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Research article

Numerical solution of unsteady elastic equations with C-Bézier basis functions

  • Received: 16 August 2023 Revised: 23 October 2023 Accepted: 13 November 2023 Published: 04 December 2023
  • MSC : 65D18, 65M60

  • In this paper, the finite element method is applied to solve the unsteady elastic equations, C-Bézier basis functions are used to construct the shape function spaces, the semi-discrete scheme of the unsteady elastic equations is obtained by Galerkin finite element method and then the fully discretized Galerkin method is obtained by further discretizing the time variable with θ-scheme finite difference. Furthermore, for several numerical examples, the accuracy of approximate solutions are improved by 1–3 order-of magnitudes compared with the Lagrange basis function in L norm, L2 norm and H1 semi-norm, and the numerical examples show that the method proposed possesses a faster convergence rate. It is fully demonstrated that the C-Bézier basis functions have a better approximation effect in simulating unsteady elastic equations.

    Citation: Lanyin Sun, Kunkun Pang. Numerical solution of unsteady elastic equations with C-Bézier basis functions[J]. AIMS Mathematics, 2024, 9(1): 702-722. doi: 10.3934/math.2024036

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  • In this paper, the finite element method is applied to solve the unsteady elastic equations, C-Bézier basis functions are used to construct the shape function spaces, the semi-discrete scheme of the unsteady elastic equations is obtained by Galerkin finite element method and then the fully discretized Galerkin method is obtained by further discretizing the time variable with θ-scheme finite difference. Furthermore, for several numerical examples, the accuracy of approximate solutions are improved by 1–3 order-of magnitudes compared with the Lagrange basis function in L norm, L2 norm and H1 semi-norm, and the numerical examples show that the method proposed possesses a faster convergence rate. It is fully demonstrated that the C-Bézier basis functions have a better approximation effect in simulating unsteady elastic equations.



    In [4], Diniz and Veloso gave the definition of Gaussian curvature for non-horizontal surfaces in sub-Riemannian Heisenberg space H1 and the proof of the Gauss-Bonnet theorem. In [1], intrinsic Gaussian curvature for a Euclidean C2-smooth surface in the Heisenberg group H1 away from characteristic points and intrinsic signed geodesic curvature for Euclidean C2-smooth curves on surfaces are defined by using a Riemannian approximation scheme. These results were then used to prove a Heisenberg version of the Gauss-Bonnet theorem. In [5], Veloso verified that Gaussian curvature of surfaces and normal curvature of curves in surfaces introduced by [4] and by [1] to prove Gauss-Bonnet theorems in Heisenberg space H1 were unequal and he applied the same formalism of [4] to get the curvatures of [1]. With the obtained formulas, the Gauss-Bonnet theorem can be proved as a straightforward application of Stokes theorem in [5].

    In [1] and [2], Balogh-Tyson-Vecchi used that the Riemannian approximation scheme may depend upon the choice of the complement to the horizontal distribution in general. In the context of H1 the choice which they have adopted is rather natural. The existence of the limit defining the intrinsic curvature of a surface depends crucially on the cancellation of certain divergent quantities in the limit. 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 [6], Wang and Wei gave sub-Riemannian limits of Gaussian curvature for a Euclidean C2-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. And they got Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane. In [7], Wang and Wei gave sub-Riemannian limits of Gaussian curvature for a Euclidean C2-smooth surface in the BCV spaces and the twisted Heisenberg group away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. And they got Gauss-Bonnet theorems in the BCV spaces and the twisted Heisenberg group.

    In this paper, we solve this problem for the generalized affine group and the generalized BCV spaces. In the case of the generalized affine group, the cancellation of certain divergent quantities in the limit happens and the limit of the Riemannian Gaussian curvature exists. In the case of the generalized BCV spaces, the result is the same as the generalized affine group. We also get Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces.

    In Section 2, we compute the sub-Riemannian limit of curvature of curves in the generalized affine group. In Section 3, we compute sub-Riemannian limits of geodesic curvature of curves on surfaces and the Riemannian Gaussian curvature of surfaces in the generalized affine group. In Section 4, we prove the Gauss-Bonnet theorem in the generalized affine group. In Section 5, we compute the sub-Riemannian limit of curvature of curves in the generalized BCV spaces. In Section 6, we compute sub-Riemannian limits of geodesic curvature of curves on surfaces and the Riemannian Gaussian curvature of surfaces in the generalized BCV spaces and get a Gauss-Bonnet theorem in the generalized BCV spaces.

    When TM=HH and gTM=gHgH, we may consider the rescaled metric gL=gHLgH, then we may consider the sub-Riemannian limit of some geometric objects like the Gauss curvature and the mean curvature , when L goes to the infinity. In this case, we call the (M,gTM) as the manifold with the splitting tangent bundle. In this paper, our main objects: the generalized affine group and the generalized BCV spaces are not sub-Riemannian manifolds (groups) in general. But they are manifolds with the splitting tangent bundle. So we can use the Riemannian approximation scheme to get the Gauss-Bonnet theorems in these spaces.

    Firstly we give some notations on the generalized affine group. Let G be the generalized affine group and choose the underlying manifold G={(x1,x2,x3)R3f(x1,x2,x3)>0}. On G, we let

    X1=fx1,X2=fx2+x3,X3=fx2. (2.1)

    where f be a smooth function with respect to x1,x2,x3. Then

    x1=1fX1,x2=1fX3,x3=X2X3, (2.2)

    and span{X1,X2,X3}=TG. Let H=span{X1,X2} be the horizontal distribution on G. Let ω1=1fdx1,ω2=dx3,ω=1fdx2dx3. Then H=Kerω. For the constant L>0, let gL=ω1ω1+ω2ω2+Lωω,g=g1 be the Riemannian metric on G. Then X1,X2,~X3:=L12X3 are orthonormal basis on TG with respect to gL. We have

    [X1,X2]=(f2+f3f)X1+f1X3,[X1,X3]=f2X1+f1X3,[X2,X3]=f3fX3. (2.3)

    where fi=fxi, for 1i3.

    Let L be the Levi-Civita connection on G with respect to gL. Then we have the following lemma,

    Lemma 2.1. Let G be the generalized affine group, then

    LX1X1=(f2+f3f)X2+f3L,LX1X2=(f2+f3f)X1+f12X3,LX2X1=f12X3,LX2X2=0,LX1X3=f2X1f1L2X2,LX3X1=f1L2X2f1X3,LX2X3=f1L2X1,LX3X2=f1L2X1f3fX3,LX3X3=f1LX1+f3LfX2. (2.4)

    Proof. By the Koszul formula, we have

    2LXiXj,XkL=[Xi,Xj],XkL[Xj,Xk],XiL+[Xk,Xi],XjL, (2.5)

    where i,j,k=1,2,3. So lemma 2.1 holds.

    Definition 2.2. Let γ:[a,b](G,gL) be a Euclidean C1-smooth curve. We say that γ is regular if ˙γ0 for every t[a,b]. Moreover we say that γ(t) is a horizontal point of γ if

    ω(˙γ(t))=˙γ2(t)f˙γ3(t)=0,

    where γ(t)=(γ1(t),γ2(t),γ3(t)) and ˙γi(t)=γi(t)t.

    Definition 2.3. Let γ:[a,b](G,gL) be a Euclidean C2-smooth regular curve in the Riemannian manifold (G,gL). The curvature kLγ of γ at γ(t) is defined as

    kLγ:=||L˙γ˙γ||2L||˙γ||4LL˙γ˙γ,˙γ2L||˙γ||6L. (2.6)

    Lemma 2.4. Let γ:[a,b](G,gL) be a Euclidean C2-smooth regular curve in the Riemannian manifold (G,gL). Then,

    kLγ={{{[¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2]+[f1L˙γ3f2˙γ1f+f1Lω(˙γ(t))]ω(˙γ(t))}2+{[¨γ3+f2˙γ21f2+f3˙γ21f3]+[f3Lω(˙γ(t))ff1L˙γ1f]ω(˙γ(t))}2+L{f2˙γ21f2L[f1˙γ1f+f3˙γ3f]ω(˙γ(t))+ddtω(˙γ(t))}2}[(˙γ1f)2+˙γ23+L(ω(˙γ(t)))2]2{[˙γ1¨γ1f2f˙γ21f3+˙γ3¨γ3]+Ldω(˙γ(t))dtω(˙γ(t))}2[(˙γ1f)2+˙γ23+L(ω(˙γ(t)))2]3}12 (2.7)

    In particular, if γ(t) is a horizontal point of γ,

    kLγ={{[¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2]2+[¨γ3+f2˙γ21f2+f3˙γ21f3]2+L[f2˙γ21f2L+ddtω(˙γ(t))]2}[(˙γ1f)2+˙γ23]2[˙γ1¨γ1f2f˙γ21f3+˙γ3¨γ3]2[(˙γ1f)2+˙γ23]3}12 (2.8)

    where f=˙γ(f)=ddtf(γ(t)).

