Gauss-Bonnet theorems in the generalized a ﬃ ne group and the generalized BCV spaces

: In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C 2 -smooth surface in the generalized a ﬃ ne group and the generalized BCV spaces away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. We get Gauss-Bonnet theorems in the generalized a ﬃ ne group and the generalized BCV spaces.


Introduction
In [4], Diniz and Veloso gave the definition of Gaussian curvature for non-horizontal surfaces in sub-Riemannian Heisenberg space H 1 and the proof of the Gauss-Bonnet theorem. In [1], intrinsic Gaussian curvature for a Euclidean C 2 -smooth surface in the Heisenberg group H 1 away from characteristic points and intrinsic signed geodesic curvature for Euclidean C 2 -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 H 1 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 H 1 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 C 2 -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 C 2smooth 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 C 2 -smooth surface in the BCV spaces and the twisted Heisenberg group away from characteristic points and signed geodesic curvature for Euclidean C 2 -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.
2. The sub-Riemannian limit of curvature of curves in the generalized affine group When T M = H H ⊥ and g T M = g H g H ⊥ , we may consider the rescaled metric g L = g H Lg H ⊥ , 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, g T M ) 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 On G, we let ω 2 + Lω ⊗ ω, g = g 1 be the Riemannian metric on G. Then X 1 , X 2 , X 3 := L − 1 2 X 3 are orthonormal basis on T G with respect to g L . We have Let ∇ L be the Levi-Civita connection on G with respect to g L . Then we have the following lemma, Lemma 2.1. Let G be the generalized affine group, then Proof. By the Koszul formula, we have where i, j, k = 1, 2, 3. So lemma 2.1 holds.
3. The sub-Riemannian limit of geodesic curvature of curves on surfaces in the generalized affine group We will say that a surface Σ ⊂ (G, g L ) is regular if Σ is a Euclidean C 2 -smooth compact and oriented surface. In particular we will assume that there exists a Euclidean C 2 -smooth function u : . Our computations will be local and away from characteristic points of Σ. Let us define first p := X 1 u, q := X 2 u, and r := X 3 u.
In particular, p 2 + q 2 = 1. These functions are well defined at every non-characteristic point. Let then v L is the Riemannian unit normal vector to Σ and e 1 , e 2 are the orthonormal basis of Σ. On T Σ we define a linear transformation J L : T Σ → T Σ such that  For every U, V ∈ T Σ, we define ∇ Σ,L U V = π∇ L U V where π : T G → T Σ is the projection. Then ∇ Σ,L is the Levi-Civita connection on Σ with respect to the metric g L . By (2.11), (3.2) and Definition 3.2. Let Σ ⊂ (G, g L ) be a regular surface. Let γ : [a, b] → Σ be a Euclidean C 2 -smooth regular curve. We define the intrinsic geodesic curvature k ∞ γ,Σ of γ at γ(t) to be k ∞ γ,Σ := lim L→+∞ k L γ,Σ , if the limit exists.
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 II L of the embedding of Σ into (G, g L ): (3.26)

Similarly to Theorem 4.3 in [3], we have
Theorem 3.7. The second fundamental form II L of the embedding of Σ into (G, g L ) is given by Proof. By e i V L , e j L − ∇ L e i V L , e j L − ∇ L e i e j , V L L = 0 and e i V L , e j L = 0, we have ∇ L e i V L , e j L = − ∇ L e i e j , V L L , i, j = 1, 2. By lemma 2.1 and (3.2), (3.28) Then Similarly, (3.30) Then Then, (3.34) Then, Proposition 3.8. Away from characteristic points, the horizontal mean curvature H ∞ of Σ ⊂ G is given by we get (3.38).
Define the curvature of a connection ∇ by Then by Lemma 2.1 and (3.39), we have the following lemma, Lemma 3.9. Let G be the affine group, then Proposition 3.10. Away from characteristic points, we have Proof. By (3.2), we have R L (e 1 , e 2 )e 1 , e 2 L (3.43) By Lemma 3.9, we have

A Gauss-Bonnet theorem in the generalized affine group
Let us first consider the case of a regular curve γ : [a, b] → (G, g L ). We define the Riemannian length measure ds L = ||γ|| L dt. Then When ω(γ(t)) 0, we have When ω(γ(t)) = 0, we have Proof. We know that similar to the proof of Lemma 6.1 in [1], we can prove (4.2). When ω(γ(t)) 0, we have Using the Taylor expansion, we can prove (4.3). From the definition of ds L and ω(γ(t)) = 0, we get (4.4).

The sub-Riemannian limit of curvature of curves in the generalized BCV spaces
We consider some notation on the generalized BCV spaces. Let f (x 2 ), f (x 1 ), F(x 1 , x 2 , x 3 ) be smooth functions. The generalized BCV spaces M is the set Then and span{X 1 , . Then H = Kerω. The generalized BCV spaces have some well-knowed special case. When F = 1 + λ 4 (x 2 1 + x 2 2 ), f = −τx 2 , f = τx 1 , we get the BCV spaces. When F = 1, f = f (x 2 ), f = f (x 1 ), we can the Heisenberg manifolds. When F = 1, f = 1 2 x 2 2 , f = 0, we get the Martinet distribution. When F = 1 x 1 , f = 0, f = −2, we get the Welyczko's example (see [5]). For the constant L > 0, let g L = ω 1 ⊗ ω 1 + ω 2 ⊗ ω 2 + Lω ⊗ ω, g = g 1 be the Riemannian metric on M. Then X 1 , X 2 , X 3 := L − 1 2 X 3 are orthonormal basis on T M with respect to g L . We have Let ∇ L be the Levi-Civita connection on M with respect to g L . Then we have the following lemma Lemma 5.1. Let M be the generalized BCV spaces, then Proof. By the Koszul formula, we have where i, j, k = 1, 2, 3. So lemma 5.1 holds.
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 II L 1 of the embedding of Σ 1 into (M, g L ) is given by Proof. By lemma 5.1 and (3.2), Similarly, (6.20) Then Then,
By (6.27) and (6.28), we have ∇ H u −1 H is locally summable around the isolated characteristic points with respect to the measure dσ Σ 1 .

Conclusions
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 K Σ,L (e 1 , e 2 ) = K L (e 1 , e 2 ) + 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.