˙ γ + ˙ γ , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 . Similar to (3.10), we have ∇ , Σ˙ γ ˙ γ = { q ¨ γ − p ¨ γ } e + (cid:26) r L p ¨ γ + r L q ¨ γ − ll L L ddt ( ω ( ˙ γ ( t ))) (cid:27) e . Similar to Definitions 2.8 and 2.9, we can define the geodesic curvature associated to ∇ , k L, ∇ γ, Σ of γ at γ ( t ) and the intrinsic geodesic curvature associated to ∇ , k ∞ , ∇ γ, Σ of γ at γ ( t ). Similar to Lemma 2.10, we have Lemma 4.3.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. Then (4.7) k ∞ , ∇ γ, Σ = 0 , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ γ, Σ = 0 if ω ( ˙ γ ( t )) = 0 , and ddt ( ω ( ˙ γ ( t ))) = 0 , (4.8) lim L → + ∞ k L, ∇ γ, Σ √ L = | ddt ( ω ( ˙ γ ( t ))) | ( q ˙ γ − p ˙ γ ) , if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 . Similar to the definitions 2.11 and 2.12, we can define the signed geodesic curvatureassociated to ∇ , k L, ∇ ,sγ, Σ of γ at γ ( t ) and the intrinsic signed geodesic curvature associatedto ∇ , k ∞ , ∇ ,sγ, Σ of γ . Similar to Lemma 2.13, we have Lemma 4.4.
Let Σ ⊂ ( H , g L ) be a regular surface. Let γ : [ a, b ] → Σ be a Euclidean C -smooth regular curve. Then (4.9) k ∞ , ∇ ,sγ, Σ = 0 , if ω ( ˙ γ ( t )) = 0 ,k ∞ , ∇ ,sγ, Σ = 0 if ω ( ˙ γ ( t )) = 0 , and ddt ( ω ( ˙ γ ( t ))) = 0 , (4.10) lim L → + ∞ k L, ∇ ,sγ, Σ √ L = ( − q ˙ γ + p ˙ γ ) ddt ( ω ( ˙ γ ( t ))) | q ˙ γ − p ˙ γ | ,if ω ( ˙ γ ( t )) = 0 and ddt ( ω ( ˙ γ ( t ))) = 0 . auss Bonnet theorems 15 Similar to (2.26), we can define the second fundamental form associated to ∇ , II ∇ ,L of the embedding of Σ into ( H , g L ). Similarly to Theorem 2.14, we have Theorem 4.5.
The second fundamental form II ∇ ,L of the embedding of Σ into ( H , g L ) is given by (4.11) II ∇ ,L = (cid:18) h , h h , h (cid:19) , where h = ll L [ X ( p ) + X ( q )] , h = − l L l h e , ∇ H ( r L ) i L ,h = − l L l h e , ∇ H ( r L ) i L − √ L √ L l l L − √ L r L ,h = − l l L h e , ∇ H ( rl ) i L + f X ( r L ) . Similar to Propositions 2.15 and 2.17, we have
Proposition 4.6.
Away from characteristic points, we have (4.12) H ∇ , ∞ = lim L → + ∞ H ∇ ,L = X ( p ) + X ( q ) . (4.13) K Σ , ∇ ( e , e ) → K Σ , ∇ , ∞ = 0 , as L → + ∞ . Acknowledgements
The author was supported in part by NSFC No.11771070.
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