    Proof. By (2.2), we have

    ˙γ(t)=˙γ1fX1+˙γ3X2+ω(˙γ(t))X3. (2.9)

    By Lemma 2.1 and (2.9), we have

    L˙γX1=[ff2˙γ1(t)+f3˙γ1(t)f2f1Lω(˙γ(t))2]X2+[f2˙γ1(t)fLf1˙γ3(t)2f1ω(˙γ(t))]X3,L˙γX2=[ff2˙γ1(t)+f3˙γ1(t)f2+f1Lω(˙γ(t))2]X1+[f3ω(˙γ(t))f+f1˙γ(t)2f]X3,L˙γX3=[f2˙γ1(t)f+Lf1˙γ3(t)2+f1Lω(˙γ(t))]X1+[f1L˙γ(t)2f+f3Lω(˙γ(t))f]X2. (2.10)

    By (2.9) and (2.10), we have

    L˙γ˙γ={[¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2]+[f1L˙γ3f2˙γ1f+f1Lω(˙γ(t))]ω(˙γ(t))}X1+{[¨γ3+f2˙γ21f2+f3˙γ21f3]+[f3Lω(˙γ(t))ff1L˙γ1f]ω(˙γ(t))}X2+{f2˙γ21f2L(f1˙γ1f+f3˙γ3f)ω(˙γ(t))+dω(˙γ(t))dt}X3 (2.11)

    By (2.6), (2.9) and (2.11), we get Lemma 2.4.

    Definition 2.5. Let γ:[a,b](G,gL) be a Euclidean C2-smooth regular curve in the Riemannian manifold (G,gL). We define the intrinsic curvature kγ of γ at γ(t) to be

    kγ:=limL+kLγ,

    if the limit exists.

    We introduce the following notation: for continuous functions f1,f2:(0,+)R,

    f1(L)f2(L),asL+limL+f1(L)f2(L)=1. (2.12)

    Lemma 2.6. Let γ:[a,b](G,gL) be a Euclidean C2-smooth regular curve in the Riemannian manifold (G,gL). Then

    kγ=[f1˙γ1f3˙γ2f+f3˙γ3]2+(f1˙γ2)2|f||ω(˙γ(t))|,ifω(˙γ(t))0, (2.13)
    kγ={{[¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2]2+[¨γ3+f2˙γ21f2+f3˙γ21f3]2+(f2˙γ21f2)2}[(˙γ1f)2+˙γ23]2[˙γ1¨γ1f2f˙γ21f3+˙γ3¨γ3]2[(˙γ1f)2+˙γ23]3}12ifω(˙γ(t))=0andddt(ω(˙γ(t)))=0, (2.14)
    limL+kLγL=|ddt(ω(˙γ(t)))|(˙γ1f)2+˙γ23,ifω(˙γ(t))=0andddt(ω(˙γ(t)))0. (2.15)

    Proof. Using the notation introduced in (2.12), when ω(˙γ(t))0, we have

    ||L˙γ˙γ||2L(ω(˙γ(t))f)2{[f1˙γ1f3˙γ2f+f3˙γ3]2+(f1˙γ2)2}L2,asL+,
    ||˙γ||2LLω(˙γ(t))2,asL+,
    L˙γ˙γ,˙γ2LO(L2)asL+.

    Therefore

    ||L˙γ˙γ||2L||˙γ||4L{[f1˙γ1f3˙γ2f+f3˙γ3)]2+(f1˙γ2)2}f2ω(˙γ(t))2,asL+,
    L˙γ˙γ,˙γ2L||˙γ||6L0,asL+.

    So by (2.6), we have (2.13). (2.14) comes from (2.8) and

    ddt(ω(˙γ(t)))=0.

    When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))0,

    we have

    ||L˙γ˙γ||2LL[ddt(ω(˙γ(t)))]2,asL+,
    ||˙γ||2L=(˙γ1f)2+˙γ23,
    L˙γ˙γ,˙γ2L=O(1)asL+.

    By (2.6), we get (2.15).

    We will say that a surface Σ(G,gL) is regular if Σ is a Euclidean C2-smooth compact and oriented surface. In particular we will assume that there exists a Euclidean C2-smooth function u:GR such that

    Σ={(x1,x2,x3)G:u(x1,x2,x3)=0}

    and ux1x1+ux2x2+ux3x30. Let Hu=X1(u)X1+X2(u)X2. A point xΣ is called characteristic if Hu(x)=0. We define the characteristic set C(Σ):={xΣ|Hu(x)=0}. Our computations will be local and away from characteristic points of Σ. Let us define first

    p:=X1u,q:=X2u,andr:=˜X3u.

    We then define

    l:=p2+q2,lL:=p2+q2+r2,¯p:=pl,¯q:=ql,¯pL:=plL,¯qL:=qlL,¯rL:=rlL. (3.1)

    In particular, ¯p2+¯q2=1. These functions are well defined at every non-characteristic point. Let

    vL=¯pLX1+¯qLX2+¯rL~X3,e1=¯qX1¯pX2,e2=¯rL¯pX1+¯rL¯qX2llL~X3, (3.2)

    then vL is the Riemannian unit normal vector to Σ and e1,e2 are the orthonormal basis of Σ. On TΣ we define a linear transformation JL:TΣTΣ such that

    JL(e1):=e2;JL(e2):=e1. (3.3)

    For every U,VTΣ, we define Σ,LUV=πLUV where π:TGTΣ is the projection. Then Σ,L is the Levi-Civita connection on Σ with respect to the metric gL. By (2.11), (3.2) and

    Σ,L˙γ˙γ=L˙γ˙γ,e1Le1+L˙γ˙γ,e2Le2, (3.4)

    we have

    Σ,L˙γ˙γ={¯q[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]¯p[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1f3Lω(˙γ(t))f)ω(˙γ(t))]}e1+{¯rL¯p[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]+¯rL¯q[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1ff3Lω(˙γ(t))f)ω(˙γ(t))]llLL12[f2˙γ21f2L(f1˙γ1f+f3˙γ3f)ω(˙γ(t))+ddt(ω(˙γ(t)))]}e2. (3.5)

    Moreover if ω(˙γ(t))=0, then

    Σ,L˙γ˙γ={¯q[¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2]¯p[¨γ3+f2˙γ21f2+f3˙γ21f3]}e1+{¯rL¯p[¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2]+¯rL¯q[¨γ3+f2˙γ21f2+f3˙γ21f3]llLL12[f2˙γ21f2Lddt(ω(˙γ(t)))]}e2. (3.6)

    Definition 3.1. Let Σ(G,gL) be a regular surface. Let γ:[a,b]Σ be a Euclidean C2-smooth regular curve. The geodesic curvature kLγ,Σ of γ at γ(t) is defined as

    kLγ,Σ:=||Σ,L˙γ˙γ||2Σ,L||˙γ||4Σ,LΣ,L˙γ˙γ,˙γ2Σ,L||˙γ||6Σ,L. (3.7)

    Definition 3.2. Let Σ(G,gL) be a regular surface. Let γ:[a,b]Σ be a Euclidean C2-smooth regular curve. We define the intrinsic geodesic curvature kγ,Σ of γ at γ(t) to be

    kγ,Σ:=limL+kLγ,Σ,

    if the limit exists.

    Lemma 3.3. Let Σ(G,gL) be a regular surface. Let γ:[a,b]Σ be a Euclidean C2-smooth regular curve. Then

    kγ,Σ=|¯p(f1˙γ1f3˙γ2f+f3˙γ3)+¯qf1˙γ2||f||ω(˙γ(t))|,ifω(˙γ(t))0, (3.8)
    kγ,Σ=0,ifω(˙γ(t))=0,andddt(ω(˙γ(t)))=0,
    limL+kLγ,ΣL=|ddt(ω(˙γ(t)))|(¯q˙γ1f¯p˙γ3)2,ifω(˙γ(t))=0andddt(ω(˙γ(t)))0. (3.9)

    Proof. we know ˙γ(t)=˙γ1(t)x1+˙γ2(t)x2+˙γ3(t)x3, then by (2.2), ˙γ(t)=˙γ1(t)γ1(t)X1+γ3(t)X2+ω(˙γ(t))X3.

    Let

    ˙γ(t)=λ1e1+λ2e2.

    Then

    {˙γ1(t)γ1(t)=λ1¯q+λ2¯rL¯p˙γ3(t)=λ1¯p+λ2¯rL¯qω(˙γ(t))=λ2llLL12 (3.10)

    We have

    {λ1=¯q˙γ1(t)γ1(t)¯p˙γ3(t)λ2=λ2lLlL12ω(˙γ(t)) (3.11)

    Thus ˙γTΣ, we have

    ˙γ=(¯q˙γ1f¯p˙γ3)e1lLlL12ω(˙γ(t))e2. (3.12)

    By (3.6), we have

    ||Σ,L˙γ˙γ||2L,Σ={¯q[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]¯p[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1f3Lω(˙γ(t))f)ω(˙γ(t))]}2+{¯rL¯p[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]+¯rL¯q[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1ff3Lω(˙γ(t))f)ω(˙γ(t))]llLL12[f2˙γ21f2L(f1˙γ1f+f3˙γ3f)ω(˙γ(t))+ddt(ω(˙γ(t)))]}2L2[¯p(f1˙γ1f3˙γ2f+f3˙γ3)+¯qf1˙γ2]2ω(˙γ(t))2f2,asL+. (3.13)

    Similarly, we have that when ω(˙γ(t))0,

    ||˙γ||Σ,L=(¯q˙γ1f¯p˙γ3)2+(lLl)2Lω(˙γ(t))2L12|ω(˙γ(t))|,asL+. (3.14)

