Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications

1.   Introduction

In ^[4], Diniz and Veloso gave the definition of Gaussian curvature for non-horizontal surfaces in sub-Riemannian Heisenberg space $\mathbb{H}^1$ and the proof of the Gauss-Bonnet theorem. In ^[1], intrinsic Gaussian curvature for a Euclidean $C^2$-smooth surface in the Heisenberg group $\mathbb{H}^1$ 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 $\mathbb{H}^1$ 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 $\mathbb{H}^1$ 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 $C^2$-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 $C^2$-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 $TM = H\bigoplus H^\bot$ and $g^{TM} = g^H\bigoplus g^{H^\bot}$, we may consider the rescaled metric $g_L = g^H\bigoplus Lg^{H^\bot}$, then we may consider the sub-Riemannian limit of some geometric objects like the Gauss curvature and the mean curvature $\cdot\cdot\cdot$, when L goes to the infinity. In this case, we call the $(M, g^{TM})$ 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 $\mathbb{G}$ be the generalized affine group and choose the underlying manifold $\mathbb{G} = \{(x_1, x_2, x_3)\in\mathbb{R}^3\mid f(x_1, x_2, x_3) > 0\}$. On $\mathbb{G}$, we let

where $f$ be a smooth function with respect to $x_1, x_2, x_3$. Then

and ${\rm span}\{X_1, X_2, X_3\} = T\mathbb{G}.$ Let $H = {\rm span}\{X_1, X_2\}$ be the horizontal distribution on $\mathbb{G}$. Let $\omega_1 = \frac{1}{f}dx_1, \; \; \omega_2 = dx_3, \; \; \omega = \frac{1}{f}dx_2-dx_3.$ Then $H = {\rm Ker}\omega$. For the constant $L > 0$, let $g_L = \omega_1\otimes \omega_1+\omega_2\otimes \omega_2+L\omega\otimes \omega, \; \; g = g_1$ be the Riemannian metric on $\mathbb{G}$. Then $X_1, X_2, \widetilde{X_3}: = L^{-\frac{1}{2}}X_3$ are orthonormal basis on $T\mathbb{G}$ with respect to $g_L$. We have

where $f_i = \frac{\partial{f}}{\partial x_i}$, for $1\leq i\leq3.$

Let $\nabla^L$ be the Levi-Civita connection on $\mathbb{G}$ with respect to $g_L$. Then we have the following lemma,

Lemma 2.1. Let $\mathbb{G}$ 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.

Definition 2.2. Let $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$ be a Euclidean $C^1$-smooth curve. We say that $\gamma$ is regular if $\dot{\gamma}\neq 0$ for every $t\in [a, b].$ Moreover we say that $\gamma(t)$ is a horizontal point of $\gamma$ if

where $\gamma(t) = (\gamma_1(t), \gamma_2(t), \gamma_3(t))$ and $\dot{\gamma}_i(t) = \frac{\partial\gamma_i(t)}{\partial t}$.

Definition 2.3. Let $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$ be a Euclidean $C^2$-smooth regular curve in the Riemannian manifold $(\mathbb{G}, g_L)$. The curvature $k^L_{\gamma}$ of $\gamma$ at $\gamma(t)$ is defined as

Lemma 2.4. Let $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$ be a Euclidean $C^2$-smooth regular curve in the Riemannian manifold $(\mathbb{G}, g_L)$. Then,

In particular, if $\gamma(t)$ is a horizontal point of $\gamma$,

where $f' = \dot{\gamma}(f) = \frac{d}{dt}f(\gamma(t)).$

Proof. By (2.2), we have

By Lemma 2.1 and (2.9), we have

By (2.9) and (2.10), we have

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

Definition 2.5. Let $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$ be a Euclidean $C^2$-smooth regular curve in the Riemannian manifold $(\mathbb{G}, g_L)$. We define the intrinsic curvature $k_{\gamma}^{\infty}$ of $\gamma$ at $\gamma(t)$ to be

if the limit exists.

