The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric

<jats:p>The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>E</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mn>1</mn>
                                    <mo>,</mo>
                                    <mn>1</mn>
                                 </mrow>
                              </mfenced>
                              <mo>,</mo>
                              <mi>g</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <msub>
                                       <mrow>
                                          <mi>λ</mi>
                                       </mrow>
                                       <mrow>
                                          <mn>1</mn>
                                       </mrow>
                                    </msub>
                                    <mo>,</mo>
                                    <msub>
                                       <mrow>
                                          <mi>λ</mi>
                                       </mrow>
                                       <mrow>
                                          <mn>2</mn>
                                       </mrow>
                                    </msub>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <msub>
                           <mrow>
                              <mi>λ</mi>
                           </mrow>
                           <mrow>
                              <mn>1</mn>
                           </mrow>
                        </msub>
                        <mo>≥</mo>
                        <msub>
                           <mrow>
                              <mi>λ</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                        <mo>></mo>
                        <mn>0</mn>
                     </math>
                  </jats:inline-formula>. It provides a natural <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mn>2</mn>
                     </math>
                  </jats:inline-formula>-parametric deformation family of the Riemannian homogeneous manifold <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <msub>
                           <mrow>
                              <mtext>Sol</mtext>
                           </mrow>
                           <mrow>
                              <mn>3</mn>
                           </mrow>
                        </msub>
                        <mo>=</mo>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>E</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mn>1</mn>
                                    <mo>,</mo>
                                    <mn>1</mn>
                                 </mrow>
                              </mfenced>
                              <mo>,</mo>
                              <mi>g</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mn>1</mn>
                                    <mo>,</mo>
                                    <mn>1</mn>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <msup>
                           <mrow>
                              <mi>C</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msup>
                     </math>
                  </jats:inline-formula>-smooth surface in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>E</mi>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <mn>1</mn>
                                    <mo>,</mo>
                                    <mn>1</mn>
                                 </mrow>
                              </mfenced>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>g</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>L</mi>
                                 </mrow>
                              </msub>
                              <mfenced open="(" close=")">
                                 <mrow>
                                    <msub>
                                       <mrow>
                                          <mi>λ</mi>
                                       </mrow>
                                       <mrow>
                                          <mn>1</mn>
                                       </mrow>
                                    </msub>
                                    <mo>,</mo>
                                    <msub>
                                       <mrow>
                                          <mi>λ</mi>
                                       </mrow>
                                       <mrow>
                                          <mn>2</mn>
                                       </mrow>
                                    </msub>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> away from characteristic points and signed geodesic curvature for the Euclidean <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <msup>
                           <mrow>
                              <mi>C</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msup>
                     </math>
                  </jats:inline-formula>-smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.</jats:p>


Introduction
In [1], Proposition 2.6 stated that any left-invariant metric on the group of rigid motions of the Minkowski plane Eð1, 1Þ is isometric to one of the metric gðλ 1 , λ 2 , λ 3 Þ with λ 1 ≥ λ 2 > 0 and λ 3 = 1/λ 1 λ 2 . In [2], the metric gðλ 1 , λ 2 , λ 3 Þ was denoted by gðλ 1 , λ 2 Þ = gðλ 1 , λ 2 , 1/λ 1 λ 2 Þ as we take in this paper, and the authors classified parallel surfaces in the groups of rigid motions of the Euclidean plane and the Minkowski plane. In [3], they completed the classification of parallel and totally geodesic surfaces in all three-dimensional homogeneous spaces by solving the problem in three-dimensional Lie groups with a left-invariant metric yielding a threedimensional isometry group. In this paper, we consider ð Eð1, 1Þ, gðλ 1 , λ 2 ÞÞ which is the group of rigid motions of the Minkowski plane with the general left-invariant metric gðλ 1 , λ 2 Þ: This group is very interesting and important for the reason that it provides a natural 2-parametric deformation family of one of the Riemannian homogeneous manifold Sol 3 = ðEð1, 1Þ, gð1, 1ÞÞ which is the model space to solve-geometry in the eight model geometries of Thurston.
In [4,5], Balogh et al. used a Riemannian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C 2 -smooth surface in the Heisenberg group H 1 away from characteristic points, and a notion of intrinsic signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. These results were then used to prove a Heisenberg version of the Gauss-Bonnet theorem. They proposed an interesting question to understand to what extent similar phenomena hold in other sub-Riemannian geometric structures. In [6,7], Wang and Wei solved this problem for the affine group, the group of rigid motions of the Minkowski plane ðEð1, 1Þ, gð1, 1ÞÞ, the BCV spaces, and the twisted Heisenberg group. Recently, we got the Gauss-Bonnet theorems in the rototranslation group and the Lorentzian Sasakian space forms [8,9]. In this paper, we try to solve this problem for the group of rigid motions of the Minkowski plane with the general left-invariant metric gðλ 1 , λ 2 Þ: We compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. We get a generalized Gauss-Bonnet theorem in ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ.
In Section 2, we provide a short introduction to ðEð1, 1Þ , g L ðλ 1 , λ 2 ÞÞ and the notions which we will use throughout the paper, such as the Levi-Civita connection in the Riemannian approximants of ðEð1, 1Þ, gðλ 1 , λ 2 ÞÞ. Furthermore, we compute the sub-Riemannian limit of the curvature of curves in ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ. In Sections 3 and 4, we compute the sub-Riemannian limits of the geodesic curvature of curves on surfaces and the Riemannian Gaussian curvature of surfaces in ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ. In Section 5, we get the Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with the general left-invariant metric.