    By (3.6) and (3.12), we have

    Σ,L˙γ˙γ,˙γΣ,L=(¯q˙γ1f¯p˙γ3){¯q[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]¯p[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1f3Lω(˙γ(t))f)ω(˙γ(t))]}lLlL12ω(˙γ(t)){¯rL¯p[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]+¯rL¯q[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1ff3Lω(˙γ(t))f)ω(˙γ(t))]llLL12[f2˙γ21f2L(f1˙γ1f+f3˙γ3f)ω(˙γ(t))+ddt(ω(˙γ(t)))]}M0L, (3.15)

    where M0 does not depend on L. By (3.7), (3.13)–(3.15), we get (3.8). When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))=0,

    we have

    ||Σ,L˙γ˙γ||2L,Σ=[¯q(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)¯p(¨γ3+f2˙γ21f2+f3˙γ21f3)]2+[¯rL¯p(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+¯rL¯q(¨γ3+f2˙γ21f2+f3˙γ21f3)llLL12f2˙γ21f2L]2[¯q(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)¯p(¨γ3+f2˙γ21f2+f3˙γ21f3)]2 (3.16)

    and

    ||˙γ||Σ,L=|¯q˙γ1f¯p˙γ3|, (3.17)
    Σ,L˙γ˙γ,˙γΣ,L=(¯q˙γ1f¯p˙γ3)[¯q(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)¯p(¨γ3+f2˙γ21f2+f3˙γ21f3)] (3.18)

    By (3.16)–(3.18) and (3.7), we get kγ,Σ=0. When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))0,

    we have

    ||Σ,L˙γ˙γ||2L,ΣL[ddt(ω(˙γ(t)))]2,
    Σ,L˙γ˙γ,˙γΣ,L=O(1),

    so we get (3.9).

    Definition 3.4. Let Σ(G,gL) be a regular surface. Let γ:[a,b]Σ be a Euclidean C2-smooth regular curve. The signed geodesic curvature kL,sγ,Σ of γ at γ(t) is defined as

    kL,sγ,Σ:=Σ,L˙γ˙γ,JL(˙γ)Σ,L||˙γ||3Σ,L, (3.19)

    where JL is defined by (3.3).

    Definition 3.5. Let Σ(G,gL) be a regular surface. Let γ:[a,b]Σ be a Euclidean C2-smooth regular curve. We define the intrinsic geodesic curvature kγ,Σ of γ at the non-characteristic point γ(t) to be

    k,sγ,Σ:=limL+kL,sγ,Σ,

    if the limit exists.

    Lemma 3.6. Let Σ(G,gL) be a regular surface. Let γ:[a,b]Σ be a Euclidean C2-smooth regular curve. Then

    k,sγ,Σ=¯p(f1˙γ1f3˙γ2f+f3˙γ3)+¯qf1˙γ2|fω(˙γ(t))|,ifω(˙γ(t))0, (3.20)
    k,sγ,Σ=0,ifω(˙γ(t))=0,andddt(ω(˙γ(t)))=0,
    limL+kL,sγ,ΣL=(¯q˙γ1f+¯p˙γ3)ddt(ω(˙γ(t)))|¯q˙γ1f¯p˙γ3|3,ifω(˙γ(t))=0andddt(ω(˙γ(t)))0. (3.21)

    Proof. By (3.3) and (3.12), we have

    JL(˙γ)=lLlL12ω(˙γ(t))e1+(¯q˙γ1f¯p˙γ3)e2. (3.22)

    By (3.5) and (3.22), we have

    Σ,L˙γ˙γ,JL(˙γ)L,Σ=lLlL12ω(˙γ(t)){¯q[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]¯p[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1f3Lω(˙γ(t))f)ω(˙γ(t))]}+(¯q˙γ1f¯p˙γ3){¯rL¯p[(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+(f1L˙γ3f2˙γ1f+f1Lω(˙γ(t)))ω(˙γ(t))]+¯rL¯q[(¨γ3+f2˙γ21f2+f3˙γ21f3)(f1L˙γ1ff3Lω(˙γ(t))f)ω(˙γ(t))]llLL12[f2˙γ21f2L(f1˙γ1f+f3˙γ3f)ω(˙γ(t))+ddt(ω(˙γ(t)))]},L32ω(˙γ(t))2¯p(f1˙γ1f3˙γ2f+f3˙γ3)+¯qf1˙γ2fasL+. (3.23)

    So we get (3.20). When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))=0, we get

    Σ,L˙γ˙γ,JL(˙γ)L,Σ=(¯q˙γ1f¯p˙γ3)[¯rL¯p(¨γ1f2˙γ1˙γ3ff˙γ1+f3˙γ1˙γ3f2)+¯rL¯q(¨γ3+f2˙γ21f2+f3˙γ21f3)llLL12f2˙γ21f2]M0L12asL+. (3.24)

    So k,sγ,Σ=0. When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))0,

    we have

    Σ,L˙γ˙γ,JL(˙γ)L,ΣL12(¯q˙γ1f+¯p˙γ3)ddt(ω(˙γ(t)))asL+. (3.25)

    So we get (3.21).

    In the following, we compute the sub-Riemannian limit of the Riemannian Gaussian curvature of surfaces in the generalized affine group. We define the second fundamental form IIL of the embedding of Σ into (G,gL):

    IIL=(Le1vL,e1L,Le1vL,e2LLe2vL,e1L,Le2vL,e2L). (3.26)

    Similarly to Theorem 4.3 in [3], we have

    Theorem 3.7. The second fundamental form IIL of the embedding of Σ into (G,gL) is given by

    IIL=(h11,h12h21,h22). (3.27)

    where

    h11=llL[X1(¯p)+X2(¯q)]¯qL(f2+f3f)¯q2f2¯rLL12,
    h12=h21=lLle1,H(¯rL)Lf1L2¯p¯qf2L12,
    h22=l2l2Le2,H(rl)L+~X3(¯rL)f1¯pL¯qLf3f¯rL¯P2f2L12.

    Proof. By eiVL,ejLLeiVL,ejLLeiej,VLL=0 and eiVL,ejL=0, we have LeiVL,ejL=Leiej,VLL, i,j=1,2.

    By lemma 2.1 and (3.2),

    Le1e1=L(¯qX1¯pX2)(¯qX1¯pX2)=[¯qX1(¯q)¯pX2(¯p)+¯p¯q(f2+f3f)]X1[¯qX1(¯p)¯pX2(¯p)¯q2(f2+f3f)]X2+¯q2f3L12~X3. (3.28)

    Then

    h11=Le1e1,VLL=¯pL[¯qX1(¯q)¯pX2(¯p)]+¯qL[¯qX1(¯p)¯pX2(¯p)]¯qL(F2+F3¯fF)¯pL(F1+F3fF)+¯rLL12F3F=llL[X1(¯p)+X2(¯q)]¯qL(F2+F3¯fF)¯pL(F1+F3fF)+¯rLL12F3F. (3.29)

    Similarly,

    Le1e2=L(¯qX1¯pX2)(¯rL¯pX1+¯rL¯qX2llL~X3)=[¯qX1(¯rL¯p)¯pX2(¯rL¯p)+f1¯pLL2¯rL¯q2(f2+f3f)+¯qLf2L12]X1+[¯qX1(¯q)¯pX2(¯q)+f1¯qLL2+¯rL¯pq(f2+f3f)]X2+[¯pX2(llL)¯qX1(llL)+f1¯rLL2+¯rL¯pqf3L12]~X3. (3.30)

    Then

    h12=Le1e2,VLL=llL[¯qX1(¯rL)¯pX2(¯rL)]+¯rL[¯qX1(llL)¯pX2(llL)]12L12(fF2F1¯f+¯fFFf)=lLle1,H(¯rL)L12L12(fF2F1¯f+¯fFFf). (3.31)

    Since

    Le2VL,e1L=Le2e1,VLL=Le1e2+[e2,e1],VLL=Le1e2,VL=Le1VL,e2L. (3.32)

    Then,

    h21=h12=lLle1,H(¯rL)L12L12(fF2F1¯f+¯fFFf). (3.33)

    Since

    Le2e2=L(¯rL¯pX1+¯rL¯qX2llL~X3)(¯rL¯pX1+¯rL¯qX2llL~X3)=[¯qX1(¯rL¯p)¯pX2(¯rL¯p)+f1¯qLL¯rL2¯pq(f2+f3f)+¯rL¯pLf2L12+(llL)2f1]X1+[¯qX1(¯q)¯pX2(¯q)+f1¯pLL+¯rL2¯pq(f2+f3f)+(llL)2f3fX2+[¯pX2(llL)¯qX1(llL)+¯rL2¯pqf3fL12+¯rL¯pLf1+¯rL¯qLf3f]~X3. (3.34)

    Then,

    h22=Le2e2,VLL=¯prlX1(¯rL)¯qrlX2(¯rL)+~X3(¯rL)¯pLfF3F¯qL¯fF3F¯rLF3FL12=l2l2Le2,H(rl)L+~X3(¯rL)¯pLfF3F¯qL¯fF3F¯rLF3FL12. (3.35)

    The Riemannian mean curvature HL of Σ is defined by

    HL:=tr(IIL).