We introduce the following notation: for continuous functions $f_1, f_2:(0, +\infty)\rightarrow \mathbb{R}$,

Lemma 2.6. Let $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$ be a Euclidean $C^2$-smooth regular curve in the Riemannian manifold $(\mathbb{G}, g_L)$. Then

Proof. Using the notation introduced in (2.12), when $\omega(\dot{\gamma}(t))\neq 0$, we have

Therefore

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

When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0,$

we have

By (2.6), we get (2.15).

3.   The sub-Riemannian limit of geodesic curvature of curves on surfaces in the generalized affine group

We will say that a surface $\Sigma\subset(\mathbb{G}, g_L)$ is regular if $\Sigma$ is a Euclidean $C^2$-smooth compact and oriented surface. In particular we will assume that there exists a Euclidean $C^2$-smooth function $u:\mathbb{G}\rightarrow \mathbb{R}$ such that

and $u_{x_1}\partial_{x_1}+u_{x_2}\partial_{x_2}+u_{x_3}\partial_{x_3}\neq 0.$ Let $\nabla_Hu = X_1(u)X_1+X_2(u)X_2.$ A point $x\in\Sigma$ is called characteristic if $\nabla_Hu(x) = 0$. We define the characteristic set $C(\Sigma): = \{x\in\Sigma|\nabla_Hu(x) = 0\}.$ Our computations will be local and away from characteristic points of $\Sigma$. Let us define first

We then define

In particular, $\overline{p}^2+\overline{q}^2 = 1$. These functions are well defined at every non-characteristic point. Let

then $v_L$ is the Riemannian unit normal vector to $\Sigma$ and $e_1, e_2$ are the orthonormal basis of $\Sigma$. On $T\Sigma$ we define a linear transformation $J_L:T\Sigma\rightarrow T\Sigma$ such that

For every $U, V\in T\Sigma$, we define $\nabla^{\Sigma, L}_UV = \pi \nabla^{L}_UV$ where $\pi:T\mathbb{G}\rightarrow T\Sigma$ is the projection. Then $\nabla^{\Sigma, L}$ is the Levi-Civita connection on $\Sigma$ with respect to the metric $g_L$. By (2.11), (3.2) and

we have

Moreover if $\omega(\dot{\gamma}(t)) = 0$, then

Definition 3.1. Let $\Sigma\subset(\mathbb{G}, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma$ be a Euclidean $C^2$-smooth regular curve. The geodesic curvature $k^L_{\gamma, \Sigma}$ of $\gamma$ at $\gamma(t)$ is defined as

Definition 3.2. Let $\Sigma\subset(\mathbb{G}, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma$ be a Euclidean $C^2$-smooth regular curve. We define the intrinsic geodesic curvature $k_{\gamma, \Sigma}^{\infty}$ of $\gamma$ at $\gamma(t)$ to be

if the limit exists.

Lemma 3.3. Let $\Sigma\subset(\mathbb{G}, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma$ be a Euclidean $C^2$-smooth regular curve. Then

Proof. we know $\dot{\gamma}(t) = \dot{\gamma_1}(t)\partial x_1+\dot{\gamma_2}(t)\partial x_2+\dot{\gamma_3}(t)\partial x_3$, then by (2.2), $\dot{\gamma}(t) = \frac{\dot{\gamma_1}(t)}{\gamma_1(t)}X_1+\gamma_3(t)X_2+\omega(\dot{\gamma}(t))X_3.$

Let

Then

We have

Thus $\dot{\gamma}\in T\Sigma$, we have

By (3.6), we have

Similarly, we have that when $\omega(\dot{\gamma}(t))\neq 0$,

By (3.6) and (3.12), we have

where $M_0$ does not depend on $L$. By (3.7), (3.13)–(3.15), we get (3.8). When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t))) = 0,$

we have

and

By (3.16)–(3.18) and (3.7), we get $k^{\infty}_{\gamma, \Sigma} = 0$. When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0,$

we have

so we get (3.9).

