Gauss-Bonnet Theorems for Surfaces in Sub-Riemannian Three-dimensional Manifolds

We prove Gauss-Bonnet formulas for surfaces in sub-Riemannian three-dimensional manifolds. This work generalizes two existent versions of the theorem. The first one is Gauss-Bonnet theorem for surfaces in H-1, given in Diniz and Veloso (J Dyn Control Syst 22(4):807-20, 2016). We consider in the sub-Riemannian manifold the adapted covariant derivative (sic) introduced in Falbel et al. (Mat. Contemp 4:119-25, 1993). If S is a surface where the sub-Riemannian distribution is transverse to the tangent space of S, the inter-section is one dimensional and we define a unitary vector field f(1) on S orthogonal to the intersection. We project (sic) along f(1) to obtain a covariant derivative (sic) on S. The curvatures defined by (sic) and the choice of appropriate measures on S are the necessary tools to prove the first Gauss-Bonnet formula. The second version was first proved by Balogh, Tyson and Vecchi in Balogh et al. (Math Z 287:1-38, 2017). They used the Riemannian approxima-tions scheme (H-1, <, > L), in the Heisenberg group, introduced by Gromov (Progress in Mathematics-Boston 144:85-324, 1996), to calculate the limits of Gaussian and normal cur-vatures defined by Levi-Civita connection on surfaces of H-1 when L -> infinity. They showed that these limits exist and they obtained Gauss-Bonnet theorem in H-1 as limit of Gauss-Bonnet theorems in (H-1, <, > L) when L goes to infinity. This construction was extended by Wang and Wei (Sci China Math 64:1843-60, 2021) to the affine group and the group of rigid motions of the Minkowski plane, by Wang and Wei (Results Math 75:126, 2020) to generalized affine group and the generalized BCV spaces . We generalized these results to surfaces in sub-Riemannian three-dimensional manifolds and proved analogous Gauss-Bonnet theorems in Veloso (2020) for surfaces in sub-Riemannian manifolds as limit of Riemannian metrics, still using the Levi-Civita connection. Afterwards, Wang verified that the results were true replacing the Levi-Civita connection by the deformed Schouten-van Kampen con-nections in the Heisenberg group and Wang and Wu verified for the affine group and the group of rigid motions of the Minkowski plane. Our objective in the second part of this paper is to generalize these results for surfaces in sub-Riemannian three-dimensional manifolds to englobe an ample class of metric connections and having, as a particular case, the Schouten-van Kampen connection. We also show that the mean curvatures defined in both approaches coincide for Levi-Civita connections and deformed Schouten-van Kampen connections.