    Let

    KΣ,L(e1,e2)=RΣ,L(e1,e2)e1,e2Σ,L,KL(e1,e2)=RL(e1,e2)e1,e2L. (3.36)

    By the Gauss equation, we have

    KΣ,L(e1,e2)=KL(e1,e2)+det(IIL). (3.37)

    Proposition 3.8. Away from characteristic points, the horizontal mean curvature H of ΣG is given by

    H=limL+HL=X1(¯p)+X2(¯q)f1¯p¯qf22¯qf3f. (3.38)

    Proof. By

    l2l2Le2,H(rl)L=¯prlX1(¯rL)+¯qrlX2(¯rL)=O(L1)
    llL[X1(¯p)+X2(¯q)]X1(¯p)+X2(¯q),~X3(¯rL)0,
    ¯q2f2¯rLL12O(L1),¯qL¯q,
    ¯rL¯P2f2L12O(L1),¯pL¯p,

    we get (3.38).

    Define the curvature of a connection by

    R(X,Y)Z=XYZYXZ[X,Y]Z. (3.39)

    Then by Lemma 2.1 and (3.39), we have the following lemma,

    Lemma 3.9. Let G be the affine group, then

     RL(X1,X2)X1=[3Lf214+(f2+f3f)2]X2+[X1(f12)X2(f2L)+f21+f22L+f2f3Lf]X3,RL(X1,X2)X2=[3Lf214+X2(f2+f3f)(f2+f3f)2]X1+f1f3fX3,RL(X1,X2)X3=[X1(f1L2)+X2(f2)f21Lf2(f2+f3f)]X1+[X2(f1Lf)f1f3Lf]X2,RL(X1,X3)X1=[X1(f1L2)+X3(f2+f3f)+f2f3ff21Lf2(f2+f3f)]X2+[f21L4+f21+f22LX3(f2L)X1(f1)+f3f(f2+f3f)]X3,RL(X1,X3)X2=[X1(f1L2)f21Lf2(f2+f3f)+X3(f2+f3f)+f2f3f]X1+[X1(f3f)+f1f2+f1f3ff1(f2+f3f)X3(f12)]X3,RL(X1,X3)X3=[X1(f1L)f3Lf(f2+f3f)+X3(f2)+f21L24f21Lf22]X1+[f1L(f2+f3f)+X1(f3Lf)f1f2Lf1f3Lf+X3(f1L2)]X2,RL(X2,X3)X1=[X2(f1L2)+f1f3Lf]X2+[X3(f12)X2(f1)+f1f3f]X3,RL(X2,X3)X2=[X2(f1L2)f1f3Lf]X1+[f23f2f21L4X2(f3f)]X3,RL(X2,X3)X3=[X2(f1L)X3(f1L2)f1f3Lf]X1+[X2(f3Lf)f23Lf2+f21L24]X2. (3.40)

    Proposition 3.10. Away from characteristic points, we have

    KΣ,L(e1,e2)A0+O(L12),asL+, (3.41)

    where

    A0:=f1e1,H(X3u|Hu|)f1f2¯p¯q¯q2f2134(X3u)2l2f21+¯p2X2(f3f)¯p[X1(¯p)+X2(¯q)¯q(f2+f3f)](f1¯p+¯qf3f)+2¯qX3ulf21+2¯qX3uX1(f12)2¯pX3ulf1f3f+¯q2X1(f1)¯q2(f2+f3f)2¯q¯pX1(f3f)2¯q¯pX3(f12)¯p2f23f2. (3.42)

    Proof. By (3.2), we have

    RL(e1,e2)e1,e2L=¯rL2RL(X1,X2)X1,X2L2llL¯qL12¯rLRL(X1,X2)X1,X3L+2llL¯pL12¯rLRL(X1,X2)X2,X3L+(llL¯q)2L1RL(X1,X3)X1,X3L2(llL)2¯p¯qL1RL(X1,X3)X2,X3L+(¯pllL)2L1RL(X2,X3)X2,X3L. (3.43)

    By Lemma 3.9, we have

    KL(e1,e2)=14l2l2Lf21L34Lf21¯rL2+2llL¯qL12¯rLf21¯q2l2l2Lf21¯rL2(f2+f3f)2+2llL¯qL12¯rLX1(f12)2llL¯qL12¯rLX2(f2L)+2llL¯qL12¯rLf22+2llL¯qL12¯rLf2f3f2¯pL12¯rLf1f3f¯q2l2l2Lf22L+¯q2l2l2LX3(f2L)+¯q2l2l2LX1(f1)¯q2l2l2L(f2+f3f)2¯qpl2l2LX1(f3f)+2¯qpl2l2Lf1f2+2¯qpl2l2Lf1f3f2¯qpl2l2Lf1(f2+f3f)2¯qpl2l2LX3(f12)¯p2l2l2Lf23f2+¯p2l2l2LX2(f3f). (3.44)

    By (3.35) and

    H(¯rL)=L12H(X3u|Hu|)+O(L1)asL+

    we get

    det(IIL)=h11h22h12h21=f21L4f1e1,H(X3u|Hu|)f1f2¯p¯q¯p[X1(¯p)+X2(¯q)¯q(f2+f3f)](f1¯p+¯qf3f)+O(L12). (3.45)

    By (3.38), (3.44), (3.45) we get (3.41).

    Let us first consider the case of a regular curve γ:[a,b](G,gL). We define the Riemannian length measure

    dsL=||˙γ||Ldt.

    Lemma 4.1. Let γ:[a,b](G,gL) be a Euclidean C2-smooth and regular curve. Let

    ds:=|ω(˙γ(t))|dt,d¯s:=121|ω(˙γ(t))|(˙γ21f2+˙γ23)dt. (4.1)

    Then

    limL+1LγdsL=bads. (4.2)

    When ω(˙γ(t))0, we have

    1LdsL=ds+d¯sL1+O(L2)asL+. (4.3)

    When ω(˙γ(t))=0, we have

    1LdsL=1L˙γ21f2+˙γ23dt. (4.4)

    Proof. We know that

    ||˙γ(t)||L=(˙γ1f)2+˙γ23+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

    1LdsL=L1((˙γ1f)2+˙γ23)+ω(˙γ(t))2dt.

    Using the Taylor expansion, we can prove (4.3). From the definition of dsL and ω(˙γ(t))=0, we get (4.4).

    Let Σ(G,gL) be a Euclidean C2-smooth surface and Σ={u=0}. Let dσΣ,L denote the surface measure on Σ with respect to the Riemannian metric gL. Then similai to Proposition 4.2 in [7], we have

    limL+1LΣdσΣ,L=dσΣ:=(¯pω2¯qω1)ω. (4.5)

    Similar to the proof of Theorem 1.1 in $ [1] $, we have

    Theorem 4.2. Let Σ(G,gL) be a regular surface with finitely many boundary components (Σ)i, i{1,,n}, given by Euclidean C2-smooth regular and closed curves γi:[0,2π](Σ)i. Let A0 be defined by (3.42) and dσΣ,d¯σΣ be defined by (4.5) and d¯s be defined by (4.1) and k,sγi,Σ be the sub-Riemannian signed geodesic curvature of γi relative to Σ. Suppose that the characteristic set C(Σ) satisfies H1(C(Σ))=0 where H1(C(Σ)) denotes the Euclidean 1-dimensional Hausdorff measure of C(Σ) and that ||Hu||1H is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set C(Σ), then

    ΣKΣ,dσΣ+ni=1γik,sγi,Σds=0. (4.6)

    Example 4.3. Let f=x21+1, then G=R3. Let u=x21+x22+x231 and =S2. is a regular surface. By (2.1), we get

    X1(u)=2(x21+1)x1;X2(u)=2(x21+1)x2+2x3. (4.7)

    Solve the equations X1(u)=X2(u)=0,

    then we get

    C(Σ)={(0,22,22),(0,22,22)}

    and H1(C(Σ))=0.

    A parametrization of Σ is

    x1=cos(ϕ)cos(θ),x2=cos(ϕ)sin(θ),x3=sin(ϕ),forϕ(π2,π2),θ[0,2π). (4.8)

    Then

    Hu2H=X1(u)2+X2(u)2=4(x21+1)2x21+4(x21+1)2x22+4x23+8(x21+1)x2x3=4(cos(ϕ)2cos(θ)2+1)2cos(ϕ)2+4sin(ϕ)2+8(cos(ϕ)2cos(θ)2+1)sin(ϕ)cos(ϕ)sin(θ). (4.9)

    By the definitions of wj for 1j3 and (4.5), we have

    dσΣ=1HuH[(X1(u))dx3(x21+1)1(X2(u))dx1]((x21+1)1dx2dx3)=1HuH2cos(ϕ)λ0dθdϕ. (4.10)

    where

    λ0=cos(ϕ)2+2(cos(ϕ)2cos(θ)2+1)1cos(ϕ)sin(ϕ)sin(θ)+(cos(ϕ)2cos(θ)2+1)2sin(ϕ)2

    is a bouned smooth function on Σ. By (4.9) and (4.10), we have Hu1H is locally summable around the isolated characteristic points with respect to the measure dσΣ.