Definition 3.4. Let $\Sigma\subset(\mathbb{G}, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma$ be a Euclidean $C^2$-smooth regular curve. The signed geodesic curvature $k^{L, s}_{\gamma, \Sigma}$ of $\gamma$ at $\gamma(t)$ is defined as

where $J_L$ is defined by (3.3).

Definition 3.5. Let $\Sigma\subset(\mathbb{G}, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma$ be a Euclidean $C^2$-smooth regular curve. We define the intrinsic geodesic curvature $k_{\gamma, \Sigma}^{\infty}$ of $\gamma$ at the non-characteristic point $\gamma(t)$ to be

if the limit exists.

Lemma 3.6. Let $\Sigma\subset(\mathbb{G}, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma$ be a Euclidean $C^2$-smooth regular curve. Then

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

By (3.5) and (3.22), we have

So we get (3.20). When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t))) = 0,$ we get

So $k^{\infty, s}_{\gamma, \Sigma} = 0.$ When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq0,$

we have

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 $II^L$ of the embedding of $\Sigma$ into $(\mathbb{G}, g_L)$:

Similarly to Theorem 4.3 in ^[3], we have

Theorem 3.7. The second fundamental form $II^L$ of the embedding of $\Sigma$ into $(\mathbb{G}, g_L)$ is given by

where

Proof. By $e_i\langle V_L, e_j\rangle_L-\langle\nabla^L_{e_i}V_L, e_j\rangle_L-\langle\nabla^L_{e_i}e_j, V_L\rangle_L = 0$ and $e_i\langle V_L, e_j\rangle_L = 0$, we have $\langle\nabla^L_{e_i}V_L, e_j\rangle_L = -\langle\nabla^L_{e_i}e_j, V_L\rangle_L,$ $i, j = 1, 2$.

By lemma 2.1 and (3.2),

Then

Similarly,

Then

Since

Then,

Since

Then,

The Riemannian mean curvature $\mathcal{H}_L$ of $\Sigma$ is defined by

Let

By the Gauss equation, we have

Proposition 3.8. Away from characteristic points, the horizontal mean curvature $\mathcal{H}_{\infty}$ of $\Sigma\subset\mathbb{G}$ is given by

Proof. By

we get (3.38).

Define the curvature of a connection $\nabla$ by

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

Lemma 3.9. Let $\mathbb{G}$ be the affine group, then

Proposition 3.10. Away from characteristic points, we have

where

Proof. By (3.2), we have

By Lemma 3.9, we have

By (3.35) and

we get

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

4.   A Gauss-Bonnet theorem in the generalized affine group

Let us first consider the case of a regular curve $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$. We define the Riemannian length measure

Lemma 4.1. Let $\gamma:[a, b]\rightarrow (\mathbb{G}, g_L)$ be a Euclidean $C^2$-smooth and regular curve. Let

Then

When $\omega(\dot{\gamma}(t))\neq 0$, we have

When $\omega(\dot{\gamma}(t)) = 0$, we have

Proof. We know that

similar to the proof of Lemma 6.1 in ^[1], we can prove (4.2). When $\omega(\dot{\gamma}(t))\neq 0$, we have

Using the Taylor expansion, we can prove (4.3). From the definition of $ds_L$ and $\omega(\dot{\gamma}(t)) = 0$, we get (4.4).

Let $\Sigma\subset(\mathbb{G}, g_L)$ be a Euclidean $C^2$-smooth surface and $\Sigma = \{u = 0\}$. Let $d\sigma_{\Sigma, L}$ denote the surface measure on $\Sigma$ with respect to the Riemannian metric $g_L$. Then similai to Proposition $4.2$ in ^[7], we have