The Sub-Riemannian Limit of Curvature of
Curves in ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ In this section, some basic notions in the motion group of the Minkowski plane will be introduced. Let Eð1, 1Þ be the motion group of the Minkowski plane: The Lie algebra ℓð1, 1Þ is given explicitly by We consider the group of rigid motions of the Minkowski plane with a general left-invariant metric, ðEð1, 1Þ, gðλ 1 , λ 2 ÞÞ . As a model of ðEð1, 1Þ, gðλ 1 , λ 2 ÞÞ, we choose the underlying manifold ℝ 3 . On ℝ 3 , we let Then, we have Let H = spanfX 1 , X 2 g be the horizontal distribution on ðEð1, 1Þ, gðλ 1 , λ 2 ÞÞ: Let be the dual coframe field. Then, H = ker ω. The Riemannian approximation scheme used in [4] can in general depend on the choice of the complement to the horizontal distribution.
In the context of ðEð1, 1Þ, gðλ 1 , λ 2 ÞÞ, the choice is similar to ðEð1, 1Þ, gð1, 1ÞÞ in [6]. Let L > 0 and define a metric g L ðλ 1 , and gðλ 1 , λ 2 Þ = g 1 ðλ 1 , λ 2 Þ be the Riemannian metric on ðEð 1, 1Þ, gðλ 1 , λ 2 ÞÞ. The approach in this paper is to define sub-Riemannian objects as limits of horizontal objects in ðE ð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ, where a family of metrics g L ðλ 1 , λ 2 Þ is essentially obtained as an anisotropic blow-up of the Riemannian metric g 1 ðλ 1 , λ 2 Þ. At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with L. In general, the metric g L ðλ 1 , λ 2 Þ does not fit in the family gðλ 1 , λ 2 Þ for the case of L ≠ 1: We have To compute the curvatures of curves and surfaces in the motion group of the Minkowski plane with respect to g L ðλ 1 , λ 2 Þ, we use the Levi-Civita connection ∇ L on ðEð1, 1Þ, g L ð λ 1 , λ 2 ÞÞ. A straightforward calculation shows the following proposition.
Theorem 13. The second fundamental form II L of the embedding of Σ into ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ is given by where Proof. Since he 1 , v L i L = 0 and he 2 , v L i L = 0, we have Using the definition of the connection, the identities in (8) and grouping terms, we have since p 2 + q 2 = 1, we have pX i p + qX i q = 0, i = 1, 2, 3. We have To compute h 12 and h 21 , using the definition of the connection, we obtain 9 Journal of Function Spaces Next, we compute the inner product of this with v L . Using the product rule and the identity q L p = p L q, we obtain The identities p L 2 + q L 2 + r L 2 = 1 and p 2 + q 2 = 1 yield Finally, we use the identity ðl/l L − l L /lÞ∇ H r L = r L ∇ H ðl/l L Þ, we have Therefore, we have Since h∇ e 2 v L , e 2 i L = −h∇ e 2 e 2 , v L i L , using the definition of connection, the identities in (9) and grouping terms, we have Taking the inner product with v L yields To simplify this, first use the product rule for the terms involving X i ð p r L Þ and X i ð q r L Þ together with the identities p X i p + qX i q = 0, r L ∇ H ðl/l L Þ = ðl/l L − l L /lÞ∇ H r L , and p 2 + q 2 = 1 . Under these simplifications, terms involving X i ð pÞ and X i ð qÞ cancel and one is left with terms involving components 10 Journal of Function Spaces of ∇ r L : We conclude by rewriting the expression X i ð r L Þ in terms of X i ðr/lÞ. Therefore, we have

☐ ☐
The Riemannian mean curvature H L of Σ is defined by Define the curvature of a connection ∇ by Let By the Gauss equation, we have Proposition 14. Away from characteristic point, the horizontal mean curvature H ∞ of Σ ∈ ðEð1, 1Þ, g L ðλ 1 , λ 2 ÞÞ is given by Proof. By