    We consider some notation on the generalized BCV spaces. Let f(x2), ¯f(x1), F(x1,x2,x3) be smooth functions. The generalized BCV spaces M is the set

    {(x1,x2,x3)R3F(x1,x2,x3)>0}

    Let

    X1=Fx1+fx3,X2=Fx2+¯fx3,X3=x3. (5.1)

    Then

    x1=1F(X1fX3),x2=1F(X2¯fX3),x3=X3, (5.2)

    and span{X1,X2,X3}=TM. Let H=span{X1,X2} be the horizontal distribution on M. Let ω1=1Fdx1,ω2=1Fdx2,ω=dx3(fdx1+¯fdx2)F. Then H=Kerω. The generalized BCV spaces have some well-knowed special case. When F=1+λ4(x21+x22),f=τx2,¯f=τx1, we get the BCV spaces. When F=1,f=f(x2),¯f=¯f(x1), we can the Heisenberg manifolds. When F=1,f=12x22,¯f=0, we get the Martinet distribution. When F=1x1,f=0,¯f=2, we get the Welyczko's example (see [5]). For the constant L>0, let gL=ω1ω1+ω2ω2+Lωω,g=g1 be the Riemannian metric on M. Then X1,X2,~X3:=L12X3 are orthonormal basis on TM with respect to gL. We have

    [X1,X2]=(F2+¯fF3F)X1+(F1+fF3F)X2+(F2fF1¯f+F¯fFf)X3,[X2,X3]=F3FX2+¯fF3FX3,[X1,X3]=F3FX1+fF3FX3. (5.3)

    where Fi=Fxi, for 1i3, f=fx2, ¯f=¯fx1. Let L be the Levi-Civita connection on M with respect to gL. Then we have the following lemma

    Lemma 5.1. Let M be the generalized BCV spaces, then

    LX1X1=(F2+F3¯fF)X2+F3LFX3,LX1X2=(F2+F3¯fF)X1+12(fF2F1¯f+F¯fFf)X3,LX1X3=F3FX1L2(fF2F1¯f+F¯fFf)X2,LX2X1=(F1+F3fF)X212(fF2F1¯f+F¯fFf)X3,LX2X2=(F1+F3fF)X1+F3FLX3,LX2X3=L2(fF2F1¯f+F¯fFf)X1F3FX2,LX3X1=L2(fF2F1¯f+F¯fFf)X2fF3FX3,LX3X2=L2(fF2F1¯f+F¯fFf)X1¯fF3FX3,LX3X3=LfF3FX1+L¯fF3FX2. (5.4)

    Proof. By the Koszul formula, we have

    2LXiXj,XkL=[Xi,Xj],XkL[Xj,Xk],XiL+[Xk,Xi],XjL, (5.5)

    where i,j,k=1,2,3. So lemma 5.1 holds.

    Definition 5.2. Let γ:[a,b](M,gL) be a Euclidean C1-smooth curve. We say that γ(t) is a horizontal point of γ if

    ω(˙γ(t))=fF˙γ1(t)¯fF˙γ2(t)+˙γ3(t)=0.

    where γ(t)=(γ1(t),γ2(t),γ3(t)) and ˙γi(t)=γi(t)t.

    Similar to the definition 2.3 and definition 2.5, we can define kLγ and kγ for the generalized BCV spaces, we have

    Lemma 5.3. Let γ:[a,b](M,gL) be a Euclidean C2-smooth regular curve in the Riemannian manifold (M,gL). Then

    kγ={[˙γ1F(F2fF1¯f+F¯fFf)+F3¯fFω(˙γ(t))]2+[˙γ2F(F2fF1¯f+F¯fFf)+F3fFω(˙γ(t))]2}12|ω(˙γ(t))|1,ifω(˙γ(t))0. (5.6)
    kγ={{[F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)]2+[F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F1+F3fF)]2}[(˙γ21+˙γ22)F2]2+{[F˙γ21F˙γ1¨γ1F3+F˙γ22F˙γ2¨γ2F3]2}[(˙γ21+˙γ22)F2]3}12ifω(˙γ(t))=0andddt(ω(˙γ(t)))=0, (5.7)

    where F=˙γ(F)=ddtF(γ(t)).

    limL+kLγL=|ddt(ω(˙γ(t)))|˙γ21F2+˙γ22F2,ifω(˙γ(t))=0andddt(ω(˙γ(t)))0. (5.8)

    Proof. By (5.2), we have

    ˙γ(t)=˙γ1FX1+˙γ2FX2+ω(˙γ(t))X3. (5.9)

    By Lemma 5.1 and (5.8), we have

    L˙γX1=[˙γ1F(F2+F3¯fF)˙γ2F(F1+F3fF)L2(F2fF1¯f+F¯fFf)ω(˙γ(t))]X2+[F3˙γ1LF2˙γ22F(F2fF1¯f+F¯fFf)F3fF(ω(˙γ(t))]X3,L˙γX2=[˙γ1F+(F2+F3¯fF)+˙γ2F(F1+F3fF)+L2(F2fF1¯f+F¯fFf)ω(˙γ(t))]X1+[F3˙γ2LF2+˙γ12F(F2fF1¯f+F¯fFf)F3¯fF(ω(˙γ(t))]X3,L˙γX3=[˙γ1F3F2+˙γ2L2F(F2fF1¯f+F¯fFf)+LF3fFω(˙γ(t))]X1+[˙γ1L2F(F2fF1¯f+F¯fFf)˙γ2F3F2+LF3¯fFω(˙γ(t))]X2. (5.10)

    By (5.8) and (5.9), we have

    L˙γ˙γ={F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)+[F3˙γ1F2+˙γ2LF(F2fF1¯f+F¯fFf)+fLF3Fω(˙γ(t))]ω(˙γ(t))}X1+{F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F2+F3fF)+[F3˙γ2F2˙γ1LF(F2fF1¯f+F¯fFf)+¯fLF3Fω(˙γ(t))]ω(˙γ(t))}X2+{(˙γ22+˙γ21)F3F3L(F3f˙γ1+F3¯f˙γ2)F2ω(˙γ(t))+ddtω(˙γ(t))}X3. (5.11)

    By (5.8) and (5.10), when ω(˙γ(t))0, we have

    ||L˙γ˙γ||2L{[˙γ2F(F2fF1¯f+F¯fFf)+F3fFω(˙γ(t))]2+[˙γ1F(F2fF1¯f+F¯fFf)+F3¯fFω(˙γ(t))]2}ω(˙γ(t))2L2,asL+,||˙γ||2LLω(˙γ(t))2,asL+,L˙γ˙γ,˙γ2LO(L2)asL+. (5.12)

    Therefore

    ||L˙γ˙γ||2L[˙γ2F(F2fF1¯f+F¯fFf)+F3fFω(˙γ(t))]2ω(˙γ(t))2+[˙γ1F(F2fF1¯f+F¯fFf)+F3¯fFω(˙γ(t))]2ω(˙γ(t))2,asL+, (5.13)
    L˙γ˙γ,˙γ2L||˙γ||6L0,asL+.

    So by (2.6), we have (5.5). (5.6) comes from (5.8), (5.10), (2.6) and ω(˙γ(t))=0 and ddt(ω(˙γ(t)))=0. When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))0, we have

    ||L˙γ˙γ||2LL[ddt(ω(˙γ(t)))]2,asL+,
    ||˙γ||2L=[˙γ21F2+˙γ22F2]2,
    L˙γ˙γ,˙γ2L=O(1)asL+.

    By (2.6), we get (5.7).

    We will consider a regular surface Σ1(M,gL) and regular curve γΣ1. We will assume that there exists a Euclidean C2-smooth function u:MR such that

    Σ1={(x1,x2,x3)M:u(x1,x2,x3)=0}.

    Similar to Section 3, we define p,q,r,l,lL,¯p,¯q,¯pL,¯qL,¯rL,vL,e1,e2,JL,kLγ,Σ1,kγ,Σ1,kL,sγ,Σ1,k,sγ,Σ1. By (3.4) and (5.10), we have

    Σ1,L˙γ˙γ={¯q[F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)]+¯q[F3˙γ1F2+˙γ2LF(F2fF1¯f+F¯fFf)+fLF3Fω(˙γ(t))]ω(˙γ(t))¯p[F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F2+F3fF)]+¯p[F3˙γ2F2˙γ1LF(F2fF1¯f+F¯fFf)+¯fLF3Fω(˙γ(t))]ω(˙γ(t))}e1+{¯rL¯p[F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)]+¯rL¯p[F3˙γ1F2+˙γ2LF(F2fF1¯f+F¯fFf)+fLF3Fω(˙γ(t))]ω(˙γ(t))+¯rL¯q[F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F2+F3fF)]+¯rL¯q[F3˙γ2F2˙γ1LF(F2fF1¯f+F¯fFf)+¯fLF3Fω(˙γ(t))]ω(˙γ(t))llLL12[(˙γ22+˙γ21)F3F3L(f˙γ1+¯f˙γ2)F3F2ω(˙γ(t))+ddt(ω(˙γ(t)))]}e2=B1e1+B2e2. (6.1)

    By (5.8) and ˙γ(t)TΣ1, we have

    ˙γ(t)=[¯q˙γ1F¯p˙γ2F]e1lLlL12ω(˙γ(t))e2. (6.2)

    We have

    Lemma 6.1. Let Σ1(M,gL) be a regular surface. Let γ:[a,b]Σ1 be a Euclidean C2-smooth regular curve. Then

    kγ,Σ1={[q˙γ2(F2fF1¯f+F¯fFf)+F3f¯qω(˙γ(t))]+[¯P˙γ1(F2fF1¯f+F¯fFf)F3f¯Pω(˙γ(t))]}|Fω(˙γ(t))|1,ifω(˙γ(t))0, (6.3)
    kγ,Σ1=0,ifω(˙γ(t))=0,andddt(ω(˙γ(t)))=0,
    limL+kLγ,Σ1L=|ddt(ω(˙γ(t)))|[¯q˙γ1F¯P˙γ2F]2,ifω(˙γ(t))=0andddt(ω(˙γ(t)))0. (6.4)

    Proof. By (6.1), we have

    ||Σ1,L˙γ˙γ||2L,Σ1=B21+B22L2ω(˙γ(t))2{[¯q˙γ2F(F2fF1¯f+F¯fFf)+F3f¯qFω(˙γ(t))]+[¯P˙γ1F(F2fF1¯f+F¯fFf)F3¯f¯PFω(˙γ(t))]}2,asL+. (6.5)

    By (6.2), we have that when ω(˙γ(t))0,

    ||˙γ||Σ1,LL12|ω(˙γ(t))|,asL+. (6.6)

    By (6.1) and (6.2), we have

    Σ1,L˙γ˙γ,˙γΣ1,LM0L, (6.7)

    where M0 does not depend on L.