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

Theorem 4.2. Let $\Sigma\subset (\mathbb{G}, g_L)$ be a regular surface with finitely many boundary components $(\partial\Sigma)_i,$ $i\in\{1, \cdots, n\}$, given by Euclidean $C^2$-smooth regular and closed curves $\gamma_i:[0, 2\pi]\rightarrow (\partial\Sigma)_i$. Let $A_0$ be defined by (3.42) and $d\sigma_\Sigma, \; \; d\overline{\sigma_\Sigma}$ be defined by (4.5) and $d\overline{s}$ be defined by (4.1) and $k^{\infty, s}_{\gamma_i, \Sigma}$ be the sub-Riemannian signed geodesic curvature of $\gamma_i$ relative to $\Sigma$. Suppose that the characteristic set $C(\Sigma)$ satisfies $\mathcal{H}^1(C(\Sigma)) = 0$ where $\mathcal{H}^1(C(\Sigma))$ denotes the Euclidean $1$-dimensional Hausdorff measure of $C(\Sigma)$ and that $||\nabla_Hu||_H^{-1}$ is locally summable with respect to the Euclidean $2$-dimensional Hausdorff measure near the characteristic set $C(\Sigma)$, then

Example 4.3. Let $f = x_1^2+1$, then $\mathbb{G} = R^3$. Let $u = x_1^2+x_2^2+x_3^2-1$ and $\sum = S^2$. $\sum$ is a regular surface. By $(2.1)$, we get

Solve the equations $X_1(u) = X_2(u) = 0$,

then we get

and $\mathcal{H}^1(C(\Sigma)) = 0.$

A parametrization of $\Sigma$ is

Then

By the definitions of $w_j$ for $1\leq j\leq3$ and $(4.5)$, we have

where

is a bouned smooth function on $\Sigma.$ By $(4.9)$ and $(4.10)$, we have $\|\nabla_Hu\|_H^{-1}$ is locally summable around the isolated characteristic points with respect to the measure $d\sigma_{\Sigma}.$

5.   The sub-Riemannian limit of curvature of curves in the generalized BCV spaces

We consider some notation on the generalized BCV spaces. Let $f(x_2)$, $\overline{f}(x_1)$, $F(x_1, x_2, x_3)$ be smooth functions. The generalized BCV spaces $M$ is the set

Let

Then

and ${\rm span}\{X_1, X_2, X_3\} = TM.$ Let $H = {\rm span}\{X_1, X_2\}$ be the horizontal distribution on $M$. Let $\omega_1 = \frac{1}{F}dx_1, \; \; \omega_2 = \frac{1}{F}dx_2, \; \; \omega = dx_3-\frac{(fdx_1+\overline{f}dx_2)}{F}.$ Then $H = {\rm Ker}\omega$. The generalized BCV spaces have some well-knowed special case. When $F = 1+\frac{\lambda}{4}(x_1^2+x_2^2), f = -\tau x_2, \overline{f} = \tau x_1,$ we get the BCV spaces. When $F = 1, f = f(x_2), \overline{f} = \overline{f}(x_1),$ we can the Heisenberg manifolds. When $F = 1, f = \frac{1}{2}x_2^2, \overline{f} = 0,$ we get the Martinet distribution. When $F = \frac{1}{x_1}, f = 0, \overline{f} = -2,$ we get the Welyczko's example (see ^[5]). For the constant $L > 0$, let $g_L = \omega_1\otimes \omega_1+\omega_2\otimes \omega_2+L\omega\otimes \omega, \; \; g = g_1$ be the Riemannian metric on $M$. Then $X_1, X_2, \widetilde{X_3}: = L^{-\frac{1}{2}}X_3$ are orthonormal basis on $TM$ with respect to $g_L$. We have

where $F_i = \frac{\partial F}{\partial x_i},$ for $1\leq i\leq3,$ $f' = \frac{\partial f}{\partial x_2},$ $\overline{f}' = \frac{\partial \overline{f}}{\partial x_1}.$ Let $\nabla^L$ 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.

Definition 5.2. Let $\gamma:[a, b]\rightarrow (M, g_L)$ be a Euclidean $C^1$-smooth curve. We say that $\gamma(t)$ is a horizontal point of $\gamma$ if

where $\gamma(t) = (\gamma_1(t), \gamma_2(t), \gamma_3(t))$ and $\dot{\gamma}_i(t) = \frac{\partial\gamma_i(t)}{\partial t}$.