    By (3.7), (6.5)–(6.7), we get (6.3). When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))=0, we have

    ||Σ1,L˙γ˙γ||2L,Σ1C0:={¯q[F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)]¯p[F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F2+F3fF)]}2,asL+. (6.8)

    and

    ||˙γ||2Σ1,L=[¯q˙γ1F¯P˙γ2F]2, (6.9)
    Σ1,L˙γ˙γ,˙γΣ1,L=[¯q˙γ1F¯P˙γ2F]C0. (6.10)

    By (6.8)–(6.10) and (3.7), we get kγ,Σ1=0. When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))0, we have

    ||Σ1,L˙γ˙γ||2L,Σ1L[ddt(ω(˙γ(t)))]2,
    Σ1,L˙γ˙γ,˙γΣ1,L=O(1),

    so we get (6.4).

    Lemma 6.2. Let Σ1(M,gL) be a regular surface. Let γ:[a,b]Σ1 be a Euclidean C2-smooth regular curve. Then

    k,sγ,Σ1={[q˙γ2(F2fF1¯f+F¯fFf)+F3f¯qω(˙γ(t))]+[¯P˙γ1(F2fF1¯f+F¯fFf)F3f¯Pω(˙γ(t))]}|Fω(˙γ(t))|1,ifω(˙γ(t))0, (6.11)
    k,sγ,Σ1=0,ifω(˙γ(t))=0,andddt(ω(˙γ(t)))=0,
    limL+kL,sγ,Σ1L=|ddt(ω(˙γ(t)))|[¯q˙γ1F¯P˙γ2F]2,ifω(˙γ(t))=0andddt(ω(˙γ(t)))0. (6.12)

    Proof. By (3.3) and (6.2), we have

    JL(˙γ)=lLlL12ω(˙γ(t))e1+[¯q˙γ1F¯p˙γ2F]e2. (6.13)

    By (6.1) and (6.13), we have

    Σ1,L˙γ˙γ,JL(˙γ)L,Σ1=lLLL12ω(˙γ(t)){¯q[F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)]+¯q[F3˙γ1F2+˙γ2LF(F2fF1¯f+F¯fFf)+fLF3Fω(˙γ(t))]ω(˙γ(t))¯p[F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F2+F3fF)]+¯p[F3˙γ2F2˙γ1LF(F2fF1¯f+F¯fFf)+¯fLF3Fω(˙γ(t))]ω(˙γ(t))}+[¯q˙γ1F¯p˙γ2F]{¯rL¯p[F˙γ1F¨γ1F2˙γ1˙γ2F2(F2+F3¯fF)+˙γ22F2(F1+F3fF)]+¯rL¯p[F3˙γ1F2+˙γ2LF(F2fF1¯f+F¯fFf)+fLF3Fω(˙γ(t))]ω(˙γ(t))+¯rL¯q[F˙γ2F¨γ2F2+˙γ21F2(F2+F3¯fF)˙γ1˙γ2F2(F2+F3fF)]+¯rL¯q[F3˙γ2F2˙γ1LF(F2fF1¯f+F¯fFf)+¯fLF3Fω(˙γ(t))]ω(˙γ(t))llLL12[(˙γ22+˙γ21)F3F3L(f˙γ1+¯f˙γ2)F3F2ω(˙γ(t))+ddt(ω(˙γ(t)))]} (6.14)

    So by (3.17), (6.6) and (6.14), we get (6.11). When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))=0, we get

    Σ1,L˙γ˙γ,JL(˙γ)L,Σ1M0L12asL+. (6.15)

    So k,sγ,Σ1=0. When ω(˙γ(t))=0 and ddt(ω(˙γ(t)))0, we have

    Σ1,L˙γ˙γ,JL(˙γ)L,Σ1L12[¯P˙γ2F¯q˙γ1F]ddt(ω(˙γ(t))),asL+. (6.16)

    So we get (6.12).

    In the following, we compute the sub-Riemannian limit of the Riemannian Gaussian curvature of surfaces in the generalized BCV spaces. Similarly to Theorem 4.3 in [3], we have

    Theorem 6.3. The second fundamental form IIL1 of the embedding of Σ1 into (M,gL) is given by

    IIL1=(h11,h12h21,h22), (6.17)

    where

    h11=llL[X1(¯p)+X2(¯q)]¯qL(F2+F3¯fF)¯pL(F1+F3fF)+¯rLL12F3F,
    h12=h21=lLle1,H(¯rL)L12L12(fF2F1¯f+¯fFFf),
    h22=l2l2Le2,H(rl)L+~X3(¯rL)¯pLfF3F¯qL¯fF3F¯rLF3FL12.

    Proof. By lemma 5.1 and (3.2),

    \begin{align} \nabla^L_{e_1}e_1& = \nabla^L_{(\overline{q}X_1-\overline{p}X_2)}(\overline{q}X_1-\overline{p}X_2)\\ & = [\overline{q}X_1(\overline{q})-\overline{p}X_2(\overline{p})+\overline{p}\overline{q}(F_2+\frac{F_3\overline{f}}{F})+\overline{p}^2(F_1+\frac{F_3f}{F})]X_1\\ &-[\overline{q}X_1(\overline{p})-\overline{p}X_2(\overline{p})-\overline{q}^2(F_2+\frac{F_3\overline{f}}{F})-\overline{p}\overline{q}(F_2+\frac{F_3f}{F})]X_2+\frac{F_3}{F}L^{-\frac{1}{2}}\widetilde{X_3}.\\ \end{align} (6.18)

    Then

    \begin{align} h_{11}& = -\langle\nabla^L_{e_1}e_1, V_L\rangle_L\\ & = -\overline{p_L}[\overline{q}X_1(\overline{q})-\overline{p}X_2(\overline{p})]+\overline{q_L}[\overline{q}X_1(\overline{p})-\overline{p}X_2(\overline{p})]\\ &-\overline{q_L}\left(F_2+\frac{F_3\overline{f}}{F}\right)-\overline{p_L}\left(F_1+\frac{F_3f}{F}\right)+\overline{r_L}L^{-\frac{1}{2}}\frac{F_3}{F}\\ & = \frac{l}{l_L}[X_1(\overline{p})+X_2(\overline{q})]-\overline{q_L}\left(F_2+\frac{F_3\overline{f}}{F}\right)-\overline{p_L}\left(F_1+\frac{F_3f}{F}\right)+\overline{r_L}L^{-\frac{1}{2}}\frac{F_3}{F}.\\ \end{align} (6.19)

    Similarly,

    \begin{align} \nabla^L_{e_1}e_2& = \nabla^L_{(\overline{q}X_1-\overline{p}X_2)}(\overline{r_L}\overline{p}X_1+\overline{r_L}\overline{q}X_2-\frac{l}{l_L}\widetilde{X_3})\\ & = [\overline{q}X_1(\overline{r_L}\overline{p})-\overline{p}X_2(\overline{r_L}\overline{p})+\frac{(fF_2-F_1\overline{f}+F\overline{f'}-Ff')\overline{p_L}\sqrt{L}}{2}\\ &-\overline{r_L}\overline{pq}(F_1+\frac{F_3f}{F})-\overline{r_L}\overline{q}^2(F_2+\frac{F_3\overline{f}}{F})+\overline{q_L}\frac{F_3}{F}L^{-\frac{1}{2}}]X_1\\ &+[\overline{q}X_1(\overline{q})-\overline{p}X_2(\overline{q})+\frac{(fF_2-F_1\overline{f}+F\overline{f'}-Ff')\overline{q_L}\sqrt{L}}{2}\\ &-\overline{r_L}\overline{pq}(F_2+\frac{F_3\overline{f}}{F})+\overline{r_L}\overline{p}^2(F_1+\frac{F_3f}{F})+\overline{p_L}\frac{F_3}{F}L^{-\frac{1}{2}}]X_2\\ &+[\overline{p}X_2(\frac{l}{l_L})-\overline{q}X_1(\frac{l}{l_L})+\frac{(fF_2-F_1\overline{f}+F\overline{f'}-Ff')\overline{r_L}\sqrt{L}}{2}]\widetilde{X_3}.\\ \end{align} (6.20)

    Then

    \begin{align} h_{12}& = -\langle\nabla^L_{e_1}e_2, V_L\rangle_L\\ & = -\frac{l}{l_L}[\overline{q}X_1(\overline{r_L})-\overline{p}X_2(\overline{r_L})]+\overline{r_L}[\overline{q}X_1(\frac{l}{l_L})-\overline{p}X_2(\frac{l}{l_L})]-\frac{1}{2}L^\frac{1}{2}(fF_2-F_1\overline{f}+\overline{f}'F-Ff')\\ & = -\frac{l_L}{l}\langle e_1, \nabla_H(\overline{r_L})\rangle_L-\frac{1}{2}L^\frac{1}{2}(fF_2-F_1\overline{f}+\overline{f}'F-Ff').\\ \end{align} (6.21)

    Since

    \begin{align} \langle\nabla^L_{e_2}V_L, e_1\rangle_L& = -\langle\nabla^L_{e_2}e_1, V_L\rangle_L = -\langle\nabla^L_{e_1}e_2+[e_2, e_1], V_L\rangle_L\\ & = -\langle\nabla^L_{e_1}e_2, V_L\rangle = \langle\nabla^L_{e_1}V_L, e_2\rangle_L.\\ \end{align} (6.22)

    Then,

    \begin{align} h_{21} = h_{12} = -\frac{l_L}{l}\langle e_1, \nabla_H(\overline{r_L})\rangle_L-\frac{1}{2}L^\frac{1}{2}(fF_2-F_1\overline{f}+\overline{f}'F-Ff').\\ \end{align} (6.23)

    Since

    \nabla^L_{e_2}e_2 = \nabla^L_{(\overline{r_L}\overline{p}X_1+\overline{r_L}\overline{q}X_2-\frac{l}{l_L}\widetilde{X_3})}(\overline{r_L}\overline{p}X_1+\overline{r_L}\overline{q}X_2-\frac{l}{l_L}\widetilde{X_3})\\ = [\overline{q}X_1(\overline{r_L}\overline{p})-\overline{p}X_2(\overline{r_L}\overline{p})+(fF_2-F_1\overline{f}+\overline{f}'F-Ff')\overline{q_L}\sqrt{L}-\overline{r_L}^2\overline{pq}(F_2+\frac{F_3\overline{f}}{F})\\ +\overline{r_L}\overline{p_L}\frac{F_3}{F}L^{-\frac{1}{2}}+\overline{r_L}^2\overline{q}^2(F_1+\frac{F_3f}{F})+(\frac{l}{l_L})^2\frac{F_3f}{F}]X_1\\ +[\overline{q}X_1(\overline{q})-\overline{p}X_2(\overline{q})+(fF_2-F_1\overline{f}+\overline{f}'F-Ff')\overline{p_L}\sqrt{L}+\overline{r_L}^2\overline{pq}(F_2+\frac{F_3\overline{f}}{F})\\ -\overline{r_L}\overline{q_L}\frac{F_3}{F}L^{-\frac{1}{2}}-\overline{r_L}^2\overline{pq}(F_1+\frac{F_3f}{F})+(\frac{l}{l_L})^2\frac{F_3\overline{f}}{F}X_2\\ +[\overline{p}X_2(\frac{l}{l_L})-\overline{q}X_1(\frac{l}{l_L})+\overline{r_L}^2\overline{pq}\frac{F_3f}{F}L^{-\frac{1}{2}}+\overline{r_L}^2\overline{q}^2\frac{F_3}{F}L^{-\frac{1}{2}}+\overline{r_L}\overline{p_L}\frac{fF_3}{F}+\overline{r_L}\overline{q_L}\frac{\overline{f}F_3}{F}]\widetilde{X_3}. (6.24)

    Then,

    \begin{align} h_{22}& = -\langle\nabla^L_{e_2}e_2, V_L\rangle_L\\ & = -\overline{p}\frac{r}{l}X_1(\overline{r_L})-\overline{q}\frac{r}{l}X_2(\overline{r_L})+\widetilde{X_3}(\overline{r_L})-\overline{p_L}\frac{fF_3}{F}-\overline{q_L}\frac{\overline{f}F_3}{F}-\overline{r_L}\frac{F_3}{F}L^{-\frac{1}{2}}\\ & = -\frac{l^2}{l_L^2}\langle e_2, \nabla_H(\frac{r}{l})\rangle_L+\widetilde{X_3}(\overline{r_L})-\overline{p_L}\frac{fF_3}{F}-\overline{q_L}\frac{\overline{f}F_3}{F}-\overline{r_L}\frac{F_3}{F}L^{-\frac{1}{2}} .\\ \end{align} (6.25)

    Similar to Proposition 3.8, we have

    Proposition 6.4. Away from characteristic points, the horizontal mean curvature \mathcal{H}_{\infty}^1 of \Sigma_1\subset M is given by

    \begin{align} \mathcal{H}_{\infty}^1& = -\left(\overline{p}\frac{fF_3}{F}+\overline{q}\frac{\overline{f}F_3}{F}\right)+X_1(\overline{p})+X_2(\overline{q})-\overline{q}\left(F_2+\frac{F_3\overline{f}}{F}\right)-\overline{p}\left(F_1+\frac{F_3f}{F}\right). \end{align} (6.26)

    By Lemma 5.1, we have

    Lemma 6.5. Let M be the the generalized BCV spaces, then

    \begin{align} &R^L(X_1, X_2)X_1 = \left[-X_1(A)-X_2(B)+\frac{3L}{4}C^2+\frac{F^3}{LF^2}+A^2+B^2\right]X_2\\ &+\left[-\frac{1}{2}X_1(C)-X_2\left(\frac{F_3}{LF}\right)+\frac{F_3fC}{F}\right]X_3, \\ &R^L(X_1, X_2)X_2 = \left[X_1(A)+X_2(B)-\frac{3L}{4}C^2-\frac{F_3^2}{LF^2}-A^2-B^2\right]X_1\\ &+\left[-\frac{1}{2}X_2(C)+X_1\left(\frac{F_3}{LF}\right)+\frac{F_3\overline{f}C}{F}\right]X_3, \\ &R^L(X_1, X_2)X_3 = \left[\frac{1}{2}X_2(LC)+X_2\left(\frac{F_3}{F}\right)-\frac{F_3fCL}{F}\right]X_1\\ &+\left[\frac{1}{2}X_2(LC)-X_1\left(\frac{F_3}{F}\right)-\frac{F_3\overline{f}CL}{F}\right]X_2, \\ &R^L(X_1, X_3)X_1 = \left[-X_1(\frac{LC}{2})-X_3(B)+\frac{BF_3}{F}-\frac{F_3^2f^2}{F^2}\right]X_2\\ &+\left[-\frac{LC^2}{4}-X_1(\frac{F_3f}{F})-X_3\left(\frac{F_3}{LF}\right)+\frac{F_3\overline{f}B}{F}+\frac{F_3^2}{LF^2}+\frac{F^3f^2}{F^2}\right]X_3, \\ &R^L(X_1, X_3)X_2 = \left[X_1(\frac{LC}{2})+X_3(B)-\frac{CF_3fL}{F}+\frac{F^3\overline{f}^2}{F^2}+\frac{BF_3}{F}\right]X_1\\ &+\left[-X_1(\frac{F_3\overline{f}}{F})-\frac{1}{2}X_3(C)+\frac{F_3C}{F}-\frac{fF_3B}{F}+\frac{F^3f\overline{f}}{F^2}\right]X_3, \\ &R^L(X_1, X_3)X_3 = \left[X_1\left(\frac{LF_3f}{F}\right)+X_3\left(\frac{F_3}{F}\right)-\frac{F_3\overline{f}L}{F}B+\frac{L^2C^2}{4}-\frac{F_3^2}{F^2}-\frac{LF_3^2f^2}{F^2}\right]X_1\\ &+\left[X_1\left(\frac{LF_3\overline{f}}{F}\right)-\frac{LCF_3}{F}+X_3\left(\frac{LC}{2}\right)+\frac{F_3fL}{F}B-\frac{LF_3^2f\overline{f}}{F^2}\right]X_2, \\ &R^L(X_2, X_3)X_1 = \left[-X_2\left(\frac{LC}{2}\right)+X_3(A)+\frac{F_3\overline{f}LC}{F}+\frac{fF_3^2}{F^2}-\frac{AF_3}{F}\right]X_2\\ &+\left[-X_2\left(\frac{fF_3}{F}\right)-\frac{CF_3}{F}+X_3\left(\frac{C}{2}\right)-\frac{F_3\overline{f}A}{F}+\frac{F_3^2f\overline{f}}{F^2}\right]X_3, \\ &R^L(X_2, X_3)X_2 = \left[X_2\left(\frac{LC}{2}\right)-X_3(A)-\frac{F_3\overline{f}LC}{F}-\frac{fF_3^2}{F^2}+\frac{AF_3}{F}\right]X_1\\ &+\left[-X_2\left(\frac{\overline{f}F_3}{F}\right)+\frac{AfF_3}{F}-X_3\left(\frac{F_3}{FL}\right)-\frac{LC^2}{4}+\frac{F_3^2}{F^2L}+\frac{F_3^2\overline{f}^2}{F^2}\right]X_3, \\ &R^L(X_2, X_3)X_3 = \left[X_2\left(\frac{LF_3f}{2}\right)-X_3(\frac{LC}{2})+\frac{AF_3\overline{f}L}{F}-\frac{Lf\overline{f}F_3^2}{F^2}+\frac{LCF_3}{F}\right]X_2\\ &+\left[X_2\left(\frac{LF_3\overline{f}}{2}\right)+X_3(\frac{F_3}{F})-\frac{AF_3fL}{F}+\frac{L^2C^2}{4}+\frac{LF_3^2\overline{f}^2}{F^2}-\frac{F_3^2}{F^2}\right]X_3. \end{align} (6.27)

    where

    \left(F_1+\frac{F_3f}{F}\right) = A, \left(F_2+\frac{F_3\overline{f}}{F}\right) = B, (F_2f-F_1\overline{f}+F\overline{f}'-Ff') = C.