Similar to the definition 2.3 and definition 2.5, we can define $k_\gamma^L$ and $k_\gamma^{\infty}$ for the generalized BCV spaces, we have

Lemma 5.3. Let $\gamma:[a, b]\rightarrow (M, g_L)$ be a Euclidean $C^2$-smooth regular curve in the Riemannian manifold $(M, g_L)$. Then

where $F' = \dot{\gamma}(F) = \frac{d}{dt}F(\gamma(t)).$

Proof. By (5.2), we have

By Lemma 5.1 and (5.8), we have

By (5.8) and (5.9), we have

By (5.8) and (5.10), when $\omega(\dot{\gamma}(t))\neq 0$, we have

Therefore

So by (2.6), we have (5.5). (5.6) comes from (5.8), (5.10), (2.6) and $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t))) = 0.$ When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0$, we have

By (2.6), we get (5.7).

6.   The sub-Riemannian limit of geodesic curvature of curves on surfaces in the generalized BCV spaces

We will consider a regular surface $\Sigma_1\subset(M, g_L)$ and regular curve $\gamma\subset\Sigma_1$. We will assume that there exists a Euclidean $C^2$-smooth function $u:M\rightarrow \mathbb{R}$ such that

Similar to Section 3, we define $p, q, r, l, l_L, \overline{p}, \overline{q}, \overline{p_L}, \overline{q_L}, \overline{r_L}, v_L, e_1, e_2, J_L, k_{\gamma, \Sigma_1}^L, k_{\gamma, \Sigma_1}^{\infty}, k_{\gamma, \Sigma_1}^{L, s}, k_{\gamma, \Sigma_1}^{\infty, s}$. By (3.4) and (5.10), we have

By (5.8) and $\dot{\gamma}(t)\in T\Sigma_1$, we have

We have

Lemma 6.1. Let $\Sigma_1\subset(M, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma_1$ be a Euclidean $C^2$-smooth regular curve. Then

Proof. By (6.1), we have

By (6.2), we have that when $\omega(\dot{\gamma}(t))\neq 0$,

By (6.1) and (6.2), we have

where $M_0$ does not depend on $L$.

By (3.7), (6.5)–(6.7), we get (6.3). When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t))) = 0,$ we have

and

By (6.8)–(6.10) and (3.7), we get $k^{\infty}_{\gamma, \Sigma_1} = 0$. When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0,$ we have

so we get (6.4).

Lemma 6.2. Let $\Sigma_1\subset(M, g_L)$ be a regular surface. Let $\gamma:[a, b]\rightarrow \Sigma_1$ be a Euclidean $C^2$-smooth regular curve. Then

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

By (6.1) and (6.13), we have

So by (3.17), (6.6) and (6.14), we get (6.11). When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t))) = 0,$ we get

So $k^{\infty, s}_{\gamma, \Sigma_1} = 0.$ When $\omega(\dot{\gamma}(t)) = 0$ and $\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0,$ we have

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 $II^L_1$ of the embedding of $\Sigma_1$ into $(M, g_L)$ is given by

where

Proof. By lemma 5.1 and (3.2),

Then

Similarly,

Then

Since

Then,

Since

Then,

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

By Lemma 5.1, we have

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

where

Proposition 6.6. Away from characteristic points, we have

where $\overline{N} = N_0+N$,

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

Similar to (3.45), we have

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

Similar to (4.2) and (4.5), for the generalized BCV spaces, 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 $\Sigma_1\subset (M, g_L)$ 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

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

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

Then

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

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 $d\sigma_{\Sigma_1}.$

7.   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 $\mathcal{K}^{\Sigma, L}(e_1, e_2) = \mathcal{K}^{L}(e_1, e_2)+{\rm det}(II^L)$, 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.

Acknowledgments

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.

Conflict of interest

The authors declare no conflict of interest.