    Proposition 6.6. Away from characteristic points, we have

    \begin{equation} \mathcal{K}^{\Sigma_1, \infty}(e_1, e_2) = -C\langle e_1, \nabla_H(\frac{X_3u}{|\nabla_Hu|})\rangle+\overline{N}+O(L^{-\frac{1}{2}}). \end{equation} (6.28)

    where \overline{N} = N_0+N ,

    N = -\left(\overline{p}\frac{fF_3}{F}+\overline{q}\frac{\overline{f}F_3}{F}\right)\left[X_1(\overline{p})+X_2(\overline{q})-\overline{q}\left(F_2+\frac{F_3\overline{f}}{F}\right)-\overline{p}\left(F_1+\frac{F_3f}{F}\right)\right],
    \begin{align*} N_0& = 2\overline{q}\left[-\frac{1}{2}X_1(C)+\frac{F_3Cf}{F}\right]-2\overline{p}\left[X_1(-\frac{1}{2}X_2(C)+\frac{F_3C\overline{f}}{F}\right]+\overline{p}^2\left[X_2(\frac{F_3\overline{f}}{F})-\frac{F_3Af}{F}-\frac{F_3^2\overline{f}^2}{F^2}\right]\\ &-2\overline{p}\overline{q}\left[X_1(\frac{F_3\overline{f}}{F})-\frac{1}{2}X_3(C)+\frac{F_3Bf}{F}-\frac{F_3C}{F}\right]+\overline{q}^2\left[X_1(\frac{F_3f}{F})-\frac{F_3B\overline{f}}{F}-\frac{F_3^2\overline{f}^2}{F^2}\right]. \end{align*}

    Proof. By (3.43) and Lemma 6.5, we have

    \begin{align} \mathcal{K}^{M, L}(e_1, e_2) = &\overline{r_L}^2\left[X_1(A)+X_2(B)-\frac{3LC^2}{4}-\frac{F_3^2}{F^2L}-A^2-B^2\right]\\ &+2\frac{l}{l_L}\overline{q}\overline{r_L}L^\frac{1}{2}\left[-\frac{1}{2}X_1(C)-X_2(\frac{F_3}{FL})+\frac{F_3Cf}{F}\right]\\ &-2\frac{l}{l_L}\overline{p}\overline{r_L}L^\frac{1}{2}\left[X_1(\frac{F_3}{FL})-\frac{1}{2}X_2(C)+\frac{F_3C\overline{f}}{F}\right]\\ &-2\frac{l^2}{l_L^2}\overline{p}\overline{q}\left[X_1(\frac{F_3\overline{f}}{F})-\frac{1}{2}X_3(C)+\frac{F_3Bf}{F}-\frac{F_3C}{F}\right]\\ &+\frac{l^2}{l_L^2}\overline{q}^2\left[\frac{LC^2}{4}+X_1(\frac{F_3f}{F})-\frac{F_3B\overline{f}}{F}+X_3(\frac{F_3}{FL})-\frac{F_3^2}{F^2L}-\frac{F_3^2\overline{f}^2}{F^2}\right]\\ &+\frac{l^2}{l_L^2}\overline{p}^2\left[\frac{LC^2}{4}+X_2(\frac{F_3\overline{f}}{F})-\frac{F_3Af}{F}+X_3(\frac{F_3}{FL})-\frac{F_3^2}{F^2L}-\frac{F_3^2\overline{f}^2}{F^2}\right]\\ &\sim \frac{LC^2}{4}+N_0, \; \; \; \; {\rm as} \; \; \; \; L\rightarrow +\infty. \end{align} (6.29)

    Similar to (3.45), we have

    \begin{align} {\rm det}(II^L_1)& = h_{11}h_{22}-h_{12}h_{21}\\ & = -\frac{LC^2}{4}-C\langle e_1, \nabla_H(\frac{X_3u}{|\nabla_Hu|})\rangle+N +O(L^{-\frac{1}{2}})\; \; {\rm as}\; \; L\rightarrow +\infty. \end{align} (6.30)

    By (6.21) and (6.22), we have (6.20).

    Similar to (4.2) and (4.5), for the generalized BCV spaces, we have

    \begin{equation} {\rm lim}_{L\rightarrow +\infty}\frac{1}{\sqrt{L}}ds_L = ds, \; \; \; \; {\rm lim}_{L\rightarrow +\infty}\frac{1}{\sqrt{L}}d\sigma_{\Sigma_1, L} = d\sigma_{\Sigma_1}. \end{equation} (6.31)

    By (6.20), (6.23) and Lemma 6.2, similar to the proof of Theorem 1 in [1], we have

    Theorem 6.7. Let \Sigma_1\subset (M, g_L) be a regular surface with finitely many boundary components (\partial\Sigma_1)_i, i\in\{1, \cdots, n\} , given by Euclidean C^2 -smooth regular and closed curves \gamma_i:[0, 2\pi]\rightarrow (\partial\Sigma_1)_i . Suppose that the characteristic set C(\Sigma_1) satisfies \mathcal{H}^1(C(\Sigma_1)) = 0 and that ||\nabla_Hu||_H^{-1} is locally summable with respect to the Euclidean 2 -dimensional Hausdorff measure near the characteristic set C(\Sigma_1) , then

    \begin{equation} \int_{\Sigma_1}\mathcal{K}^{\Sigma_1, \infty}d\sigma_{\Sigma_1}+\sum\limits_{i = 1}^n\int_{\gamma_i}k^{\infty, s}_{\gamma_i, \Sigma_1}d{s} = 0. \end{equation} (6.32)

    Example 6.8. Let F = 1, f = -x_2^2, \overline{f} = x_1^2. Consider M = \{(x_1, x_2, x_3)\in \mathbb{R}^3\mid F > 0\} = \mathbb{R}^3 , let u = x_1^2+x_2^2+x_3^2-1 and \sum_1 = S^2 . \sum_1 is a regular surface. By (4.1) , we get

    \begin{equation} X_1(u) = 2x_1-2x_2^2x_3;\; \; \; X_2(u) = 2x_2+2x_1^2x_3. \end{equation} (6.33)

    Solve the equations X_1(u) = X_2(u) = 0 , then we get C(\Sigma) = \{(0, 0, 1), (0, 0, -1)\} and \mathcal{H}^1(C(\Sigma_1)) = 0 . A parametrization of \Sigma is

    \begin{align} &x_1 = cos(\phi)cos(\theta), \; \; x_2 = cos(\phi)sin(\theta), \\ &x_3 = sin(\phi), \; \; for \; \; \phi\in (-\frac{\pi}{2}, \; \; \frac{\pi}{2}), \; \; \theta \in[0, 2\pi). \end{align} (6.34)

    Then

    \begin{align} \|\nabla_Hu\|_H^2& = X_1(u)^2+X_2(u)^2 = 4(x_1^2+x_2^2)+4(x_1^4+x_2^4)x_3^2\\ & = 4cos(\phi)^2+4sin(\phi)^2cos(\phi)^4(cos(\theta)^4+sin(\theta)^4). \end{align} (6.35)

    By the definitions of w_j for 1\leq j\leq3 and (6.23) , we have

    \begin{align} d\sigma_{\Sigma_1}& = \frac{1}{\|\nabla_Hu\|_H}[(X_1(u))dx_2-(X_2(u))dx_1]\wedge(dx_3+x_2^2dx_1-x_1^2dx_2)\\ & = \frac{1}{\|\nabla_Hu\|_H}[2cos(\phi)^3+2sin(\phi)^2cos(\phi)^5(cos(\theta)^4+sin(\theta)^4)\\ &-4cos(\phi)^4sin(\theta)^2sin(\phi)cos(\theta)+4cos(\phi)^4sin(\theta)sin(\phi)cos(\theta)^2]d\theta\wedge d\phi. \end{align} (6.36)

    By (6.27) and (6.28) , we have \|\nabla_Hu\|_H^{-1} is locally summable around the isolated characteristic points with respect to the measure d\sigma_{\Sigma_1}.

    Firstly, We give some basic definitions of two kinds of spaces, such as 2.3, 2.4 and 2.5. By computation, we get sub-Riemannian limits of Gaussian curvature for a Euclidean C^2 -smooth surface in the generalized affine group and the generalized BCV spaces away from characteristic points and signed geodesic curvature for Euclidean C^2 -smooth curves on surfaces, respectively. Then, by the second fundamental form II^L and the Gauss equation \mathcal{K}^{\Sigma, L}(e_1, e_2) = \mathcal{K}^{L}(e_1, e_2)+{\rm det}(II^L) , we find the gauss curvature on the surface is convergent in two cases. Therefore, a good result is obtained. Finally, we give the proof of Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces.

    The second author was supported in part by NSFC No.11771070. The authors are deeply grateful to the referees for their valuable comments and helpful suggestions.

    The authors declare no conflict of interest.



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